HLIBMathematics

Mathematics: Applications and Interpretation

The Higher Level pathway applies advanced mathematical models through statistics, functions, geometry, and calculus in real-world contexts.

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Course Overview

IB Mathematics: Applications and Interpretation HL is organized into 5 units and 77 focused topics with 1584 Key Concepts sections, 931 questions, and 1540 flashcards.

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Begin with Topic 1.1: Operations with numbers and scientific notation.

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Course units

Each unit groups focused topics. Every topic follows the same clear rhythm: review Key Concepts, use the Question Bank, then strengthen recall.

Unit 1
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Number and Algebra
AI HL begins with the AI SL foundations of scientific notation, sequences, financial mathematics and technology-supported equation solving, then extends into logarithm laws, infinite geometric series, complex numbers, matrices and eigenvalue methods. This topic supports finance, transformations, Markov models, phase portraits and repeated-state systems later in the course.
15 topics
Unit 1 progress0%
Topics in this unit
Topic 1.1Not started
Operations with numbers and scientific notation
Scientific notation provides a stable way to work with very large and very small values and is frequently used in applied contexts.
180 min estimated
Topic 1.1 progress0%
Key Concepts0/19 sections complete Question Bank0/14 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Write non-zero numbers in normalized form a x $10^{k}$ with $1 \le a < 10$ and integer k.
Perform arithmetic using scientific notation and interpret calculator E-notation correctly.
Compare orders of magnitude and use appropriate units.
Maintain sufficient precision when scientific notation appears inside a longer calculation.
Key formulas
Normalized scientific notation
$N=a\times10^k,\quad 1\le |a|<10,\quad k\in\mathbb Z$
Product of powers of ten
$10^m\times10^n=10^{m+n}$
Quotient of powers of ten
$\frac{10^m}{10^n}=10^{m-n}$
Power of a power
$(10^m)^r=10^{mr}$
Relative-size ratio
$\text{factor}=\frac{\text{larger quantity}}{\text{smaller quantity}}$
Multiplication rule
$(a\times10^m)(b\times10^n)=(ab)\times10^{m+n}$
Division rule
$\frac{a\times10^m}{b\times10^n}=\frac{a}{b}\times10^{m-n},\quad b\ne0$
Addition with common exponent
$a\times10^k+b\times10^k=(a+b)\times10^k$
Power rule
$(a\times10^k)^n=a^n\times10^{kn}$
Calculator translation
$a\text{E}k=a\times10^k$
Common metric prefixes
$\text{kilo}=10^3,\quad \text{milli}=10^{-3},\quad \text{micro}=10^{-6},\quad \text{nano}=10^{-9}$
Normalized form
$N=a\times10^k,\quad 1\le |a|<10,\quad k\in\mathbb Z$
Product
$(a\times10^m)(b\times10^n)=ab\times10^{m+n}$
Quotient
$\frac{a\times10^m}{b\times10^n}=\frac ab\times10^{m-n}$
Power
$(a\times10^m)^r=a^r\times10^{mr}$
Common-exponent sum
$a\times10^k+b\times10^k=(a+b)\times10^k$
Topic 1.2Not started
Arithmetic sequences and series
Arithmetic sequences model approximately constant additive change. AI students should be able to calculate with the model and judge how well it fits real data.
190 min estimated
Topic 1.2 progress0%
Key Concepts0/19 sections complete Question Bank0/16 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Identify the first term and common difference.
Find a required term and the sum of a finite arithmetic series.
Use sigma notation for finite sums.
Apply arithmetic models to prediction and interpret deviations from perfect arithmetic behaviour.
Key formulas
First n terms as a series
$S_n=u_1+u_2+\cdots+u_n$
Common difference
$d=u_{n+1}-u_n$
Arithmetic recurrence
$u_1=a,\qquad u_{n+1}=u_n+d$
Arithmetic nth term
$u_n=a+(n-1)d$
Linear form
$u_n=dn+(a-d)$
Difference from two terms
$d=\frac{u_q-u_p}{q-p},\quad q\ne p$
Sum using first and last terms
$S_n=\frac{n}{2}(a+u_n)$
Sum using first term and difference
$S_n=\frac{n}{2}[2a+(n-1)d]$
Number times average term
$S_n=n\left(\frac{a+u_n}{2}\right)$
Arithmetic sum in sigma notation
$\sum\limits_{r=1}^{n}[a+(r-1)d]=S_n$
Number of indexed terms
$\text{count}=q-p+1$
Simple-interest amount
$A_n=P(1+nr)$
Simple interest earned
$I=Prn$
Approximate difference from two positions
$d_{\text{approx}}=\frac{u_q-u_p}{q-p}$
Residual
$e_n=\text{observed}_n-\text{predicted}_n$
Nth term
$u_n=a+(n-1)d$
Sum from first and last terms
$S_n=\frac n2(a+u_n)$
Sum from first term and difference
$S_n=\frac n2[2a+(n-1)d]$
Sigma form
$\sum\limits_{r=1}^{n}[a+(r-1)d]=S_n$
Topic 1.3Not started
Geometric sequences and series
Geometric sequences model repeated proportional change, making them central to growth, decay and financial applications.
200 min estimated
Topic 1.3 progress0%
Key Concepts0/21 sections complete Question Bank0/7 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Identify the first term and common ratio.
Find a specified term.
Find the sum of a finite geometric series.
Use sigma notation and interpret repeated percentage change.
Key formulas
Consecutive-term ratio
$r=\frac{u_{n+1}}{u_n}$
Growth multiplier
$r=1+\frac{p}{100}$
Decay multiplier
$r=1-\frac{p}{100}$
Geometric recurrence
$u_1=a,\qquad u_{n+1}=ru_n$
Geometric nth term
$u_n=ar^{n-1}$
Term number from logarithms
$n=1+\frac{\ln(\frac{u_n}{a})}{\ln r}$
Ratio from two terms
$r^{q-p}=\frac{u_q}{u_p}$
First term from a known term
$a=\frac{u_p}{r^{p-1}}$
First n geometric terms
$S_n=a+ar+ar^2+\cdots+ar^{n-1}$
Finite geometric sum
$S_n=\frac{a(1-r^n)}{1-r}=\frac{a(r^n-1)}{r-1}$
Constant-ratio special case
$S_n=na\quad\text{when }r=1$
Finite geometric sum in sigma notation
$\sum\limits_{k=0}^{n-1}ar^k=\frac{a(1-r^n)}{1-r}$
Inclusive index count
$\text{number of terms}=n-m+1$
Infinite geometric sum
$S_\infty=\frac{a}{1-r},\qquad |r|<1$
Infinite sigma form
$\sum\limits_{k=0}^{\infty}ar^k=\frac{a}{1-r},\qquad |r|<1$
Repeated-change model
$A_t=A_0r^t$
Percentage change from ratio
$\text{percentage change}=100(r-1)\%$
Approximate ratio from endpoints
$r\approx\left(\frac{u_q}{u_p}\right)^{\frac{1}{q-p}}$
Residual
$e_n=\text{observed value}-\text{predicted value}$
Common ratio
$r=\frac{u_{n+1}}{u_n}$
Nth term
$u_n=ar^{n-1}$
Finite sum
$S_n=\frac{a(1-r^n)}{1-r}$
Infinite sum
$S_\infty=\frac{a}{1-r}$
Topic 1.4Not started
Financial applications
Financial mathematics connects exponentials, repeated percentage change and technology. Students should understand what the variables mean rather than treating finance as a calculator-only topic.
190 min estimated
Topic 1.4 progress0%
Key Concepts0/13 sections complete Question Bank0/26 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Calculate present and future values under compound growth.
Model depreciation.
Interpret nominal rate, compounding frequency and number of periods.
Use financial technology for realistic savings and loan problems.
State answers in context with appropriate currency and time units.
Key formulas
Simple interest earned
$I=Prt$
Simple-interest balance
$A=P(1+rt)$
Percentage multiplier
$\text{growth: }1+\frac{p}{100},\qquad \text{decay: }1-\frac{p}{100}$
Compound future value
$FV=PV(1+i)^n$
Periodic rate from two values
$i=\left(\frac{FV}{PV}\right)^{\frac{1}{n}}-1,\quad n>0,\ PV\ne0,\frac{\ FV}{PV}>0$
Number of periods
$\begin{gathered}n=\frac{\ln(FV/PV)}{\ln(1+i)}\\ PV\ne0,\quad FV/PV>0,\quad i>-1,\ i\ne0\end{gathered}$
Nominal-rate compound interest
$FV=PV\left(1+\frac{r}{m}\right)^{mt}$
Periodic rate and period count
$i=\frac{r}{m},\qquad n=mt$
Effective annual rate
$EAR=\left(1+\frac{r}{m}\right)^m-1$
Present value of one future amount
$PV=\frac{FV}{(1+i)^n}=FV(1+i)^{-n}$
Appreciation
$V_n=V_0(1+a)^n$
Reducing-balance depreciation
$V_n=V_0(1-d)^n$
Future value of an ordinary annuity
$FV=\begin{cases}R\frac{(1+i)^n-1}{i},&i\ne0\\ nR,&i=0\end{cases}$
Future value of an annuity due
$FV_{\mathrm{due}}=\begin{cases}R\frac{(1+i)^n-1}{i}(1+i),&i\ne0\\ nR,&i=0\end{cases}$
Deposit required for a target
$R=\begin{cases}FV\frac{i}{(1+i)^n-1},&i\ne0\\ FV/n,&i=0\end{cases}$
Present value of loan repayments
$PV=\begin{cases}R\frac{1-(1+i)^{-n}}{i},&i\ne0\\ nR,&i=0\end{cases}$
Periodic loan repayment
$R=\begin{cases}PV\frac{i}{1-(1+i)^{-n}},&i\ne0\\ PV/n,&i=0\end{cases}$
Outstanding balance after k payments
$B_k=\begin{cases}PV(1+i)^k-R\frac{(1+i)^k-1}{i},&i\ne0\\ PV-kR,&i=0\end{cases}$
Total interest on equal repayments
$I_{\mathrm{total}}=Rn-PV$
Future price under inflation
$C_t=C_0(1+f)^t$
Real value after inflation
$V_{\mathrm{real}}=\frac{M}{(1+f)^t}$
Exact real annual rate
$r_{\mathrm{real}}=\frac{1+r}{1+f}-1$
Simple interest
$I=Prt,\qquad A=P(1+rt)$
Compound amount
$FV=PV(1+i)^N$
Nominal compounding
$i=\frac{r}{m},\qquad N=mt$
Present value
$PV=\frac{FV}{(1+i)^N}$
Depreciation
$V_N=V_0(1-d)^N$
Regular savings
$FV=PMT\left(\frac{(1+i)^N-1}{i}\right)$
Loan payment
$PMT=PV\left(\frac{i}{1-(1+i)^{-N}}\right)$
Topic 1.5Not started
Exponents and introductory logarithms
The current shared AI syllabus includes exponent laws and numerical use of common and natural logarithms. At HL, logarithmic ideas are later developed more formally.
170 min estimated
Topic 1.5 progress0%
Key Concepts0/12 sections complete Question Bank0/3 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Apply integer exponent laws.
Evaluate powers, roots, log10(x) and ln(x) using technology.
Recognize logarithms as inverses of exponentials.
Use numerical logarithms in modelling and equation solving.
Key formulas
Positive integer exponent
$a^n=\underbrace{a\cdot a\cdots a}_{n\text{ factors}},\quad n\in\mathbb N$
Zero exponent
$a^0=1,\quad a\ne0$
Negative integer exponent
$a^{-n}=\frac{1}{a^n},\quad a\ne0$
Product law
$a^m a^n=a^{m+n}$
Quotient law
$\frac{a^m}{a^n}=a^{m-n},\quad a\ne0$
Power of a power
$(a^m)^n=a^{mn}$
Power of a product
$(ab)^n=a^n b^n$
Power of a quotient
$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n},\quad b\ne0$
Moving a negative power across a fraction bar
$\frac{a^{-m}}{b^{-n}}=\frac{b^n}{a^m},\quad a,b\ne0$
Exponential-logarithmic equivalence
$a^c=b\quad\Longleftrightarrow\quad\log_a b=c$
Common and natural logarithms
$\log x=\log_{10}x,\qquad\ln x=\log_e x$
Inverse identities
$\log_a(a^x)=x,\qquad a^{\log_a x}=x\quad(x>0)$
Product property
$\log_a(xy)=\log_a x+\log_a y$
Quotient property
$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$
Power property
$\log_a(x^k)=k\log_a x$
Useful exact values
$\log_a1=0,\qquad\log_a a=1$
Common-base principle
$a^{f(x)}=a^{g(x)}\Longrightarrow f(x)=g(x),\quad a>0,\ a\ne1$
Solving a basic exponential equation
$a^x=b\Longrightarrow x=\frac{\ln b}{\ln a},\quad a>0,\ a\ne1,\ b>0$
Single logarithm equation
$\log_a f(x)=c\Longleftrightarrow f(x)=a^c,\quad f(x)>0$
Equality of logarithms
$\log_a f(x)=\log_a g(x)\Longrightarrow f(x)=g(x),\quad f(x),g(x)>0$
Exponential growth or decay model
$Q(t)=Q_0b^t$
Percentage multiplier
$b=1\pm\frac{r}{100}$
Time to a target
$t=\frac{\ln(\frac{T}{Q_0})}{\ln b}$
pH model
$\mathrm{pH}=-\log_{10}[\mathrm H^+]$
Sound-intensity level
$L=10\log_{10}\left(\frac{I}{I_0}\right)$
Ratio from a base-10 scale difference
$\text{ratio}=10^{\Delta}$
Change-of-base formula
$\log_a b=\frac{\log b}{\log a}=\frac{\ln b}{\ln a}$
Product and quotient laws
$a^ma^n=a^{m+n},\qquad\frac{a^m}{a^n}=a^{m-n}$
Power laws
$(a^m)^n=a^{mn},\qquad(ab)^n=a^nb^n$
Zero and negative exponents
$a^0=1,\qquad a^{-n}=\frac1{a^n}\quad(a\ne0)$
Exponential-logarithmic equivalence
$a^x=b\Longleftrightarrow\log_a b=x$
Change of base
$\log_a b=\frac{\log b}{\log a}=\frac{\ln b}{\ln a}$
Growth or decay model
$Q(t)=Q_0b^t$
Topic 1.6Not started
Approximation, bounds and percentage error
Approximation is treated as part of mathematical judgement. Rounded measurements represent intervals, and an applied conclusion may depend on how much error is acceptable.
180 min estimated
Topic 1.6 progress0%
Key Concepts0/13 sections complete Question Bank0/20 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Round to decimal places and significant figures.
Determine upper and lower bounds for rounded quantities.
Calculate absolute and percentage error.
Carry enough precision through technology calculations.
Explain whether the resulting accuracy is meaningful in context.
Key formulas
Rounding unit for decimal places
$u=10^{-d}$
General rounding error interval
$r-\frac{u}{2}\le x<r+\frac{u}{2}$
Lower and upper bounds
$x_{\mathrm L}=r-\frac{u}{2},\qquad x_{\mathrm U}=r+\frac{u}{2}$
Interval notation
$x\in[x_{\mathrm L},x_{\mathrm U})$
Sum bounds
$S_{\mathrm L}=a_{\mathrm L}+b_{\mathrm L},\qquad S_{\mathrm U}=a_{\mathrm U}+b_{\mathrm U}$
Difference bounds
$D_{\mathrm L}=a_{\mathrm L}-b_{\mathrm U},\qquad D_{\mathrm U}=a_{\mathrm U}-b_{\mathrm L}$
Product bounds for positive values
$P_{\mathrm L}=a_{\mathrm L}b_{\mathrm L},\qquad P_{\mathrm U}=a_{\mathrm U}b_{\mathrm U}$
Quotient bounds for positive values
$Q_{\mathrm L}=\frac{a_{\mathrm L}}{b_{\mathrm U}},\qquad Q_{\mathrm U}=\frac{a_{\mathrm U}}{b_{\mathrm L}}$
Absolute error
$E_{\mathrm a}=\left|x_{\mathrm{approx}}-x_{\mathrm{exact}}\right|$
Relative error
$E_{\mathrm r}=\frac{E_{\mathrm a}}{|x_{\mathrm{exact}}|},\qquad x_{\mathrm{exact}}\ne0$
Percentage error
$E_{\%}=\frac{|x_{\mathrm{approx}}-x_{\mathrm{exact}}|}{|x_{\mathrm{exact}}|}\times100\%$
Maximum absolute rounding error
$E_{\max}=\frac{u}{2}$
Maximum output percentage error
$E_{\%,\max}=\frac{\max\left(|y_{\mathrm U}-y_0|,|y_0-y_{\mathrm L}|\right)}{|y_0|}\times100\%$
Symmetric tolerance
$t-\delta\le x\le t+\delta$
Decimal-place unit
$u=10^{-d}$
Error interval
$r-\frac{u}{2}\le x<r+\frac{u}{2}$
Maximum rounding error
$E_{\max}=\frac{u}{2}$
Absolute error
$E_{\mathrm a}=|x_{\mathrm{approx}}-x_{\mathrm{exact}}|$
Relative error
$E_{\mathrm r}=\frac{E_{\mathrm a}}{|x_{\mathrm{exact}}|}$
Percentage error
$E_{\%}=100E_{\mathrm r}\%$
Positive product bounds
$P_{\mathrm L}=a_{\mathrm L}b_{\mathrm L},\qquad P_{\mathrm U}=a_{\mathrm U}b_{\mathrm U}$
Positive quotient bounds
$Q_{\mathrm L}=\frac{a_{\mathrm L}}{b_{\mathrm U}},\qquad Q_{\mathrm U}=\frac{a_{\mathrm U}}{b_{\mathrm L}}$
Topic 1.7Not started
Amortization and annuities
Regular payments and deposits are normally handled with financial technology. The key skill is setting up the cash flows and interpreting the result.
185 min estimated
Topic 1.7 progress0%
Key Concepts0/13 sections complete Question Bank0/13 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Calculate regular loan payments and outstanding balances.
Analyze annuities and savings plans.
Recognize present value, future value, payment, interest rate and number of payment periods.
Interpret sign conventions used by GDC finance applications.
Compare financial options using mathematically justified criteria.
Key formulas
Annuity-due timing adjustment
$V_{\mathrm{due}}=V_{\mathrm{ordinary}}(1+i)$
Periodic rate
$i=\frac{r}{m}$
Number of periods
$n=mt$
Effective annual rate
$r_{\mathrm{eff}}=(1+i)^m-1$
Future value of an ordinary annuity
$FV=R\frac{(1+i)^n-1}{i}$
Future value of an annuity due
$FV_{\mathrm{due}}=\begin{cases}R\frac{(1+i)^n-1}{i}(1+i),&i\ne0\\ nR,&i=0\end{cases}$
Regular deposit for a target
$R=FV\frac{i}{(1+i)^n-1}$
Interest earned on a savings annuity
$I=FV-nR$
Present value of an ordinary annuity
$PV=R\frac{1-(1+i)^{-n}}{i}$
Present value of an annuity due
$PV_{\mathrm{due}}=R\frac{1-(1+i)^{-n}}{i}(1+i)$
Loan as a present-value annuity
$PV=R\frac{1-(1+i)^{-n}}{i}$
Periodic loan repayment
$R=\begin{cases}PV\frac{i}{1-(1+i)^{-n}},&i\ne0\\ PV/n,&i=0\end{cases}$
Total scheduled repayments
$T=nR$
Total interest without fees
$I_{\mathrm{total}}=nR-PV$
Outstanding balance from original loan
$B_k=PV(1+i)^k-R\frac{(1+i)^k-1}{i}$
Outstanding balance from remaining payments
$B_k=R\frac{1-(1+i)^{-(n-k)}}{i}$
Principal repaid by time k
$P_{\mathrm{repaid}}=PV-B_k$
Interest component
$I_j=iB_{j-1}$
Principal component
$P_j=R-I_j$
Balance recurrence
$B_j=B_{j-1}(1+i)-R$
Periodic rate and period count
$i=\frac{r}{m},\qquad n=mt$
Ordinary-annuity future value
$FV=R\frac{(1+i)^n-1}{i}$
Annuity-due adjustment
$V_{\mathrm{due}}=V_{\mathrm{ordinary}}(1+i)$
Ordinary-annuity present value
$PV=R\frac{1-(1+i)^{-n}}{i}$
Loan repayment
$R=PV\frac{i}{1-(1+i)^{-n}}$
Outstanding balance
$B_k=PV(1+i)^k-R\frac{(1+i)^k-1}{i}$
Amortization row
$I_j=iB_{j-1},\qquad P_j=R-I_j,\qquad B_j=B_{j-1}-P_j$
Total interest
$I_{\mathrm{total}}=nR-PV$
Topic 1.8Not started
Solving equations with technology
AI uses numerical technology to solve systems and polynomial equations efficiently, while the student remains responsible for creating the correct equations.
205 min estimated
Topic 1.8 progress0%
Key Concepts0/17 sections complete Question Bank0/3 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Solve systems of up to three linear equations.
Solve polynomial equations numerically.
Use graphical or numerical methods when appropriate.
Interpret and filter solutions according to domain and context.
Check computed results when a substitution or graph makes verification practical.
Key formulas
Root definition
$h(r)=0$
Intersection-to-zero conversion
$f(x)=g(x)\iff f(x)-g(x)=0$
Absolute residual
$R=\left|\text{LHS}-\text{RHS}\right|$
Polynomial factor and root
$(x-r)^m\text{ is a factor}\Longrightarrow r\text{ is a root of multiplicity }m$
Break-even equation
$R(x)=C(x)$
Difference function for intersections
$h(x)=f(x)-g(x)$
Sign-change bracket
$h(a)h(b)\lt0\Longrightarrow\text{at least one root in }(a,b)\text{ if }h\text{ is continuous}$
Two-variable linear system
$a_1x+b_1y=c_1,\qquad a_2x+b_2y=c_2$
Three-variable linear system
$a_ix+b_iy+c_iz=d_i\qquad(i=1,2,3)$
Polynomial standard form
$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0,\qquad a_n\ne0$
Maximum number of distinct real roots
$\text{number of distinct real roots}\le n$
Even-root restriction
$\sqrt{u(x)}\text{ requires }u(x)\ge0$
Logarithm restriction
$\log_b(u(x))\text{ requires }u(x)\gt0$
Denominator restriction
$\frac{p(x)}{q(x)}\text{ requires }q(x)\ne0$
Exponential solution check
$ab^x=c\Longrightarrow x=\frac{\ln(\frac{c}{a})}{\ln b}$
Logarithm argument condition
$\log_b u\text{ is real only when }u\gt0$
Approximate equality symbol
$x\approx1.732$
Exact equality symbol
$x=\sqrt3$
Zero residual
$R=|h(r)|$
Intersection residual
$R=|f(r)-g(r)|$
Zero and intersection equivalence
$f(x)=g(x)\iff f(x)-g(x)=0$
Sign-change bracket
$h(a)h(b)\lt0$
Residual
$R=|\text{LHS}-\text{RHS}|$
Polynomial form
$a_nx^n+\cdots+a_1x+a_0=0$
Two-variable system
$a_1x+b_1y=c_1,\quad a_2x+b_2y=c_2$
Topic 1.9Not started
Laws of logarithms
HL students manipulate logarithms algebraically and use their laws to simplify expressions, solve equations and support linearization methods.
230 min estimated
Topic 1.9 progress0%
Key Concepts0/12 sections complete Question Bank0/14 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Translate confidently between exponential and logarithmic forms and state the conditions for real logarithms.
Use and justify the product, quotient, and power laws while preserving domain restrictions.
Expand and condense logarithmic expressions and use change of base accurately.
Solve logarithmic equations, reject extraneous roots, and solve exponential equations using logarithms.
Apply logarithms to growth, decay, threshold, and logarithmic-scale problems.
Linearize exponential and power models, interpret transformed gradients and intercepts, and back-transform parameters.
Use a GDC to evaluate logarithms, solve equations, and fit transformed models while communicating the method.
Present exact and approximate answers with correct rounding, units, constraints, and contextual interpretation.
Key formulas
Logarithm definition
$\log_a x=y\quad\Longleftrightarrow\quad a^y=x$
Logarithm of one
$\log_a1=0$
Logarithm of its base
$\log_a a=1$
Inverse relationships
$a^{\log_a x}=x,\qquad \log_a(a^x)=x$
Product-law derivation
$xy=a^ua^v=a^{u+v}$
Quotient-law derivation
$\frac{x}{y}=\frac{a^u}{a^v}=a^{u-v}$
Product law
$\log_a(xy)=\log_a x+\log_a y$
Quotient law
$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$
Power law
$\log_a(x^r)=r\log_a x$
Reciprocal result
$\log_a\left(\frac1x\right)=-\log_a x$
Change of base
$\log_a x=\frac{\log_b x}{\log_b a}$
Calculator forms
$\log_a x=\frac{\ln x}{\ln a}=\frac{\log x}{\log a}$
Condensing pattern
$r\log_a x-s\log_a y=\log_a\left(\frac{x^r}{y^s}\right)$
Real logarithm conditions
$a>0,\qquad a\ne1,\qquad g(x)>0$
Power identity for unrestricted nonzero x
$\log_a(x^2)=2\log_a|x|$
Same-base principle
$a^u=a^v\quad\Longrightarrow\quad u=v$
Logarithmic solution step
$a^{f(x)}=k\quad\Longrightarrow\quad f(x)=\frac{\ln k}{\ln a}$
Log equality principle
$\log_a u=\log_a v\quad\Longrightarrow\quad u=v$
Exponential growth or decay model
$Q(t)=Q_0b^t$
Percentage multiplier
$b=1\pm\frac{r}{100}$
Time to a target
$t=\frac{\ln(\frac{T}{Q_0})}{\ln b}$
pH model
$\mathrm{pH}=-\log_{10}[\mathrm H^+]$
Sound-intensity level
$L=10\log_{10}\left(\frac{I}{I_0}\right)$
Ratio from a base-10 scale difference
$\text{ratio}=10^{\Delta}$
Exponential linearization
$y=Ab^x\quad\Longrightarrow\quad \ln y=\ln A+x\ln b$
Power-model linearization
$y=Ax^n\quad\Longrightarrow\quad \ln y=\ln A+n\ln x$
Back-transform an intercept
$c=\ln A\quad\Longrightarrow\quad A=e^c$
Change of base for a GDC
$\log_a x=\frac{\ln x}{\ln a}=\frac{\log x}{\log a}$
Equation workflow
$\text{DOMAIN}\to\text{TRANSFORM}\to\text{SOLVE}\to\text{CHECK}\to\text{INTERPRET}$
Exponential threshold time
$t=\frac{\ln(\frac{T}{Q_0})}{\ln b}$
Definition
$\log_a x=y\iff a^y=x$
Quotient law
$\log_a(\frac{x}{y})=\log_a x-\log_a y$
Change of base
$\log_a x=\frac{\ln x}{\ln a}$
Valid real logarithm
$a>0,\ a\ne1,\ x>0$
Topic 1.10Not started
Expressions with rational exponents
Rational exponents allow roots and powers to be handled in one algebraic language and appear in power models and transformations.
220 min estimated
Topic 1.10 progress0%
Key Concepts0/11 sections complete Question Bank0/2 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Interpret rational exponents and convert fluently between power and radical notation.
Apply the exponent laws to products, quotients, and powers while respecting real-domain conditions.
Evaluate positive, negative, and zero rational powers exactly where possible.
Analyze negative bases, principal roots, and absolute-value requirements accurately.
Simplify multi-factor expressions and present final answers with positive exponents when requested.
Solve equations involving rational exponents, substitutions, multiple branches, and domain checks.
Use rational exponents in power models, geometric scaling, and other applied contexts.
Use a GDC for evaluation and numerical solutions while communicating syntax, domains, accuracy, and interpretation.
Key formulas
Rational exponent definition
$a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$
Negative rational exponent
$a^{\frac{-m}{n}}=\frac{1}{a^{\frac{m}{n}}}$
Special root form
$a^{\frac{1}{n}}=\sqrt[n]{a}$
Radical–exponent conversion
$\sqrt[n]{x^m}=x^{\frac{m}{n}},\qquad x>0,\ m\in\mathbb Z,\ n\in\mathbb Z_{>0}$
Principal square root
$\sqrt{x^2}=|x|$
Odd root of a power
$\sqrt[2k+1]{x^{2k+1}}=x$
Product law
$a^r a^s=a^{r+s}$
Quotient law
$\frac{a^r}{a^s}=a^{r-s}$
Power of a power
$(a^r)^s=a^{rs}$
Power of a product
$(ab)^r=a^r b^r$
Power of a quotient
$\left(\frac ab\right)^r=\frac{a^r}{b^r}$
Negative exponent conversion
$x^{-r}=\frac{1}{x^r}$
Common-denominator strategy
$x^{\frac{p}{q}}x^{\frac{r}{s}}=x^{\frac{(ps+rq)}{(qs)}}$
Exact before decimal
$\sqrt[n]{a^m}=a^{\frac{m}{n}}$
Negative base with odd reduced denominator
$(-a)^{\frac{m}{n}}=\left(\sqrt[n]{-a}\right)^m\quad(n\text{ odd})$
Principal square root
$(x^2)^{\frac{1}{2}}=\sqrt{x^2}=|x|$
Zero restriction
$0^r\text{ is defined for }r>0\text{ and undefined for }r<0$
Direct positive-domain pattern
$x^{\frac{m}{n}}=k\quad\Longrightarrow\quad x=k^{\frac{n}{m}}$
Useful substitution
$u=x^{\frac{1}{n}}\quad\Longrightarrow\quad x^{\frac{m}{n}}=u^m$
Two-branch warning
$u^2=k\quad\Longrightarrow\quad u=\pm\sqrt{k}$
Approximation notation
$\text{exact value}\approx\text{rounded decimal}$
Factor a perfect nth power
$\sqrt[n]{c^n d}=c\sqrt[n]{d}\quad(c\ge0\text{ if }n\text{ is even})$
Power model
$y=Ax^p$
Scaling response
$x\mapsto kx\quad\Longrightarrow\quad y\mapsto k^p y$
Cube side from volume
$s=V^{\frac{1}{3}}$
Safe calculator entry
$a^{\frac{m}{n}}\;\longrightarrow\;a^{(\frac{m}{n})}$
Real negative-base evaluation
$(-a)^{\frac{m}{n}}=\left(\sqrt[n]{-a}\right)^m\quad(n\text{ odd})$
Equation workflow
$\text{DOMAIN}\to\text{SUBSTITUTE}\to\text{SOLVE}\to\text{CHECK}$
Scaling rule
$y=Ax^p,\quad x\mapsto kx\Rightarrow y\mapsto k^py$
Rational exponent
$a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$
Negative rational exponent
$a^{\frac{-m}{n}}=\frac1{a^{\frac{m}{n}}}$
Product law
$a^ra^s=a^{r+s}$
Topic 1.11Not started
Infinite geometric sequences and series
A geometric series can converge to a finite limit when the magnitude of the common ratio is less than one.
230 min estimated
Topic 1.11 progress0%
Key Concepts0/11 sections complete Question Bank0/11 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Review geometric sequences and series and distinguish terms from partial sums.
Identify the first term and common ratio, including alternating geometric structures.
Explain convergence through partial sums and apply the condition |r|<1 correctly.
Derive and use the finite-sum and sum-to-infinity formulas.
Find unknown first terms, common ratios, terms, and parameters while rejecting invalid candidates.
Calculate and interpret remainders after a finite number of terms.
Model repeated processes, recurring decimals, bouncing motion, steady states, and perpetuities.
Use a GDC to inspect partial sums and solve parameter equations while communicating convergence and context.
Key formulas
Infinite geometric series
$a+ar+ar^2+ar^3+\cdots$
Partial sum notation
$S_n=\sum\limits_{k=0}^{n-1}ar^k$
Common ratio
$r=\frac{u_{n+1}}{u_n}$
Geometric term
$u_n=ar^{n-1}$
Geometric convergence condition
$|r|<1\quad\Longleftrightarrow\quad -1<r<1$
Vanishing powers
$|r|<1\quad\Longrightarrow\quad r^n\to0$
Finite geometric sum
$S_n=\frac{a(1-r^n)}{1-r}$
Sum to infinity
$S_\infty=\frac{a}{1-r},\qquad |r|<1$
First term from the sum
$a=S_\infty(1-r)$
Ratio from first term and sum
$r=1-\frac{a}{S_\infty},\qquad S_\infty\ne0$
Term condition
$u_n=ar^{n-1}$
Remainder after n terms
$R_n=S_\infty-S_n=\frac{ar^n}{1-r}$
Approximation relation
$S_\infty=S_n+R_n$
Bouncing-ball distance
$D=h+\frac{2hq}{1-q}=h\left(\frac{1+q}{1-q}\right)$
Repeated fixed contribution
$A_\infty=\frac{d}{1-r},\qquad |r|<1$
Two-digit recurring decimal
$0.\overline{ab}=\frac{10a+b}{99},\qquad a,b\in\{0,1,\ldots,9\}$
Retain-and-add recurrence
$A_{n+1}=rA_n+d$
Stable long-term level
$L=\frac{d}{1-r},\qquad |r|<1$
Present value of an end-of-period perpetuity
$PV=\frac{P}{i},\qquad i>0$
Partial-sum calculator entry
$S_n=\frac{a(1-r^n)}{1-r}$
Solver interval
$-1<r<1$
Convergence and sum
$|r|<1\quad\Longrightarrow\quad S_\infty=\frac{a}{1-r}$
Term and remainder
$u_n=ar^{n-1},\qquad R_n=\frac{ar^n}{1-r}$
nth term
$u_n=ar^{n-1}$
Finite sum
$S_n=\frac{a(1-r^n)}{1-r}$
Convergence condition
$|r|<1$
Remainder after n terms
$R_n=\frac{ar^n}{1-r}$
Unknown first term
$a=S_\infty(1-r)$
Unknown common ratio
$r=1-\frac{a}{S_\infty}$
Topic 1.12Not started
Complex numbers in Cartesian form
Complex numbers extend the real number system and allow polynomial equations with negative discriminant to be solved. AI HL develops both algebraic operations and geometric interpretation in the complex plane.
300 min estimated
Topic 1.12 progress0%
Key Concepts0/16 sections complete Question Bank0/10 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Define the imaginary unit and write complex numbers accurately in Cartesian form.
Identify real and imaginary parts and use component equality to find unknowns.
Add, subtract, multiply, and divide complex numbers in exact Cartesian form.
Reduce integer powers of i using the four-term cycle.
Use complex conjugates to simplify products, perform division, and establish identities.
Calculate modulus and distance and interpret both geometrically on an Argand diagram.
Solve linear, quadratic, conjugate, and Cartesian square-root equations involving complex numbers.
Apply Cartesian complex numbers to component models such as impedance and planar displacement.
Use a GDC in complex mode to verify results while preserving exact reasoning and contextual interpretation.
Key formulas
Imaginary unit
$i^2=-1$
Cartesian form
$z=a+bi$
Set of complex numbers
$\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}$
Real part
$\operatorname{Re}(a+bi)=a$
Imaginary part
$\operatorname{Im}(a+bi)=b$
Real-number condition
$z\in\mathbb{R}\iff\operatorname{Im}(z)=0$
Complex equality
$a+bi=c+di\iff a=c\text{ and }b=d$
Addition
$(a+bi)+(c+di)=(a+c)+(b+d)i$
Subtraction
$(a+bi)-(c+di)=(a-c)+(b-d)i$
Four-power cycle
$i^0=1,\quad i^1=i,\quad i^2=-1,\quad i^3=-i,\quad i^4=1$
Periodicity
$i^{n+4}=i^n$
Inverse of i
$i^{-1}=-i$
Complex multiplication
$(a+bi)(c+di)=(ac-bd)+(ad+bc)i$
Complex square
$(a+bi)^2=(a^2-b^2)+2abi$
Conjugate
$\overline{a+bi}=a-bi$
Sum with conjugate
$z+\overline z=2\operatorname{Re}(z)$
Difference from conjugate
$z-\overline z=2i\operatorname{Im}(z)$
Conjugate product
$z\overline z=a^2+b^2$
Conjugate of a product
$\overline{zw}=\overline z\,\overline w$
Complex quotient
$\frac{a+bi}{c+di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$
Complex reciprocal
$\frac1{a+bi}=\frac{a-bi}{a^2+b^2}$
Modulus
$|a+bi|=\sqrt{a^2+b^2}$
Modulus squared
$|z|^2=z\overline z$
Distance
$d(z_1,z_2)=|z_1-z_2|$
Product modulus
$|zw|=|z||w|$
Triangle inequality
$|z+w|\le |z|+|w|$
Argand coordinate
$z=a+bi\longleftrightarrow(a,b)$
Conjugate reflection
$a+bi\longleftrightarrow a-bi$
Displacement
$z_1-z_2$
Square in Cartesian form
$(a+bi)^2=(a^2-b^2)+2abi$
Quadratic formula
$z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Conjugate factor pair
$(x-(a+bi))(x-(a-bi))=(x-a)^2+b^2$
Quadratic from conjugate roots
$x^2-2ax+(a^2+b^2)=0$
Circle locus
$|z-c|=r$
Equidistance locus
$|z-z_1|=|z-z_2|$
Impedance model
$Z=R+Xi$
Simplified AC current
$I=\frac{V}{Z}$
Resultant displacement
$z_{\mathrm{new}}=z_{\mathrm{old}}+w$
Universal arithmetic check
$\text{computed answer}\xrightarrow{\text{substitution}}\text{original equation}$
Core Cartesian workflow
$\text{expand}\to\text{use }i^2=-1\to\text{collect real and imaginary parts}$
Conjugate
$\overline z=a-bi$
Modulus
$|z|=\sqrt{a^2+b^2}$
Division
$\frac{a+bi}{c+di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$
Topic 1.13Not started
Polar and exponential forms of complex numbers
Polar and exponential forms make multiplication, division and powers of complex numbers more natural by separating magnitude from angle.
330 min estimated
Topic 1.13 progress0%
Key Concepts0/18 sections complete Question Bank0/4 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Find the modulus and a quadrant-correct argument of a non-zero complex number.
Convert accurately among Cartesian, polar, cis and exponential forms.
Distinguish a principal argument from the general family of arguments.
Multiply and divide complex numbers by combining moduli and arguments.
Raise complex numbers to integer powers using polar or exponential form.
Interpret multiplication geometrically as a dilation and rotation on an Argand plane.
Use polar complex numbers in applied phasor and repeated-transformation models.
Use calculator/GDC complex-number functions while preserving exact values where required.
Communicate complete IB AI HL solutions with correct radians, quadrants and argument conventions.
Key formulas
Modulus
$r=|z|=\sqrt{x^2+y^2}$
Cartesian projections
$x=r\cos\theta,\qquad y=r\sin\theta$
Tangent relation
$\tan\theta=\frac{y}{x}$
Polar form
$z=r(\cos\theta+i\sin\theta)$
Equivalent arguments
$\theta_k=\theta+2k\pi,\qquad k\in\mathbb Z$
Cartesian reconstruction
$z=r\cos\theta+i r\sin\theta$
Cis definition
$\operatorname{cis}\theta=\cos\theta+i\sin\theta$
Cis form
$z=r\operatorname{cis}\theta$
Cis periodicity
$\operatorname{cis}(\theta+2k\pi)=\operatorname{cis}\theta$
Euler's relation
$e^{i\theta}=\cos\theta+i\sin\theta$
Exponential form
$z=re^{i\theta}$
Equivalent notation
$re^{i\theta}=r\operatorname{cis}\theta$
Principal-argument convention
$-\pi<\operatorname{Arg}z\le\pi$
General arguments
$\arg z=\operatorname{Arg}z+2k\pi,\qquad k\in\mathbb Z$
Cartesian-to-polar modulus
$r=\sqrt{x^2+y^2}$
Cartesian-to-polar argument
$\theta=\operatorname{atan2}(y,x)$
Equivalent output forms
$r(\cos\theta+i\sin\theta)=r\operatorname{cis}\theta=re^{i\theta}$
Real component
$x=r\cos\theta$
Imaginary component
$y=r\sin\theta$
Polar-to-Cartesian conversion
$r\operatorname{cis}\theta=r\cos\theta+i r\sin\theta$
Product in cis form
$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$
Product in exponential form
$z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}$
Product modulus
$|z_1z_2|=|z_1||z_2|$
Power in cis form
$\left(r\operatorname{cis}\theta\right)^n=r^n\operatorname{cis}(n\theta)$
Power in exponential form
$\left(re^{i\theta}\right)^n=r^n e^{in\theta}$
Modulus of a power
$|z^n|=|z|^n$
Quotient in cis form
$\frac{z_1}{z_2}=\frac{r_1}{r_2}\operatorname{cis}(\theta_1-\theta_2)$
Reciprocal
$\frac1{r\operatorname{cis}\theta}=\frac1r\operatorname{cis}(-\theta)$
Conjugate in polar form
$\overline{r\operatorname{cis}\theta}=r\operatorname{cis}(-\theta)$
Quotient modulus
$\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$
Scaling and rotation
$z\mapsto k\operatorname{cis}\phi\,z$
Modulus transformation
$|k\operatorname{cis}\phi\,z|=k|z|$
Argument transformation
$\arg(k\operatorname{cis}\phi\,z)=\arg z+\phi\pmod{2\pi}$
Phasor quotient model
$I=\frac{V}{Z}$
Repeated scale-and-rotate model
$z_n=z_0\left(ke^{i\phi}\right)^n=z_0k^n e^{in\phi}$
Phase difference
$\Delta\theta=\operatorname{Arg}(z_2)-\operatorname{Arg}(z_1)$
Axis arguments
$\operatorname{Arg}(a)=0\ (a>0),\quad \operatorname{Arg}(a)=\pi\ (a<0)$
Imaginary-axis arguments
$\operatorname{Arg}(bi)=\frac{\pi}{2}\ (b>0),\quad \operatorname{Arg}(bi)=-\frac{\pi}{2}\ (b<0)$
Positive-modulus conversion
$-r\operatorname{cis}\theta=r\operatorname{cis}(\theta+\pi)$
Operation choice
$\text{add or subtract}\to\text{Cartesian},\qquad \text{multiply or divide}\to\text{polar}$
Quadrant-aware argument
$\theta=\operatorname{atan2}(y,x)$
Post-operation check
$|z_1z_2|=|z_1||z_2|,\qquad \arg(z_1z_2)=\arg z_1+\arg z_2\pmod{2\pi}$
Principal normalization
$\theta\mapsto\theta+2k\pi\in(-\pi,\pi]$
Conversion-operation workflow
$\text{identify form}\to\text{find }r,\theta\to\text{operate}\to\text{normalize}\to\text{verify}$
Modulus
$r=\sqrt{x^2+y^2}$
Product
$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$
Quotient
$\frac{z_1}{z_2}=\frac{r_1}{r_2}\operatorname{cis}(\theta_1-\theta_2)$
Topic 1.14Not started
Matrices
Matrices provide a compact way to represent transformations, simultaneous equations and repeated-state systems. They become especially important in graph theory, Markov chains and differential-equation models.
330 min estimated
Topic 1.14 progress0%
Key Concepts0/17 sections complete Question Bank0/9 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Interpret matrix notation, dimensions, elements, rows and columns in mathematical and applied contexts.
Perform matrix addition, subtraction and scalar multiplication with compatible matrices.
Multiply matrices using row-column products and justify the order from dimensions and context.
Use zero and identity matrices and calculate powers of square matrices.
Calculate determinants and identify singular and non-singular matrices.
Find and verify inverse matrices, including the exact two-by-two formula.
Solve matrix equations and systems of two or three linear equations.
Interpret two-dimensional matrix transformations and their composition.
Construct and interpret transition, production and repeated-state matrix models.
Use calculator/GDC matrix functions accurately while showing mathematical setup and contextual conclusions.
Key formulas
General matrix notation
$A=(a_{ij})_{m\times n}$
Example element
$a_{ij}=\text{entry in row }i\text{ and column }j$
Dimensions
$A_{m\times n}=\text{matrix with }m\text{ rows and }n\text{ columns}$
Matrix addition
$(A+B)_{ij}=a_{ij}+b_{ij}$
Matrix subtraction
$(A-B)_{ij}=a_{ij}-b_{ij}$
Scalar multiplication
$(kA)_{ij}=k a_{ij}$
Additive identity
$A+O=A$
Product dimensions
$A_{m\times n}B_{n\times p}=(AB)_{m\times p}$
Product entry
$(AB)_{ij}=\sum\limits_{k=1}^{n}a_{ik}b_{kj}$
Identity matrix
$I_2=\begin{pmatrix}1&0\\0&1\end{pmatrix}$
Identity property
$AI=IA=A$
Matrix power
$A^n=\underbrace{AA\cdots A}_{n\text{ factors}}$
Two-by-two determinant
$\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$
Invertibility condition
$A^{-1}\text{ exists}\iff\det(A)\ne0$
Product determinant
$\det(AB)=\det(A)\det(B)$
Two-by-two inverse
$\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$
Inverse definition
$AA^{-1}=A^{-1}A=I$
Inverse of a product
$(AB)^{-1}=B^{-1}A^{-1}$
Left-sided unknown
$AX=B\implies X=A^{-1}B$
Right-sided unknown
$XA=B\implies X=BA^{-1}$
Matrix form of a system
$A\mathbf{x}=\mathbf{b}$
Unique inverse solution
$\mathbf{x}=A^{-1}\mathbf{b},\qquad\det(A)\ne0$
Augmented matrix
$[A\mid\mathbf{b}]$
Elementary row replacement
$R_i\leftarrow R_i+kR_j$
Row scaling
$R_i\leftarrow kR_i,\qquad k\ne0$
Coordinate transformation
$\mathbf{x}'=A\mathbf{x}$
Horizontal shear
$\begin{pmatrix}x'\\y'\end{pmatrix}=\begin{pmatrix}1&k\\0&1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x+ky\\y\end{pmatrix}$
Composition order
$\mathbf{x}''=B(A\mathbf{x})=(BA)\mathbf{x}$
One-step state model
$\mathbf{x}_{n+1}=T\mathbf{x}_n$
Repeated state model
$\mathbf{x}_n=T^n\mathbf{x}_0$
Steady-state condition
$T\mathbf{s}=\mathbf{s}$
Technology system check
$A(A^{-1}\mathbf b)=\mathbf b$
Repeated GDC model
$\mathbf{x}_n=T^n\mathbf{x}_0$
Core workflow
$\text{dimensions}\to\text{operation}\to\text{condition}\to\text{calculation}\to\text{interpretation}$
Practice workflow
$\text{check size}\to\text{choose operation}\to\text{calculate}\to\text{verify}\to\text{interpret}$
Product compatibility
$A_{m\times n}B_{n\times p}=(AB)_{m\times p}$
Determinant
$\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$
Inverse
$A^{-1}=\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$
System solution
$\mathbf{x}=A^{-1}\mathbf b$
Transformation
$\mathbf{x}'=A\mathbf{x}$
Repeated model
$\mathbf{x}_n=T^n\mathbf{x}_0$
Topic 1.15Not started
Eigenvalues, eigenvectors and diagonalization
Eigenvectors identify directions or states that are only scaled by a transformation. In AI HL they support powers of matrices and later help interpret coupled systems and long-term behaviour.
360 min estimated
Topic 1.15 progress0%
Key Concepts0/18 sections complete Question Bank0/4 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Interpret eigenvalues and eigenvectors through the equation A v = λv and matrix geometry.
Form and solve the characteristic equation to find exact eigenvalues.
Find and verify corresponding eigenvectors and describe eigenspaces.
Distinguish algebraic and geometric multiplicity for repeated eigenvalues.
Determine whether a matrix has enough independent eigenvectors to be diagonalizable.
Construct and verify a diagonalization $A = PDP^{-1}$ with correctly matched columns and diagonal entries.
Use diagonalization to calculate powers of matrices efficiently.
Decompose an initial state into eigencomponents and analyze repeated transformations.
Interpret eigenvalues in dynamic, stability and transition-matrix models.
Use calculator/GDC eigenvalue and matrix methods while preserving exact reasoning and contextual conclusions.
Key formulas
Eigenvector equation
$A\mathbf v=\lambda\mathbf v,\qquad\mathbf v\ne\mathbf0$
Equivalent homogeneous system
$(A-\lambda I)\mathbf v=\mathbf0$
Length scaling
$\|A\mathbf v\|=|\lambda|\,\|\mathbf v\|$
Repeated eigenvector action
$A^n\mathbf v=\lambda^n\mathbf v$
Characteristic equation
$\det(A-\lambda I)=0$
Two-by-two characteristic polynomial
$\lambda^2-\operatorname{tr}(A)\lambda+\det(A)=0$
Eigenvalue checks
$\lambda_1+\lambda_2=\operatorname{tr}(A),\qquad\lambda_1\lambda_2=\det(A)$
Two-by-two expansion
$\det(A-\lambda I)=(a-\lambda)(d-\lambda)-bc$
Quadratic formula
$\lambda=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Eigenvector system
$(A-\lambda I)\mathbf v=\mathbf0$
Eigenspace
$E_{\lambda}=\ker(A-\lambda I)$
Eigenspace span
$E_{\lambda}=\operatorname{span}\{\mathbf v_1,\ldots,\mathbf v_k\}$
Multiplicity bound
$1\le m_g(\lambda)\le m_a(\lambda)$
Diagonalizability count
$\dim(E_{\lambda_1})+\cdots+\dim(E_{\lambda_k})=n$
Multiplicity condition
$m_g(\lambda)=m_a(\lambda)\text{ for every eigenvalue}$
Diagonalization
$A=PDP^{-1}$
Equivalent relation
$AP=PD$
Eigenvector and eigenvalue arrangement
$P=[\mathbf v_1\ \cdots\ \mathbf v_n],\qquad D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$
Construction
$P=[\mathbf v_1\ \cdots\ \mathbf v_n],\qquad D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$
Reconstruction check
$PDP^{-1}=A$
Power of a diagonalizable matrix
$A^n=PD^nP^{-1}$
Power of D
$D^n=\operatorname{diag}(\lambda_1^n,\ldots,\lambda_n^n)$
Eigenvector power rule
$A^n\mathbf v=\lambda^n\mathbf v$
Eigenbasis coordinates
$\mathbf c=P^{-1}\mathbf{x}_0$
State decomposition
$\mathbf{x}_0=P\mathbf c=\sum\limits_{i=1}^{k}c_i\mathbf v_i$
Repeated state
$A^n\mathbf{x}_0=\sum\limits_{i=1}^{k}c_i\lambda_i^n\mathbf v_i$
Discrete iteration
$\mathbf{x}_{n+1}=A\mathbf{x}_n$
State after n steps
$\mathbf{x}_n=A^n\mathbf{x}_0$
Spectral radius
$\rho(A)=\max_i|\lambda_i|$
Discrete asymptotic stability
$\rho(A)<1$
Steady-state eigen-equation
$T\mathbf{s}=\mathbf{s}$
Normalized stationary state
$\sum\limits_{i=1}^{k}s_i=1$
Steady plus transient form
$\mathbf{x}_n=\mathbf{x}_{\mathrm{steady}}+c\lambda_{\mathrm{transient}}^n\mathbf v$
Eigenpair residual
$\mathbf r=A\mathbf v-\lambda\mathbf v$
Diagonalization check
$AP=PD$
Solution workflow
$\det(A-\lambda I)=0\to\lambda\to(A-\lambda I)\mathbf v=0\to P,D\to\text{verify}\to\text{apply}$
Practice workflow
$A\to\det(A-\lambda I)=0\to\lambda_i\to\mathbf v_i\to P,D\to\text{power or model}$
Eigenpair
$A\mathbf v=\lambda\mathbf v$
Eigenvectors
$(A-\lambda I)\mathbf v=\mathbf0$
Power
$A^n=PD^nP^{-1}$
Repeated state
$\mathbf{x}_n=A^n\mathbf{x}_0$
Unit 2
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Functions
Functions are the modelling language of AI. The shared course covers straight lines, graph interpretation and common real-world models; HL extends this into composite and inverse functions, transformations, logistic and logarithmic models, piecewise structures and logarithmic linearization.
10 topics
Unit 2 progress0%
Topics in this unit
Topic 2.1Not started
Equations of a straight line
Straight lines model constant rate of change and also provide the transformed relationships used in regression and linearization.
170 min estimated
Topic 2.1 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Calculate and interpret gradient.
Construct line equations from points, gradient or intercepts.
Recognize parallel and perpendicular lines.
Interpret slope and intercept with units in context.
Move between common line forms.
Key formulas
Gradient from two points
$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1},\qquad x_2\ne x_1$
Gradient-intercept form
$y=mx+c$
Intercepts of y = mx + c
$y\text{-intercept}: (0,c),\qquad x\text{-intercept}: \left(-\frac{c}{m},0\right),\ m\ne0$
Point-slope form
$y-y_1=m(x-x_1)$
General form
$ax+by+d=0$
General-form gradient and intercept
$m=-\frac{a}{b},\qquad c=-\frac{d}{b}\qquad(b\ne0)$
Two-point equation
$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1),\qquad x_2\ne x_1$
Parallel-line condition
$m_1=m_2$
Perpendicular-line condition
$m_1m_2=-1\qquad\Longleftrightarrow\qquad m_2=-\frac{1}{m_1}$
Intercept method for $ax+by=d$
$x\text{-intercept}=\left(\frac{d}{a},0\right),\qquad y\text{-intercept}=\left(0,\frac{d}{b}\right)$
Generic linear model
$\text{output}=\text{rate}\times\text{input}+\text{initial value}$
Units of gradient
$\text{units of }m=\frac{\text{output units}}{\text{input units}}$
Equal-model condition
$f(x)=g(x)$
Gradient from two points
$m=\frac{y_2-y_1}{x_2-x_1}$
Parallel condition
$m_1=m_2$
Perpendicular condition
$m_1m_2=-1$
Topic 2.2Not started
Concept of a function
A function connects an input to a unique output. AI emphasizes the meaning of the variables and the domain in which the relationship is valid.
205 min estimated
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Learning objectives
Use function notation in abstract and contextual forms.
Identify domain and range.
Interpret independent and dependent variables.
Understand an inverse as reversing a one-to-one relationship.
Relate inverse graphs by reflection in $y = x$ .
Key formulas
Function mapping
$f:A\to B,\qquad x\mapsto f(x)$
Output notation
$y=f(x)$
Evaluating a function
$f(a)=\left.f(x)\right|_{x=a}$
Domain notation
$\operatorname{Dom}(f)=\{x:\ f(x)\text{ is defined}\}$
Range notation
$\operatorname{Ran}(f)=\{f(x):\ x\in\operatorname{Dom}(f)\}$
Graph coordinates
$(x,y)=(x,f(x))$
Sideways parabola example
$x=y^2\quad\Longrightarrow\quad y=\pm\sqrt{x}$
Inverse relationship
$f(a)=bquadLongleftrightarrowquad f^{-1}(b)=a$
Inverse check
$f\!\left(f^{-1}(x)\right)=x\quad\text{and}\quad f^{-1}\!\left(f(x)\right)=x$
Function notation
$y=f(x)$
Graph point
$(x,y)=(x,f(x))$
Domain
$\operatorname{Dom}(f)=\{x:f(x)\text{ is defined}\}$
Range
$\operatorname{Ran}(f)=\{f(x):x\in\operatorname{Dom}(f)\}$
Inverse relationship
$f(a)=b\Longleftrightarrow f^{-1}(b)=a$
Topic 2.3Not started
Graphing functions with technology
Graphing technology is used to explore behaviour, but exam answers still need a clear mathematical sketch or stated features when requested.
230 min estimated
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Learning objectives
Choose an appropriate graphing window.
Sketch from a formula, context or data.
Graph sums and differences of functions.
Identify hidden features that a poor window might miss.
Transfer relevant coordinates and asymptotic behaviour from technology to written work.
Key formulas
Zero condition
$f(r)=0$
Intersection condition
$f(x)=g(x)$
Fraction entry
$\frac{x+1}{x-2}\quad\longrightarrow\quad \frac{(x+1)}{(x-2)}$
Exponential entry
$ab^x\quad\longrightarrow\quad a(b)^x$
Displayed x-range
$X_{\min}\le x\le X_{\max}$
Sign-change clue
$f(a)f(b)<0\quad\Longrightarrow\quad\text{a crossing root may lie in }(a,b)$
Useful quadratic form
$y=a(x-h)^2+k$
Useful reciprocal form
$y=\frac{a}{x-h}+k$
Point on a function graph
$(a,b)\text{ lies on }y=f(x)\iff b=f(a)$
Domain from a graph
$\operatorname{Dom}(f)=\{x\in\mathbb{R}:\text{the graph contains a point with x-coordinate }x\}$
Range from a graph
$\operatorname{Ran}(f)=\{y\in\mathbb{R}:y=f(x)\text{ for some }x\in\operatorname{Dom}(f)\}$
x-intercepts and roots
$f(x)=0\quad\Longleftrightarrow\quad (x,0)\text{ is an x-intercept}$
y-intercept
$x=0\quad\Longrightarrow\quad (0,f(0))$
Quadratic vertex form
$f(x)=a(x-h)^2+k\quad\Longrightarrow\quad\text{vertex }(h,k)$
Sign from graph position
$f(x)>0\iff\text{graph above x-axis},\qquad f(x)<0\iff\text{graph below x-axis}$
Root as an intercept
$f(r)=0\iff (r,0)\text{ lies on }y=f(x)$
Horizontal-level solution
$f(x)=k\iff y=f(x)\text{ intersects }y=k$
Intersection condition
$f(x)=g(x)\quad\Longleftrightarrow\quad f(x)-g(x)=0$
Profit from two models
$P(x)=R(x)-C(x)$
Point on graph
$(a,b)\text{ on }y=f(x)\iff f(a)=b$
Root or x-intercept
$f(x)=0$
y-intercept
$(0,f(0))$
Intersection
$f(x)=g(x)$
Sign
$f(x)>0\text{ above the x-axis},\qquad f(x)<0\text{ below the x-axis}$
Maximum or minimum point
$(x_{\mathrm{ext}},f(x_{\mathrm{ext}}))$
Context check
$x\in\text{stated model domain}$
Topic 2.4Not started
Key features of graphs
Students should extract information from graphs rather than treating them as pictures. Intersections, intercepts and turning behaviour often have contextual meaning.
220 min estimated
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Learning objectives
Find intersections numerically.
Identify roots and intercepts.
Locate maxima and minima where relevant.
Interpret intervals of increase/decrease and important changes in shape.
Relate graph features to the quantities being modelled.
Key formulas
Point on a graph
$(a,b)\text{ lies on }y=f(x)\iff f(a)=b$
Even-root condition
$\sqrt{u(x)}\text{ is real}\iff u(x)\geq0$
Denominator condition
$\frac{p(x)}{q(x)}\text{ is defined only when }q(x)\ne0$
Logarithm condition
$\log_a(u(x))\text{ is defined only when }u(x)>0$
Root condition
$f(r)=0\iff (r,0)\text{ is an x-intercept}$
y-intercept
$(0,f(0))$
Quadratic vertex x-coordinate
$x_{\mathrm v}=-\frac{b}{2a}\quad\text{for }f(x)=ax^2+bx+c$
Positive region
$f(x)>0\iff\text{the graph is above the x-axis}$
Negative region
$f(x)<0\iff\text{the graph is below the x-axis}$
Intersection condition
$f(x)=g(x)\iff f(x)-g(x)=0$
Root
$f(x)=0$
Target output
$f(x)=k$
Intersection
$f(x)=g(x)$
Difference-function form
$f(x)-g(x)=0$
Zero command target
$Y_1=f(x),\qquad Y_1=0$
Intersection command target
$Y_1=f(x),\quad Y_2=g(x),\qquad Y_1=Y_2$
Break-even condition
$R(x)=C(x)$
Profit model
$P(x)=R(x)-C(x)$
Break-even as a root
$P(x)=0$
Root or zero
$f(r)=0$
x-intercept
$(r,0)$
Break-even
$R(x)=C(x)\iff P(x)=0$
Topic 2.5Not started
Modelling with common functions
AI HL uses a broad library of functions for applied modelling. Students should recognize characteristic shapes and connect parameters to real features.
250 min estimated
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Learning objectives
Model with linear, quadratic, exponential, power/direct-inverse, cubic and sinusoidal functions.
Use technology to determine parameters from data.
Identify features such as vertex, roots, intercepts, asymptotes, amplitude and period.
Choose a model that is plausible for the context and observed shape.
Use the model for interpolation or prediction with an appropriate domain.
Key formulas
General modelling statement
$y=f(x;\,\text{parameters})$
Linear model
$y=mx+c$
Gradient from two points
$m=\frac{y_2-y_1}{x_2-x_1}$
Intercept from a point
$c=y_1-mx_1$
Quadratic model
$y=ax^2+bx+c,\qquad a\ne0$
Vertex input
$x_{\mathrm v}=-\frac{b}{2a}$
Vertex form
$y=a(x-h)^2+k$
Exponential model
$y=ab^x$
Continuous form
$y=ae^{kx}$
Growth factor
$b=1+r$
Decay factor
$b=1-r$
Power model
$y=ax^n$
Exponent from two positive points
$n=\frac{\ln(\frac{y_2}{y_1})}{\ln(\frac{x_2}{x_1})}$
Inverse proportionality
$y=\frac{a}{x}$
Cubic model
$y=ax^3+bx^2+cx+d$
Sinusoidal model
$y=a\sin(b(x-c))+d$
Sinusoidal period
$T=\frac{2\pi}{|b|}$
Sinusoidal range
$[d-|a|,\ d+|a|]$
First difference
$\Delta y=y_{k+1}-y_k$
Successive ratio
$\frac{y_{k+1}}{y_k}$
Residual
$e_i=y_i-\hat y_i$
Exponential base from two points
$b=\left(\frac{y_2}{y_1}\right)^{\frac{1}{(x_2-x_1)}}$
Exponential coefficient
$a=\frac{y_1}{b^{x_1}}$
Quadratic from roots
$y=a(x-r_1)(x-r_2)$
Linear regression form
$y=mx+c$
Exponential regression form
$y=ab^x$
Power regression form
$y=ax^b$
Prediction
$\hat y=f(x_0)$
Shifted exponential model
$y=c+ab^x$
Long-term level for decay
$0<b<1\quad\Longrightarrow\quad y\to c\text{ as }x\to\infty$
Regression prediction
$\hat y=f_{\mathrm{reg}}(x_0)$
Linear
$y=mx+c$
Quadratic
$y=ax^2+bx+c$
Exponential
$y=ab^x$
Power
$y=ax^n$
Cubic
$y=ax^3+bx^2+cx+d$
Sinusoidal
$y=a\sin(b(x-c))+d$
Residual
$e=y-\hat y$
Prediction
$\hat y=f_{\mathrm{model}}(x_0)$
Topic 2.6Not started
The modelling cycle
A strong AI solution does more than fit a curve. Students should be able to develop, test and reflect on a model as a representation of reality.
260 min estimated
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Learning objectives
Define variables and the intended domain.
Select a model based on data, shape and context.
Fit parameters using appropriate technology.
Compare model values with observations.
Evaluate assumptions, residual behaviour and practical limitations.
Refine the model or restrict the domain when necessary.
Key formulas
General model
$\hat y=f(x;\,\text{parameters})$
Linear regression form
$\hat y=ax+b$
Exponential regression form
$\hat y=ab^x$
Residual
$e_i=y_i-\hat y_i$
Fitted value from residual
$\hat y_i=y_i-e_i$
Sum of squared errors
$\mathrm{SSE}=\sum\limits_{i=1}^{n}(y_i-\hat y_i)^2$
Root mean square error
$\mathrm{RMSE}=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}(y_i-\hat y_i)^2}$
Linear change per unit
$\Delta\hat y=m\quad\text{when }\Delta x=1$
Exponential percentage change
$r=(b-1)\times100\%$
Shifted exponential cooling model
$\hat T=A+Cq^t$
Fitted model
$\hat y=f(x;\,\text{parameters})$
Residual
$e=y-\hat y$
SSE
$\mathrm{SSE}=\sum\limits_{i=1}^{n}e_i^2$
RMSE
$\mathrm{RMSE}=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}e_i^2}$
Linear parameter meaning
$\hat y=mx+c$
Exponential parameter meaning
$\hat y=ab^x$
Topic 2.7Not started
Composite and inverse functions
Composition represents processes performed in sequence, while an inverse reverses a one-to-one process on an appropriate domain.
350 min estimated
Topic 2.7 progress0%
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Learning objectives
Interpret composite functions as ordered multistage processes.
Evaluate composites from formulas, tables and mappings.
Construct and simplify composite-function formulas.
Determine domains of composite functions without losing inherited restrictions.
Interpret inverse functions as input-output reversals rather than reciprocals.
Test one-to-one behavior and restrict domains when necessary.
Find inverse functions algebraically and state their domains and ranges.
Verify inverse functions using both compositions.
Interpret inverse graphs as reflections in the line y equals x.
Use GDC methods and inverse models accurately in IB AI HL contexts.
Key formulas
Composite notation
$(f\circ g)(x)=f(g(x))$
Reverse composition
$(g\circ f)(x)=g(f(x))$
Three-function composition
$(f\circ g\circ h)(x)=f(g(h(x)))$
Table-chain condition
$g(x)\in\operatorname{Dom}(f)$
Formula substitution
$f(x)=F(x),\ g(x)=G(x)\implies(f\circ g)(x)=F(G(x))$
Order-sensitive example
$(2x-3)^2+1\ne2(x^2+1)-3$
Composite-domain rule
$\operatorname{Dom}(f\circ g)=\{x\in\operatorname{Dom}(g):g(x)\in\operatorname{Dom}(f)\}$
Even-root condition
$R(x)\ge0$
Denominator condition
$D(x)\ne0$
Two-stage model
$x\xmapsto{g}g(x)\xmapsto{f}f(g(x))$
Percentage multiplier
$x\mapsto(1+r)x$
Inverse input-output reversal
$f(a)=b\iff f^{-1}(b)=a$
Inverse identities
$f^{-1}(f(x))=x,\qquad f(f^{-1}(x))=x$
One-to-one condition
$f(a)=f(b)\implies a=b$
Derivative test for strict increase
$f'(x)\gt0\text{ throughout an interval}\implies f\text{ is one-to-one there}$
Positive square branch
$f(x)=x^2,\ x\ge0\implies f^{-1}(x)=\sqrt x$
Negative square branch
$f(x)=x^2,\ x\le0\implies f^{-1}(x)=-\sqrt x$
Algebraic inverse workflow
$y=f(x)\longrightarrow x=F(y)\longrightarrow f^{-1}(x)=F(x)$
Linear inverse
$f(x)=ax+b,\ a\ne0\implies f^{-1}(x)=\frac{x-b}{a}$
Inverse-domain swap
$\operatorname{Dom}(f^{-1})=\operatorname{Ran}(f)$
Inverse-range swap
$\operatorname{Ran}(f^{-1})=\operatorname{Dom}(f)$
Point reversal
$(a,b)\text{ on }f\iff(b,a)\text{ on }f^{-1}$
Left inverse check
$f^{-1}(f(x))=x,\quad x\in\operatorname{Dom}(f)$
Right inverse check
$f(f^{-1}(x))=x,\quad x\in\operatorname{Dom}(f^{-1})$
Graph reflection
$(a,b)\mapsto(b,a)$
Reflection line
$y=x$
Fixed-point equation
$f(x)=x$
Inverse-value equivalence
$f^{-1}(x)=a\iff x=f(a)$
Identity simplification
$f^{-1}(f(x))=x$
Exponential-model inverse
$P(t)=P_0e^{kt}\implies t=\frac1k\ln\left(\frac{P}{P_0}\right)$
Linear calibration inverse
$R(x)=ax+b\implies R^{-1}(y)=\frac{y-b}{a}$
GDC inverse check
$Y_1=f(x),\quad Y_2=f^{-1}(x),\quad Y_3=x$
Numerical identity check
$f^{-1}(f(x))\approx x$
Core workflow
$\text{order}\to\text{substitute}\to\text{restrict}\to\text{solve}\to\text{verify}\to\text{interpret}$
Practice workflow
$\text{identify notation}\to\text{form expression}\to\text{apply restrictions}\to\text{solve}\to\text{check}$
Composite
$(f\circ g)(x)=f(g(x))$
Composite domain
$\{x\in\operatorname{Dom}(g):g(x)\in\operatorname{Dom}(f)\}$
Inverse identities
$f^{-1}\circ f=I,\qquad f\circ f^{-1}=I$
Domain-range swap
$\operatorname{Dom}(f^{-1})=\operatorname{Ran}(f),\quad\operatorname{Ran}(f^{-1})=\operatorname{Dom}(f)$
Graph reflection
$(a,b)\leftrightarrow(b,a)\text{ across }y=x$
Topic 2.8Not started
Transformations of functions
Transform parent functions systematically, track every key feature, and interpret translations, reflections and scale changes in mathematical and applied models.
370 min estimated
Topic 2.8 progress0%
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Learning objectives
Describe vertical and horizontal translations using correct function notation.
Distinguish reflections in the x-axis and y-axis.
Apply and interpret vertical and horizontal stretches and compressions.
Analyze the general transformed form y=a f(b(x-h))+k.
Map points and key graph features from a parent function to its image.
Determine transformed domains, ranges, intercepts and asymptotes.
Identify transformations from equations and construct accurate sketches.
Combine transformations in a reliable order without inside-outside sign errors.
Interpret transformed functions in technology-supported real-world models.
Use a GDC to verify transformations while preserving exact mathematical reasoning.
Key formulas
Parent and image
$y=f(x)\quad\longrightarrow\quad y=g(x)$
Inside versus outside
$g(x)=a\,f(b(x-h))+k$
Vertical translation
$g(x)=f(x)+k$
Point mapping
$(x,y)\longmapsto(x,y+k)$
Horizontal translation
$g(x)=f(x-h)$
Point mapping
$(x,y)\longmapsto(x+h,y)$
x-axis reflection
$g(x)=-f(x),\qquad(x,y)\longmapsto(x,-y)$
y-axis reflection
$g(x)=f(-x),\qquad(x,y)\longmapsto(-x,y)$
Vertical scale
$g(x)=a f(x)$
Point mapping
$(x,y)\longmapsto(x,ay)$
Horizontal scale
$g(x)=f(bx)$
Point mapping
$(x,y)\longmapsto\left(\frac{x}{b},y\right),\qquad b\ne0$
General transformed function
$g(x)=a f(b(x-h))+k,\qquad a\ne0,\ b\ne0$
Complete point mapping
$(u,v)\longmapsto\left(h+\frac{u}{b},k+av\right)$
Scale factors
$\text{vertical}=|a|,\qquad\text{horizontal}=\frac{1}{|b|}$
Anchor-point map
$(u,v)\longmapsto\left(h+\frac{u}{b},k+av\right)$
Stationary feature map
$(u_s,v_s)\longmapsto\left(h+\frac{u_s}{b},k+av_s\right)$
Standard grouped input
$bx+c=b\left(x+\frac{c}{b}\right)$
Direct combined map
$(u,v)\longmapsto\left(h+\frac{u}{b},k+av\right)$
Domain condition
$\operatorname{Dom}(g)=\{x:b(x-h)\in\operatorname{Dom}(f)\}$
Range transformation
$\operatorname{Ran}(g)=\{av+k:v\in\operatorname{Ran}(f)\}$
Interval endpoint map
$u\longmapsto h+\frac{u}{b},\qquad v\longmapsto k+av$
x-intercept equation
$a f(b(x-h))+k=0$
y-intercept
$g(0)=a f(-bh)+k$
Stationary-point map
$(u_s,v_s)\longmapsto\left(h+\frac{u_s}{b},k+av_s\right)$
Vertical-asymptote map
$x=c\quad\longmapsto\quad x=h+\frac{c}{b}$
Horizontal-asymptote map
$y=L\quad\longmapsto\quad y=aL+k$
Hole map
$(c,L)\longmapsto\left(h+\frac{c}{b},k+aL\right)$
General identification form
$g(x)=a f(b(x-h))+k$
Quadratic vertex form
$y=a(x-h)^2+k$
Construction map
$(u,v)\longmapsto\left(h+\frac{u}{b},k+av\right)$
Feature checklist
$\text{domain, range, intercepts, extrema, asymptotes, endpoints}$
Delayed and rescaled model
$g(t)=a f(b(t-h))+k$
Transformed period
$T_g=\frac{T_f}{|b|}$
GDC comparison
$Y_1=f(X),\qquad Y_2=aY_1(b(X-h))+k$
Feature equation
$g(x)=c\quad\Longleftrightarrow\quad a f(b(x-h))+k=c$
Reliable coordinate rule
$(u,v)\longmapsto\left(h+\frac{u}{b},k+av\right)$
Practice master form
$g(x)=a f(b(x-h))+k$
Vertical translation
$f(x)+k: (x,y)\longmapsto(x,y+k)$
Horizontal translation
$f(x-h): (x,y)\longmapsto(x+h,y)$
Axis reflections
$-f(x): (x,y)\longmapsto(x,-y),\qquad f(-x): (x,y)\longmapsto(-x,y)$
Scale changes
$a f(x): (x,y)\longmapsto(x,ay),\qquad f(bx): (x,y)\longmapsto\left(\frac{x}{b},y\right)$
Combined transformation
$a f(b(x-h))+k: (u,v)\longmapsto\left(h+\frac{u}{b},k+av\right)$
Topic 2.9Not started
Logarithmic, sinusoidal, logistic and piecewise models
Choose, fit and critique logarithmic, sinusoidal, logistic and piecewise models by connecting their parameters and graph features to real data and context.
430 min estimated
Topic 2.9 progress0%
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Learning objectives
Recognize when logarithmic, sinusoidal, logistic or piecewise models suit a data pattern.
Interpret every parameter in standard logarithmic, sinusoidal and logistic forms.
Determine domains, ranges, intercepts and asymptotes for the four model families.
Use amplitude, period, phase shift and vertical shift to construct sinusoidal models.
Analyze carrying capacity, inflection and growth behaviour in logistic models.
Construct and evaluate piecewise models with correct boundary notation.
Fit model parameters from exact features and from regression technology.
Use residuals and contextual evidence to compare candidate models.
Distinguish interpolation from extrapolation and evaluate prediction reliability.
Present GDC-supported modelling conclusions with appropriate accuracy, units and limitations.
Key formulas
Model-selection evidence
$\text{shape}+\text{context}+\text{residuals}+\text{purpose}$
Prediction principle
$\text{fit quality}\ne\text{guaranteed future validity}$
Logarithm definition
$y=\log_bx\iff b^y=x,\qquad b\gt0,\ b\ne1,\ x\gt0$
Natural-log model
$L(x)=a+b\ln(x-h)$
Logarithmic domain
$x\gt h$
Logarithmic asymptote
$x=h$
Reference value
$L(h+1)=a$
x-intercept
$x=h+e^{\frac{-a}{b}},\qquad b\ne0$
Linearized form
$y=a+bX,\qquad X=\ln x$
Target input
$x=\exp\!\left(\frac{y-a}{b}\right)$
Sinusoidal model
$S(t)=d+a\sin(b(t-c))$
Cosine model
$S(t)=d+a\cos(b(t-c))$
Amplitude, period and range
$A=|a|,\qquad T=\frac{2\pi}{|b|},\qquad d-|a|\le S\le d+|a|$
Amplitude and midline
$A=\frac{M-m}{2},\qquad d=\frac{M+m}{2}$
Frequency parameter
$|b|=\frac{2\pi}{T}$
Maximum-anchored model
$S(t)=d+A\cos\!\left(\frac{2\pi}{T}(t-c)\right)$
Sinusoidal regression form
$y=a\sin(bx+c)+d$
Residual
$r_i=y_i-\widehat y_i$
Standard logistic model
$P(t)=\frac{K}{1+Ae^{-kt}},\qquad K,A,k\gt0$
Initial value
$P(0)=\frac{K}{1+A}$
Initial parameter
$A=\frac{K-P(0)}{P(0)}$
Full-domain logistic range
$0\lt P(t)\lt K$
Contextual range for t non-negative
$P(0)\le P(t)\lt K$
Shifted logistic form
$P(t)=\frac{K}{1+e^{-k(t-t_0)}}$
Inflection time and level
$t_{\mathrm{inf}}=\frac{\ln A}{k},\qquad P(t_{\mathrm{inf}})=\frac K2$
Logistic rate equation
$\frac{dP}{dt}=kP\left(1-\frac PK\right)$
Maximum growth rate
$\left(\frac{dP}{dt}\right)_{\max}=\frac{kK}{4}$
Known-capacity linearization
$\ln\!\left(\frac{P}{K-P}\right)=kt-\ln A$
Parameter from initial value
$A=\frac{K-P_0}{P_0}$
Piecewise notation
$f(x)=\begin{cases}f_1(x),&x\in I_1,\\f_2(x),&x\in I_2.\end{cases}$
Branch selection
$x\in I_j\quad\Longrightarrow\quad f(x)=f_j(x)$
Continuous tiered branch
$C(x)=C(c)+m(x-c),\qquad x\gt c$
Accumulated-distance branch
$D(t)=D(t_0)+v(t-t_0)$
Continuity condition
$\lim_{x\to c^-}f(x)=f(c)=\lim_{x\to c^+}f(x)$
Jump size
$J=\left|\lim_{x\to c^+}f(x)-\lim_{x\to c^-}f(x)\right|$
Logarithmic domain
$x-h\gt0$
Sinusoidal range
$d-|a|\le S\le d+|a|$
Logistic range
$0\lt P\lt K$
Model-comparison goal
$\text{small, patternless residuals}+\text{plausible parameters}$
Stored prediction
$\widehat y=f_{\mathrm{reg}}(x)$
Residual list
$L_{\mathrm{res}}=L_y-\widehat L_y$
Observed interval
$x_{\min}\le x\le x_{\max}$
Extrapolation condition
$x\lt x_{\min}\quad\text{or}\quad x\gt x_{\max}$
Relative residual
$\text{relative residual}=\frac{y-\widehat y}{y}$
Reliability evidence
$\text{coverage}+\text{residuals}+\text{robust parameters}+\text{context}$
Trend plus seasonality
$M(t)=T(t)+A\sin(b(t-c))$
Piecewise hybrid
$M(t)=\begin{cases}M_1(t),&t\le c,\\M_2(t),&t\gt c.\end{cases}$
Exam response
$\text{equation}+\text{parameters}+\text{prediction}+\text{units}+\text{limitation}$
Practice decision rule
$\text{identify behaviour}\to\text{choose form}\to\text{fit}\to\text{validate}\to\text{interpret}$
Logarithmic model
$L(x)=a+b\ln(x-h),\qquad x\gt h$
Sinusoidal model
$S(t)=d+a\sin(b(t-c)),\qquad A=|a|,\ T=\frac{2\pi}{|b|}$
Logistic model
$P(t)=\frac{K}{1+Ae^{-kt}},\qquad P_{\mathrm{inf}}=\frac K2$
Piecewise model
$f(x)=\begin{cases}f_1(x),&x\in I_1,\\f_2(x),&x\in I_2.\end{cases}$
Residual
$r=\text{observed}-\text{predicted}$
Topic 2.10Not started
Logarithmic scaling and linearization
Use semi-log and log-log transformations to reveal straight-line relationships, fit exponential and power models, and recover interpretable parameters.
230 min estimated
Topic 2.10 progress0%
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Learning objectives
Explain how equal distances on a logarithmic axis represent equal ratios rather than equal differences.
Distinguish linear, semi-log and log-log plots and state the positivity conditions for logarithms.
Linearize exponential models using natural or common logarithms.
Linearize power models by transforming both variables.
Interpret the gradient and intercept of a transformed straight line.
Recover original model parameters correctly for the chosen logarithm base.
Create transformed data columns and perform linear regression with technology.
Use transformed plots and residuals to identify and compare plausible models.
Interpret multiplicative change, doubling time, half-life and scale exponents in context.
Recognize limitations caused by non-positive data, shifts, outliers and transformed error structures.
Distinguish interpolation from extrapolation and evaluate the reliability of predictions.
Communicate an IB AI HL modelling solution with equations, parameters, units and contextual conclusions.
Key formulas
Linear-axis spacing
$\Delta x=x_2-x_1$
Log-axis spacing
$\Delta(\log_bx)=\log_b\left(\frac{x_2}{x_1}\right)$
Order-of-magnitude difference
$\log_{10}\left(\frac{x_2}{x_1}\right)$
Change of base
$\log_bx=\frac{\ln x}{\ln b}$
Inverse of natural log
$\ln A=c\Rightarrow A=e^c$
Inverse of common log
$\log_{10}A=c\Rightarrow A=10^c$
Semi-log coordinates
$X=x,\qquad Y=\ln y$
Exponential straight line
$Y=kX+\ln A$
One-unit growth factor
$\frac{y(x+1)}{y(x)}=e^k$
Log-log coordinates
$X=\ln x,\qquad Y=\ln y$
Power straight line
$Y=nX+\ln A$
Scaling rule
$\frac{y(qx)}{y(x)}=q^n$
Straight-line target
$Y=mX+c$
Exponential choice
$X=x,\quad Y=\ln y$
Power choice
$X=\ln x,\quad Y=\ln y$
Natural exponential linearization
$y=Ae^{kx}\Rightarrow\ln y=kx+\ln A$
Recovered parameters
$k=m,\qquad A=e^c$
Percentage change
$100(e^k-1)\%$
General exponential linearization
$\ln y=(\ln b)x+\ln A$
Natural-log recovery
$A=e^c,\qquad b=e^m$
Common-log recovery
$A=10^c,\qquad b=10^m$
Power linearization
$y=Ax^n\Rightarrow\ln y=n\ln x+\ln A$
Power recovery with ln
$n=m,\qquad A=e^c$
Elasticity
$\frac{\Delta y}{y}\approx n\frac{\Delta x}{x}$
Exponential intercept
$x=0\Rightarrow y=A$
Power intercept
$\ln x=0\Rightarrow x=1\Rightarrow y=A$
Common-log multiplier
$b=10^m$
Exponential ratio
$\frac{y_2}{y_1}=b^{x_2-x_1}$
Power ratio
$\frac{y_2}{y_1}=\left(\frac{x_2}{x_1}\right)^n$
Power exponent
$n=\frac{\ln(\frac{y_2}{y_1})}{\ln(\frac{x_2}{x_1})}$
Semi-log table
$(x,y)\mapsto(x,\ln y)$
Log-log table
$(x,y)\mapsto(\ln x,\ln y)$
Positivity check
$\ln z\text{ is real only for }z\gt0$
Linear signature
$y=a+bx$
Exponential signature
$\ln y=c+mx$
Power signature
$\ln y=c+m\ln x$
Transformed residual
$e_i=Y_i-(mX_i+c)$
Coefficient of determination
$R^2=1-\frac{\sum\limits_{i=1}^{n}e_i^2}{\sum\limits_{i=1}^{n}(Y_i-\overline{Y})^2}$
Back-transformed prediction
$\widehat y=e^{\widehat Y}$
Log residual as ratio
$\ln y-\ln\widehat y=\ln\left(\frac{y}{\widehat y}\right)$
Observed-to-fitted ratio
$\frac{y}{\widehat y}=e^{e_{\log}}$
Original residual
$r_i=y_i-\widehat y_i$
Direct back-transform
$\widehat y=\exp(\widehat{\ln y})$
Nonlinearity warning
$e^{E(Z)}\ne E(e^Z)\text{ in general}$
Log-normal mean adjustment
$E(Y)=e^{\mu+\frac{\sigma^2}{2}}$
Positive-domain conditions
$y\gt0\quad\text{and, for log-log,}\quad x\gt0$
Shifted exponential linearization
$y=C+Ae^{kx}\Rightarrow\ln(y-C)=kx+\ln A$
Shift requirement
$y-C\gt0$
pH scale
$\mathrm{pH}=-\log_{10}[H^+]$
Decibel level
$L=10\log_{10}\left(\frac I{I_0}\right)$
Ratio from index difference
$\frac{Q_2}{Q_1}=10^{\frac{\Delta L}{k}}$
Doubling time
$T_d=\frac{\ln2}{k},\qquad k\gt0$
Half-life
$T_{\frac{1}{2}}=\frac{\ln2}{|k|},\qquad k\lt0$
Base-form doubling time
$T_d=\frac{\ln2}{\ln b}$
Geometric scaling
$V\propto L^3$
Area scaling
$S\propto L^2$
Factor response
$x\mapsto qx\Rightarrow y\mapsto q^ny$
Interpolation condition
$x_{\min}\le x_{\text{pred}}\le x_{\max}$
Extrapolation condition
$x_{\text{pred}}\lt x_{\min}\text{ or }x_{\text{pred}}\gt x_{\max}$
Prediction
$\widehat y=f(x_{\text{pred}})$
Absolute error
$|y-\widehat y|$
Absolute percentage error
$100\left|\frac{y-\widehat y}{y}\right|\%$
Root mean square error
$\operatorname{RMSE}=\sqrt{\frac1n\sum\limits_{i=1}^n(y_i-\widehat y_i)^2}$
Dimensionless logarithm
$\ln\left(\frac Q{Q_0}\right)$
Exponent units
$[k]=[x]^{-1}$
Power coefficient units
$[A]=[y][x]^{-n}$
Exponential prediction from line
$\widehat y=e^{mx+c}$
Power prediction from line
$\widehat y=e^c x^m$
Unrounded calculation principle
$\text{store full }m,c\Rightarrow\text{round final }\widehat y$
Valid product law
$\ln(ab)=\ln a+\ln b$
Invalid sum split
$\ln(a+b)\ne\ln a+\ln b$
Exam chain
$\text{transform}\to\text{regress}\to\text{recover}\to\text{predict}\to\text{evaluate}$
Practice check: exponential
$\ln y\text{ vs }x\text{ linear}\Rightarrow y=Ae^{kx}$
Practice check: power
$\ln y\text{ vs }\ln x\text{ linear}\Rightarrow y=Ax^n$
Practice check: prediction
$\text{fit}\ne\text{guaranteed extrapolation}$
Exponential
$y=Ae^{kx}\Longleftrightarrow\ln y=kx+\ln A$
Power
$y=Ax^n\Longleftrightarrow\ln y=n\ln x+\ln A$
Base recovery
$\ln A=c\Rightarrow A=e^c,\qquad\log_{10}A=c\Rightarrow A=10^c$
Unit 3
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Geometry and Trigonometry
AI HL begins with measurement, triangle trigonometry, circles, perpendicular bisectors and Voronoi diagrams, then extends into unit-circle ideas, transformation matrices, vectors, kinematics and a full graph-theory/network-algorithm strand.
16 topics
Unit 3 progress0%
Topics in this unit
Topic 3.1Not started
Distance, midpoint and three-dimensional solids
Coordinate geometry and mensuration support practical measurement in two and three dimensions.
260 min estimated
Topic 3.1 progress0%
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Learning objectives
Calculate distance in 2D or 3D.
Find the midpoint of two points.
Use surface-area and volume formulas for cones, pyramids, spheres, hemispheres and composite solids.
Find relevant angles between geometric objects from a diagram or coordinate model.
Use units and significant figures appropriately.
Key formulas
Pythagorean structure
$d^2=(\text{change in }x)^2+(\text{change in }y)^2$
2D distance
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
2D midpoint
$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
Missing x-coordinate
$x_2=2m_x-x_1$
Scaled distance
$D_{\mathrm{actual}}=kD_{\mathrm{diagram}}$
Pythagorean theorem
$c^2=a^2+b^2$
Cuboid space diagonal
$d=\sqrt{l^2+w^2+h^2}$
3D distance
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
3D midpoint
$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$
Prism volume
$V=A_{\mathrm{cross}}L$
Cuboid volume
$V=lwh$
Cuboid surface area
$S=2(lw+lh+wh)$
Cylinder volume
$V=\pi r^2h$
Closed cylinder surface area
$S=2\pi r^2+2\pi rh$
Pyramid volume
$V=\frac13Bh$
Cone volume
$V=\frac13\pi r^2h$
Cone slant height
$s=\sqrt{r^2+h^2}$
Cone total surface area
$S=\pi r^2+\pi rs$
Regular pyramid surface area
$S=B+\frac12ps$
Sphere volume
$V=\frac43\pi r^3$
Sphere surface area
$S=4\pi r^2$
Hemisphere volume
$V=\frac23\pi r^3$
Hemisphere curved area
$S_{\mathrm{curved}}=2\pi r^2$
Hemisphere total area
$S_{\mathrm{total}}=3\pi r^2$
Composite volume
$V_{\mathrm{total}}=\sum\limits_{i=1}^{n}V_i-\sum\limits_{j=1}^{m}V_{\mathrm{removed},j}$
Line-plane angle
$\tan\theta=\frac{\text{perpendicular height}}{\text{projection length}}$
Cuboid diagonal
$d=\sqrt{l^2+w^2+h^2}$
Cylinder
$V=\pi r^2h$
Cone or pyramid
$V=\frac13Bh$
Sphere
$V=\frac43\pi r^3,\quad S=4\pi r^2$
Topic 3.2Not started
Trigonometry in triangles
Students solve right and non-right triangles and connect the calculation to a physical or spatial context.
230 min estimated
Topic 3.2 progress0%
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Learning objectives
Use sine, cosine and tangent in right triangles.
Apply the sine rule and cosine rule.
Use the included-angle area formula.
Recognize when a triangle problem may have more than one valid configuration.
Give angles in the requested unit.
Key formulas
Interior angle sum
$A+B+C=180^\circ$
Standard opposite-side convention
$a\leftrightarrow A,\quad b\leftrightarrow B,\quad c\leftrightarrow C$
Sine ratio
$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
Cosine ratio
$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$
Tangent ratio
$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$
Pythagorean theorem
$a^2+b^2=c^2$
Missing shorter side
$a=\sqrt{c^2-b^2}$
Complementary acute angles
$\alpha+\beta=90^\circ$
Sine rule
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Reciprocal sine-rule form
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$
Supplementary sine solution
$B_2=180^\circ-B_1$
SSA comparison height
$h=b\sin A$
Cosine rule for a side
$c^2=a^2+b^2-2ab\cos C$
Cosine rule for an angle
$\cos C=\frac{a^2+b^2-c^2}{2ab}$
Triangle area from two sides and included angle
$K=\frac{1}{2}ab\sin C$
Equivalent area forms
$K=\frac{1}{2}bc\sin A=\frac{1}{2}ca\sin B$
Method-selection guide
$\begin{aligned}\text{right triangle}&\Rightarrow\text{SOH-CAH-TOA or Pythagoras}\\\text{known opposite pair}&\Rightarrow\text{sine rule}\\\text{SAS or SSS}&\Rightarrow\text{cosine rule}\end{aligned}$
Triangle inequality
$a+b>c,\quad b+c>a,\quad c+a>b$
Right-triangle ratios
$\sin\theta=\frac{O}{H},\quad\cos\theta=\frac{A}{H},\quad\tan\theta=\frac{O}{A}$
Cosine rule
$c^2=a^2+b^2-2ab\cos C$
Triangle area
$K=\frac12 ab\sin C$
Topic 3.3Not started
Applications of trigonometry
The applied skill is translating a verbal spatial situation into a labelled mathematical diagram before calculating.
230 min estimated
Topic 3.3 progress0%
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Learning objectives
Use Pythagoras in 2D and 3D contexts.
Solve elevation and depression problems.
Construct a useful labelled diagram from written information.
Combine several triangles when one calculation feeds another.
Interpret final lengths and angles in context.
Key formulas
Triangle angle sum
$A+B+C=180^\circ$
Height from elevation angle
$h=d\tan\theta$
Horizontal distance from depression angle
$d=\frac{h}{\tan\theta}$
Two-position height relation
$x\tan\beta=(x+s)\tan\alpha$
Reverse bearing
$\theta_{\text{reverse}}=(\theta+180^\circ)\bmod 360^\circ$
Bearing from east and north components
$\theta=\tan^{-1}\left(\frac{E}{N}\right)$
Cosine rule for a route displacement
$d^2=p^2+q^2-2pq\cos C$
Sine rule for triangulation
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Sine rule
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Second possible sine-rule angle
$B_2=180^\circ-B_1$
Cosine rule for a side
$c^2=a^2+b^2-2ab\cos C$
Cosine rule for an angle
$C=\cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right)$
Triangle area
$K=\frac{1}{2}ab\sin C$
Equivalent area forms
$K=\frac12 bc\sin A=\frac12 ca\sin B$
Cuboid space diagonal
$d=\sqrt{l^2+w^2+h^2}$
Line-plane angle
$\tan\theta=\frac{\text{perpendicular height}}{\text{projection length}}$
Three-dimensional distance
$d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}$
Route displacement bounds
$|p-q|\le d\le p+q$
Right-triangle ratios
$\sin\theta=\frac{O}{H},\quad\cos\theta=\frac{A}{H},\quad\tan\theta=\frac{O}{A}$
Pythagorean theorem
$a^2+b^2=c^2$
Cosine rule
$c^2=a^2+b^2-2ab\cos C$
Triangle area
$K=\frac12 ab\sin C$
Reverse bearing
$\theta_{\text{reverse}}=(\theta+180^\circ)\bmod360^\circ$
Topic 3.4Not started
Circular measure
Arc length and sector area connect angle, radius and geometry. Radian measure is the natural unit for these formulas and later calculus/trigonometry.
225 min estimated
Topic 3.4 progress0%
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Learning objectives
Convert between degrees and radians.
Find arc length.
Find sector area.
Use radians consistently in circular-measure calculations.
Solve reverse problems for angle or radius.
Key formulas
Radian definition
$\theta=\frac{s}{r}$
Full turn
$360^\circ=2\pi\text{ rad}$
Half turn
$180^\circ=\pi\text{ rad}$
Coterminal angles
$\theta_{\text{coterminal}}=\theta+2k\pi,\quad k\in\mathbb{Z}$
Degrees to radians
$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180}$
Radians to degrees
$\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$
One-radian approximation
$1\text{ rad}=\frac{180^\circ}{\pi}\approx57.3^\circ$
Standard exact angles
$30^\circ=\frac{\pi}{6},\quad45^\circ=\frac{\pi}{4},\quad60^\circ=\frac{\pi}{3},\quad90^\circ=\frac{\pi}{2}$
Arc length in radians
$s=r\theta$
Arc length in degrees
$s=\frac{\theta^\circ}{360^\circ}(2\pi r)$
Major central angle
$\theta_{\text{major}}=2\pi-\theta_{\text{minor}}$
Sector area in radians
$K=\frac12r^2\theta$
Sector area in degrees
$K=\frac{\theta^\circ}{360^\circ}\pi r^2$
Major sector area
$K_{\text{major}}=\pi r^2-K_{\text{minor}}$
Radius from an arc
$r=\frac{s}{\theta}$
Angle from an arc
$\theta=\frac{s}{r}$
Radius from sector area
$r=\sqrt{\frac{2K}{\theta}}$
Angle from sector area
$\theta=\frac{2K}{r^2}$
Sector area from radius and arc
$K=\frac12rs$
Sector perimeter
$P=2r+r\theta$
Chord length
$c=2r\sin\left(\frac{\theta}{2}\right)$
Central triangle area
$K_{\triangle}=\frac12r^2\sin\theta$
Minor segment area
$K_{\text{segment}}=\frac12r^2(\theta-\sin\theta)$
Annular sector area
$K_{\text{annular}}=\frac12(R^2-r^2)\theta$
Annular-sector perimeter
$P=(R+r)\theta+2(R-r)$
Angular displacement after turns
$\theta=2\pi n$
Rolling distance
$s=r\theta=2\pi rn$
Angle conversions
$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180},\quad\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$
Arc length
$s=r\theta$
Sector area
$K=\frac12r^2\theta=\frac12rs$
Topic 3.5Not started
Perpendicular bisectors
Perpendicular bisectors divide the plane into nearest-location regions and are the geometric foundation of Voronoi diagrams.
235 min estimated
Topic 3.5 progress0%
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Learning objectives
Find the midpoint of a segment.
Calculate the segment gradient and perpendicular gradient.
Write the equation of the perpendicular bisector.
Handle vertical/horizontal special cases.
Use a bisector to identify points equidistant from two sites.
Key formulas
Equidistance property
$PA=PB$
Midpoint
$M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
Gradient
$m_{AB}=\frac{y_2-y_1}{x_2-x_1}$
Distance
$AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Perpendicular gradients
$m_1m_2=-1$
Negative reciprocal
$m_\perp=-\frac1m$
Point-gradient form
$y-y_0=m(x-x_0)$
Equal squared distances
$(x-x_1)^2+(y-y_1)^2=(x-x_2)^2+(y-y_2)^2$
General bisector equation
$2(x_2-x_1)x+2(y_2-y_1)y=x_2^2+y_2^2-x_1^2-y_1^2$
Region nearer A
$(x-x_A)^2+(y-y_A)^2<(x-x_B)^2+(y-y_B)^2$
Region no farther from A
$d_A^2\le d_B^2$
Three-site verification
$PA=PB=PC$
Scale conversion
$d_{\text{real}}=d_{\text{map}}\times\text{scale factor}$
Reasonableness check
$m_{AB}m_\perp=-1$
Segment gradient
$m=\frac{y_2-y_1}{x_2-x_1}$
Perpendicular gradient
$m_\perp=-\frac1m$
Point-gradient line
$y-y_M=m_\perp(x-x_M)$
Equal squared distances
$(x-x_A)^2+(y-y_A)^2=(x-x_B)^2+(y-y_B)^2$
Nearest-site comparison
$d_A^2<d_B^2\Longrightarrow\text{nearer to }A$
Topic 3.6Not started
Voronoi diagrams
A Voronoi diagram partitions a region according to the nearest site, making it useful for service areas, facilities and location decisions.
240 min estimated
Topic 3.6 progress0%
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Learning objectives
Identify sites, cells, edges and vertices.
Explain why edges lie on perpendicular bisectors.
Add a new site to an existing diagram.
Use nearest-neighbour interpolation.
Analyze facility-location problems such as maximizing distance from existing sites or choosing a service region.
Key formulas
Cell of site i
$V_i=\{P:d(P,S_i)\le d(P,S_j)\text{ for every site }S_j\}$
Edge condition for A and B
$PA=PB\le PC\text{ for every other site }C$
Typical vertex condition
$PA=PB=PC$
Equal-distance boundary
$(x-x_A)^2+(y-y_A)^2=(x-x_B)^2+(y-y_B)^2$
Midpoint
$M\left(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2}\right)$
Perpendicular gradient
$m_\perp=-\frac1{m_{AB}}$
Distance check
$d(P,S)=\sqrt{(x-x_S)^2+(y-y_S)^2}$
Nearest-site rule
$d(P,S_{\text{nearest}})^2=\min_i d(P,S_i)^2$
Empty-circle radius
$r_V=d(V,A)=d(V,B)=d(V,C)$
New-site cell
$V_D=\{P:d(P,D)\le d(P,S_i)\text{ for every old site }S_i\}$
Nearest-neighbour estimate
$\widehat f(P)=f(S_{\text{nearest}})$
Maximin objective
$\text{maximize }\min_i d(P,S_i)\text{ over feasible }P$
Squared distance to site S
$D_S(x,y)=(x-x_S)^2+(y-y_S)^2$
Map-scale conversion
$d_{\text{real}}=d_{\text{coordinate}}\times\text{scale factor}$
Distance
$d(P,S)=\sqrt{(x-x_S)^2+(y-y_S)^2}$
Pairwise boundary
$d(P,A)=d(P,B)$
Cell membership
$P\in V_A\Longleftrightarrow d(P,A)\le d(P,S_i)\text{ for every site }S_i$
Typical vertex
$d(V,A)=d(V,B)=d(V,C)$
Nearest-neighbour interpolation
$\widehat f(P)=f(S_{\text{nearest}})$
Maximin location
$\text{maximize }\min_i d(P,S_i)$
Topic 3.7Not started
Radian measure at Higher Level
Use radians fluently in advanced circle geometry, trigonometric equations and angular-motion models, with exact reasoning and reliable GDC methods.
220 min estimated
Topic 3.7 progress0%
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Learning objectives
Explain a radian as the ratio of arc length to radius.
Convert fluently between degrees and radians, including exact multiples of pi.
Identify exact and coterminal angles on the unit circle.
Use radian formulas for arc length, sector area, chords, segments and annular sectors.
Solve multi-stage circle problems for unknown radius, angle, length or area.
Model angular displacement, angular speed, period and frequency.
Connect angular speed to tangential speed and coupled rotational systems.
Use radians with trigonometric functions, graphs and equations over restricted intervals.
Apply small-angle approximations appropriately and state their limitations.
Use GDC radian mode, exact pi input and numerical solvers accurately.
Interpret units, direction and physical constraints in applied problems.
Communicate complete IB AI HL solutions with diagrams, exact values, working and justified rounding.
Key formulas
Radian definition
$\theta=\frac sr$
Arc relation
$s=r\theta$
Full turn
$2\pi\text{ rad}=360^\circ$
Degrees to radians
$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180}$
Radians to degrees
$\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$
Half turn
$\pi\text{ rad}=180^\circ$
Unit-circle point
$P=(\cos\theta,\sin\theta)$
Tangent ratio
$\tan\theta=\frac{\sin\theta}{\cos\theta}$
Pythagorean identity
$\sin^2\theta+\cos^2\theta=1$
Coterminal family
$\theta=\theta_0+2\pi k,\qquad k\in\mathbb Z$
Common principal interval
$0\le\theta\lt2\pi$
Negative-direction convention
$\theta\lt0\Rightarrow\text{clockwise rotation}$
Arc length
$s=r\theta$
Angle from arc
$\theta=\frac sr$
Major angle
$\theta_{\mathrm{major}}=2\pi-\theta_{\mathrm{minor}}$
Sector area
$A=\frac12r^2\theta$
Fraction of circle
$\frac{A}{\pi r^2}=\frac{\theta}{2\pi}$
Major-sector area
$A_{\mathrm{major}}=\pi r^2-\frac12r^2\theta_{\mathrm{minor}}$
Chord length
$c=2r\sin\left(\frac\theta2\right)$
Angle from chord
$\theta=2\sin^{-1}\left(\frac{c}{2r}\right)$
Arc-chord comparison
$2r\sin(\frac{\theta}{2})\lt r\theta\quad\text{for }0\lt\theta\lt2\pi$
Central triangle area
$A_\triangle=\frac12r^2\sin\theta$
Minor-segment area
$A_{\mathrm{seg}}=\frac12r^2(\theta-\sin\theta)$
Major-segment area
$A_{\mathrm{major}}=\pi r^2-A_{\mathrm{minor}}$
Annular-sector area
$A=\frac12(R^2-r^2)\theta$
Total curved length
$s_{\mathrm{curved}}=(R+r)\theta$
Annular-sector perimeter
$P=(R+r)\theta+2(R-r)$
Sector perimeter
$P=r\theta+2r$
Angle from perimeter
$\theta=\frac{P-2r}{r}$
Exposed-boundary principle
$P=\sum\limits_{i=1}^{n}\ell_i$
Area using arc
$A=\frac12rs$
Radius from area and arc
$r=\frac{2A}{s}$
Angle from area and radius
$\theta=\frac{2A}{r^2}$
Angles around a point
$\sum\limits_{i=1}^{n}\theta_i=2\pi$
Composite area
$A_{\mathrm{total}}=\sum\limits_{i=1}^{n}A_i-\sum\limits_{j=1}^{m}A_j^{\mathrm{removed}}$
Mixed boundary
$P=\sum\limits_{i=1}^{n}s_i+\sum\limits_{j=1}^{m}\ell_j$
Revolutions to radians
$\theta=2\pi N$
Revolutions from angle
$N=\frac{\theta}{2\pi}$
Linear displacement
$s=r\theta$
Average angular velocity
$\omega_{\mathrm{avg}}=\frac{\Delta\theta}{\Delta t}$
Frequency and period
$f=\frac1T$
Angular speed relation
$\omega=2\pi f=\frac{2\pi}{T}$
Tangential speed
$v=r\omega$
Circular distance
$s=r\theta$
Distance after N turns
$s=2\pi rN$
rpm to radians per second
$\omega=\frac{2\pi n}{60}$
Radians per second to rpm
$n=\frac{60\omega}{2\pi}$
Rim speed from rpm
$v=\frac{2\pi rn}{60}$
No-slip speed
$r_1|\omega_1|=r_2|\omega_2|$
Gear-tooth relation
$N_1|\omega_1|=N_2|\omega_2|$
Shared belt displacement
$r_1|\theta_1|=r_2|\theta_2|$
Angular acceleration
$\alpha_{\mathrm{avg}}=\frac{\Delta\omega}{\Delta t}$
Tangential acceleration
$a_t=r\alpha$
Centripetal acceleration
$a_c=r\omega^2=\frac{v^2}{r}$
Sine and cosine period
$T=2\pi$
Tangent period
$T=\pi$
Transformed sine period
$T=\frac{2\pi}{|b|}$
Sine general solutions
$\theta=\alpha+2\pi k\quad\text{or}\quad\theta=\pi-\alpha+2\pi k$
Cosine general solutions
$\theta=\pm\alpha+2\pi k$
Tangent general solution
$\theta=\alpha+\pi k$
Small-angle sine
$\sin\theta\approx\theta$
Small-angle tangent
$\tan\theta\approx\theta$
Small-angle cosine
$\cos\theta\approx1-\frac{\theta^2}{2}$
Radian-mode check
$\sin\left(\frac\pi6\right)=\frac12$
Root condition
$f(x)=g(x)\iff f(x)-g(x)=0$
Verification
$|f(x)-g(x)|\approx0$
Constant-speed travel
$s=vt=r\omega t$
Angle over time
$\theta=\omega t$
Wheel distance
$s=2\pi rN$
Core circle formulas
$s=r\theta,\qquad A=\frac12r^2\theta$
Core motion formulas
$\omega=\frac{\Delta\theta}{\Delta t},\qquad v=r\omega$
Exam workflow
$\text{sketch}\to\text{convert}\to\text{model}\to\text{solve}\to\text{check}$
Practice circle check
$s=r\theta,\quad A=\frac12r^2\theta,\quad c=2r\sin(\frac{\theta}{2})$
Practice motion check
$\theta=2\pi N,\quad\omega=2\pi f,\quad v=r\omega$
Practice mode check
$\sin(\frac{\pi}{6})=\frac{1}{2}$
Circle core
$\theta=\frac sr,\quad s=r\theta,\quad A=\frac12r^2\theta$
Motion core
$\omega=\frac{\Delta\theta}{\Delta t},\quad\omega=2\pi f,\quad v=r\omega$
Conversion core
$180^\circ=\pi\text{ rad}$
Topic 3.8Not started
Unit circle, identities and trigonometric equations
Use the unit circle to evaluate exact trigonometric values, apply identities and solve periodic equations completely in radians or degrees.
175 min estimated
Topic 3.8 progress0%
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Learning objectives
Define sine, cosine and tangent from coordinates on the unit circle.
Convert accurately between degree and radian measure.
Recall and derive exact values for standard angles.
Use quadrants, reference angles and coterminal angles to evaluate trigonometric ratios.
Apply periodicity, odd/even symmetry and related-angle relationships.
Use reciprocal, quotient and Pythagorean identities with appropriate domain restrictions.
Simplify trigonometric expressions and prove elementary identities.
Solve sine, cosine and tangent equations on finite intervals and for all real values.
Solve quadratic, factorized and multiple-angle trigonometric equations without losing roots.
Use a GDC to find and verify graphical or numerical solutions.
Interpret trigonometric solutions and periods in applied models.
Communicate exact and approximate answers in an IB AI HL exam-ready form.
Key formulas
Unit-circle equation
$x^2+y^2=1$
Terminal point
$P(\cos\theta,\sin\theta)$
Tangent as slope
$\tan\theta=\frac{\sin\theta}{\cos\theta}$
Degree-radian equivalence
$180^\circ=\pi\text{ rad}$
Degrees to radians
$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180}$
Radians to degrees
$\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$
Special-angle sine pattern
$\sin\theta=\frac{\sqrt0,\sqrt1,\sqrt2,\sqrt3,\sqrt4}{2}$
Special-angle cosine pattern
$\cos\theta=\frac{\sqrt4,\sqrt3,\sqrt2,\sqrt1,\sqrt0}{2}$
Tangent from sine and cosine
$\tan\theta=\frac{\sin\theta}{\cos\theta}$
Quadrant I
$\sin\theta\gt0,\quad\cos\theta\gt0,\quad\tan\theta\gt0$
Quadrants II and III
$\mathrm{QII}:\sin\theta\gt0,\qquad\mathrm{QIII}:\tan\theta\gt0$
Quadrant IV
$\mathrm{QIV}:\cos\theta\gt0$
Quadrant II reference angle
$\alpha=\pi-\theta$
Quadrant III reference angle
$\alpha=\theta-\pi$
Quadrant IV reference angle
$\alpha=2\pi-\theta$
Radian coterminal family
$\theta+2\pi k,\quad k\in\mathbb Z$
Degree coterminal family
$\theta+360^\circ k,\quad k\in\mathbb Z$
Principal reduction
$\theta\equiv\theta+2\pi k\pmod{2\pi}$
Sine periodicity
$\sin(\theta+2\pi k)=\sin\theta$
Cosine periodicity
$\cos(\theta+2\pi k)=\cos\theta$
Tangent periodicity
$\tan(\theta+\pi k)=\tan\theta$
Sine odd symmetry
$\sin(-\theta)=-\sin\theta$
Cosine even symmetry
$\cos(-\theta)=\cos\theta$
Tangent odd symmetry
$\tan(-\theta)=-\tan\theta$
Quadrant-II sine
$\sin(\pi-\theta)=\sin\theta$
Quadrant-II cosine
$\cos(\pi-\theta)=-\cos\theta$
Quadrant-IV relations
$\sin(2\pi-\theta)=-\sin\theta,\quad\cos(2\pi-\theta)=\cos\theta$
Reciprocal identities
$\sec\theta=\frac1{\cos\theta},\quad\csc\theta=\frac1{\sin\theta},\quad\cot\theta=\frac1{\tan\theta}$
Quotient identities
$\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cot\theta=\frac{\cos\theta}{\sin\theta}$
Reciprocal products
$\sin\theta\csc\theta=1,\quad\cos\theta\sec\theta=1$
Fundamental Pythagorean identity
$\sin^2\theta+\cos^2\theta=1$
Tangent-sec identity
$1+\tan^2\theta=\sec^2\theta$
Cotangent-cosecant identity
$1+\cot^2\theta=\csc^2\theta$
Difference of squares
$1-\cos^2x=\sin^2x$
Secant rearrangement
$\sec^2x-1=\tan^2x$
Cosecant rearrangement
$\csc^2x-1=\cot^2x$
Conjugate product
$(1-\sin x)(1+\sin x)=1-\sin^2x$
Pythagorean substitution
$1-\sin^2x=\cos^2x$
Secant square
$\frac1{\cos^2x}=\sec^2x$
Sine range
$-1\le\sin x\le1$
Sine principal interval
$-\frac\pi2\le\sin^{-1}(a)\le\frac\pi2$
Sine solution pair
$x=\alpha+2\pi k\quad\text{or}\quad x=\pi-\alpha+2\pi k$
Cosine range
$-1\le\cos x\le1$
Cosine principal interval
$0\le\cos^{-1}(a)\le\pi$
Cosine general solution
$x=\pm\arccos(a)+2\pi k$
Tangent range
$\tan x\in\mathbb R$
Tangent principal interval
$-\frac\pi2\lt\tan^{-1}(a)\lt\frac\pi2$
Tangent general solution
$x=\arctan(a)+k\pi$
Sine families
$x=\alpha+2\pi k\quad\text{or}\quad x=\pi-\alpha+2\pi k$
Cosine families
$x=\pm\alpha+2\pi k$
Tangent family
$x=\alpha+k\pi$
Quadratic substitution
$au^2+bu+c=0,\qquad u=\sin x,\cos x\text{ or }\tan x$
Sine-cosine range test
$-1\le u\le1$
Difference of squares
$u^2-1=(u-1)(u+1)$
Zero-product form
$A(x)B(x)=0\Longrightarrow A(x)=0\text{ or }B(x)=0$
Sine-square replacement
$\sin^2x=1-\cos^2x$
Cosine-square replacement
$\cos^2x=1-\sin^2x$
Input substitution
$u=nx+c$
Transformed interval
$a\le x\lt b\Longrightarrow na+c\le u\lt nb+c\quad(n\gt0)$
Back-substitution
$x=\frac{u-c}{n}$
Interval filtering
$x\in I$
Exact-answer principle
$\sin x=\frac12\Longrightarrow x\text{ written as exact multiples of }\pi$
Verification test
$L(x)=R(x)$
Equation as a zero
$f(x)=g(x)\Longleftrightarrow f(x)-g(x)=0$
Sinusoidal period
$T=\frac{2\pi}{|B|}$
Sinusoidal range
$D-|A|\le y\le D+|A|$
Principal sine input
$\alpha=\sin^{-1}(a)$
Principal cosine input
$\alpha=\cos^{-1}(a)$
Residual check
$\text{residual}=L(x)-R(x)$
Final checklist
$\text{mode}\to\text{reference angle}\to\text{quadrants}\to\text{interval}\to\text{check}$
Sine-cosine feasibility
$|a|\le1$
Integer parameter
$k\in\mathbb Z$
Practice equation workflow
$\text{isolate}\to\text{solve ratio}\to\text{generate angles}\to\text{filter}\to\text{verify}$
Model period
$T=\frac{2\pi}{|B|}$
Graphical root condition
$f(x)-g(x)=0$
Unit-circle point
$(x,y)=(\cos\theta,\sin\theta)$
Three Pythagorean identities
$\sin^2\theta+\cos^2\theta=1,\quad1+\tan^2\theta=\sec^2\theta,\quad1+\cot^2\theta=\csc^2\theta$
Periods
$T_{\sin}=T_{\cos}=2\pi,\qquad T_{\tan}=\pi$
Topic 3.9Not started
Geometric transformations using matrices
Represent reflections, rotations, scalings, stretches, shears and affine motions with matrices; compose them in the correct order and interpret their geometry.
180 min estimated
Topic 3.9 progress0%
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Learning objectives
Represent planar coordinates as column vectors and apply two-by-two transformation matrices.
Interpret matrix columns as images of the standard basis vectors.
Use and derive matrices for reflections in axes and diagonal lines.
Use standard and general matrices for clockwise and anticlockwise rotations.
Model uniform enlargements, non-uniform scalings, stretches and shears.
Transform individual points and complete shapes using vertex matrices.
Describe preserved features, fixed points and geometric effects of transformations.
Use determinants to determine area scale, orientation and invertibility.
Use inverse matrices to recover object points from images.
Compose transformations using matrix multiplication in the correct order.
Find an unknown transformation matrix from basis images or independent vector correspondences.
Use affine vector terms for translations and transformations about non-origin centres.
Use a GDC to calculate matrix products, determinants and inverses and interpret the output.
Solve and communicate IB AI HL transformation problems in geometric and applied contexts.
Key formulas
Point vector
$P(x,y)\longleftrightarrow\mathbf p=\begin{pmatrix}x\\ y\end{pmatrix}$
Linear image
$\mathbf p'=A\mathbf p$
Vector between points
$\overrightarrow{AB}=\mathbf b-\mathbf a$
General transformation
$\begin{pmatrix}x'\\ y'\end{pmatrix}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}$
Coordinate rule
$x'=ax+by,\qquad y'=cx+dy$
Dimension rule
$(2\times2)(2\times1)=(2\times1)$
Standard basis
$\mathbf e_1=\begin{pmatrix}1\\0\end{pmatrix},\qquad\mathbf e_2=\begin{pmatrix}0\\1\end{pmatrix}$
Matrix from columns
$A=\begin{pmatrix}|&|\\ A\mathbf e_1&A\mathbf e_2\\|&|\end{pmatrix}$
Linearity from basis
$A(x\mathbf e_1+y\mathbf e_2)=xA\mathbf e_1+yA\mathbf e_2$
Reflection in x-axis
$M_x=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$
Reflection in y-axis
$M_y=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$
Reflection in origin
$M_O=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}=-I$
Reflection in y=x
$M_{y=x}=\begin{pmatrix}0&1\\1&0\end{pmatrix}$
Reflection in y=-x
$M_{y=-x}=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$
Reflection in a line at angle phi
$M_\phi=\begin{pmatrix}\cos2\phi&\sin2\phi\\\ sin2\phi&-\cos2\phi\end{pmatrix}$
90-degree anticlockwise rotation
$R_{\pi/2}=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$
90-degree clockwise rotation
$R_{-\pi/2}=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$
180-degree rotation
$R_\pi=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$
General anticlockwise rotation
$R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\ sin\theta&\cos\theta\end{pmatrix}$
Clockwise rotation
$R_{-\theta}=\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}$
Rotation inverse
$R_\theta^{-1}=R_{-\theta}$
Uniform scale matrix
$S_k=\begin{pmatrix}k&0\\0&k\end{pmatrix}=kI$
Length scale
$L'=|k|L$
Area scale
$A'=k^2A$
Axis scaling
$S_{a,b}=\begin{pmatrix}a&0\\0&b\end{pmatrix}$
Coordinate rule
$(x,y)\mapsto(ax,by)$
Area scale factor
$|\det S_{a,b}|=|ab|$
x-shear
$H_x(k)=\begin{pmatrix}1&k\\0&1\end{pmatrix}$
y-shear
$H_y(k)=\begin{pmatrix}1&0\\ k&1\end{pmatrix}$
Shear determinant
$\det H_x(k)=\det H_y(k)=1$
Point image
$\mathbf p'=A\mathbf p$
Rotation coordinates
$x'=x\cos\theta-y\sin\theta,\qquad y'=x\sin\theta+y\cos\theta$
Diagonal scaling coordinates
$x'=ax,\qquad y'=by$
Vertex transformation
$X'=AX$
Vertex storage
$X=\begin{pmatrix}x_1&x_2&\cdots&x_n\\ y_1&y_2&\cdots&y_n\end{pmatrix}$
Linearity of an edge
$A((1-t)\mathbf p+t\mathbf q)=(1-t)A\mathbf p+tA\mathbf q$
Fixed-vector condition
$A\mathbf v=\mathbf v$
Origin fixed
$A\mathbf0=\mathbf0$
Length preservation test
$\|A\mathbf v\|=\|\mathbf v\|$
Two-by-two determinant
$\det A=ad-bc$
Area scaling
$\operatorname{Area}(\text{image})=|\det A|\operatorname{Area}(\text{object})$
Determinant of a product
$\det(BA)=\det B\det A$
Inverse matrix
$A^{-1}=\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$
Undo property
$A^{-1}A=AA^{-1}=I$
Invertibility condition
$A^{-1}\text{ exists}\Longleftrightarrow\det A\ne0$
Two-step composition
$\mathbf p''=B(A\mathbf p)=(BA)\mathbf p$
Three-step composition
$\mathbf p'''=CBA\mathbf p$
Associativity
$C(BA)=(CB)A$
Non-commutativity
$BA\ne AB\text{ in general}$
Rightmost first
$BA\mathbf p=B(A\mathbf p)$
Uniform scale commutes
$(kI)A=A(kI)=kA$
Affine transformation
$\mathbf p'=A\mathbf p+\mathbf t$
Transformation about c
$\mathbf p'=A(\mathbf p-\mathbf c)+\mathbf c$
Equivalent affine form
$\mathbf p'=A\mathbf p+(I-A)\mathbf c$
Basis-column construction
$A=\begin{pmatrix}|&|\\ T(\mathbf e_1)&T(\mathbf e_2)\\|&|\end{pmatrix}$
First column
$A\mathbf e_1=\operatorname{col}_1(A)$
Second column
$A\mathbf e_2=\operatorname{col}_2(A)$
Correspondence equation
$AV=W$
Matrix recovery
$A=WV^{-1}$
Independence condition
$\det V\ne0$
Column orthogonality
$\mathbf c_1\cdot\mathbf c_2=0$
Unit-column test
$\|\mathbf c_1\|=\|\mathbf c_2\|=1$
Orientation test
$\det A\gt0\text{ preserves orientation},\qquad\det A\lt0\text{ reverses it}$
GDC vertex calculation
$X'=AX$
GDC inverse check
$AA^{-1}\approx I$
GDC composition
$\text{second matrix}\times\text{first matrix}$
Graphics pipeline
$X'=AX+\mathbf t\mathbf1^{\mathsf T}$
Repeated transformation
$\mathbf p_n=A^n\mathbf p_0$
Area-density relation
$\text{image area}=|\det A|\times\text{object area}$
Final check
$\text{dimensions}\to\text{order}\to\text{basis images}\to\det\to\text{geometry}$
Composition wording
$A\text{ then }B\Longrightarrow BA$
Translation test
$T(\mathbf0)\ne\mathbf0\Longrightarrow T\text{ is not represented by a }2\times2\text{ matrix alone}$
Practice workflow
$\text{identify}\to\text{write matrix}\to\text{order product}\to\text{multiply}\to\text{interpret}$
Area check
$A'=|\det A|A$
Non-origin centre
$\mathbf p'=A(\mathbf p-\mathbf c)+\mathbf c$
Core linear rule
$\mathbf p'=A\mathbf p$
Core rotation
$R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\ sin\theta&\cos\theta\end{pmatrix}$
Core affine rule
$\mathbf p'=A\mathbf p+\mathbf t$
Topic 3.10Not started
Vector concepts and components
Use 2D and 3D components to represent, combine and interpret displacement, velocity, force and geometric relationships with vectors.
180 min estimated
Topic 3.10 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Distinguish scalar quantities from vector quantities in mathematical and applied contexts.
Use bold, arrow, component, column and i-j-k vector notation consistently.
Interpret and construct vector components in two and three dimensions.
Use position vectors to represent points relative to an origin.
Calculate vector magnitudes and distances in 2D and 3D.
Normalize nonzero vectors and construct vectors with specified magnitudes.
Find direction angles and direction cosines from components.
Add, subtract and multiply vectors by scalars algebraically and geometrically.
Find vectors between points, midpoints and points dividing a segment.
Test whether vectors are parallel and whether points are collinear.
Express resultants and other vectors as linear combinations.
Use vectors in navigation, relative velocity and force-equilibrium models.
Use a GDC appropriately for magnitude, normalization and direction calculations.
Communicate complete IB AI HL vector solutions with axes, units and interpretation.
Key formulas
Scalar description
$q=\text{magnitude with unit}$
Vector description
$\mathbf v=\text{magnitude and direction}$
Zero displacement
$\mathbf r_{\mathrm{final}}-\mathbf r_{\mathrm{initial}}=\mathbf0$
2D component notation
$\mathbf v=\langle a,b\rangle=\begin{pmatrix}a\\ b\end{pmatrix}=a\mathbf i+b\mathbf j$
3D component notation
$\mathbf v=\langle a,b,c\rangle=a\mathbf i+b\mathbf j+c\mathbf k$
Directed segment
$\overrightarrow{AB}=\mathbf b-\mathbf a$
Vector equality
$\langle a,b,c\rangle=\langle p,q,r\rangle\Longleftrightarrow a=p,\ b=q,\ c=r$
Additive inverse
$\mathbf v+(-\mathbf v)=\mathbf0$
Zero vector
$\mathbf0=\langle0,0,0\rangle,\qquad|\mathbf0|=0$
2D component vector
$\mathbf v=\langle v_x,v_y\rangle$
Components from magnitude and direction
$v_x=|\mathbf v|\cos\theta,\qquad v_y=|\mathbf v|\sin\theta$
Reconstruction
$\mathbf v=v_x\mathbf i+v_y\mathbf j$
3D vector
$\mathbf v=\langle a,b,c\rangle$
Unit-vector form
$\mathbf v=a\mathbf i+b\mathbf j+c\mathbf k$
Component equality
$\langle a,b,c\rangle=\langle p,q,r\rangle\Longleftrightarrow a=p,\ b=q,\ c=r$
Position vector
$\mathbf p=\overrightarrow{OP}$
2D position
$P(x,y)\Longrightarrow\mathbf p=\langle x,y\rangle$
3D position
$P(x,y,z)\Longrightarrow\mathbf p=\langle x,y,z\rangle$
2D magnitude
$|\mathbf v|=\sqrt{a^2+b^2}$
3D magnitude
$|\mathbf v|=\sqrt{a^2+b^2+c^2}$
Magnitude under negation
$|-\mathbf v|=|\mathbf v|$
Normalization
$\widehat{\mathbf v}=\frac{\mathbf v}{|\mathbf v|},\qquad\mathbf v\ne\mathbf0$
Unit magnitude
$|\widehat{\mathbf v}|=1$
Specified magnitude
$\mathbf u=m\frac{\mathbf v}{|\mathbf v|}$
Direction ratio
$\tan\theta=\frac{v_y}{v_x}$
Quadrant-aware direction
$\theta=\operatorname{atan2}(v_y,v_x)$
Polar reconstruction
$\mathbf v=|\mathbf v|\langle\cos\theta,\sin\theta\rangle$
Direction cosines
$\cos\alpha=\frac a{|\mathbf v|},\quad\cos\beta=\frac b{|\mathbf v|},\quad\cos\gamma=\frac c{|\mathbf v|}$
Direction-cosine identity
$\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$
Unit direction
$\widehat{\mathbf v}=\langle\cos\alpha,\cos\beta,\cos\gamma\rangle$
Componentwise addition
$\langle a,b,c\rangle+\langle p,q,r\rangle=\langle a+p,b+q,c+r\rangle$
Commutativity
$\mathbf u+\mathbf v=\mathbf v+\mathbf u$
Associativity
$(\mathbf u+\mathbf v)+\mathbf w=\mathbf u+(\mathbf v+\mathbf w)$
Componentwise subtraction
$\langle a,b,c\rangle-\langle p,q,r\rangle=\langle a-p,b-q,c-r\rangle$
Subtract as add negative
$\mathbf u-\mathbf v=\mathbf u+(-\mathbf v)$
Change
$\Delta\mathbf r=\mathbf r_{\mathrm{final}}-\mathbf r_{\mathrm{initial}}$
Scalar multiplication
$k\langle a,b,c\rangle=\langle ka,kb,kc\rangle$
Magnitude scaling
$|k\mathbf v|=|k|\,|\mathbf v|$
Distributivity
$k(\mathbf u+\mathbf v)=k\mathbf u+k\mathbf v$
Point-to-point vector
$\overrightarrow{AB}=\mathbf b-\mathbf a$
Reverse direction
$\overrightarrow{BA}=-\overrightarrow{AB}$
Distance
$AB=|\overrightarrow{AB}|$
Midpoint vector
$\mathbf m=\frac{\mathbf a+\mathbf b}{2}$
Internal division
$AP:PB=m:n\Longrightarrow\mathbf p=\frac{n\mathbf a+m\mathbf b}{m+n}$
Segment parameter
$\mathbf r=(1-t)\mathbf a+t\mathbf b,\quad0\le t\le1$
Parallel test
$\mathbf v\parallel\mathbf u\Longleftrightarrow\mathbf v=k\mathbf u\text{ for some }k\ne0$
Same direction
$k\gt0$
Opposite direction
$k\lt0$
Collinearity test
$\overrightarrow{AB}=k\overrightarrow{AC}$
Parametric line point
$\mathbf r=\mathbf a+t(\mathbf c-\mathbf a)$
Between condition
$0\le t\le1$
Linear combination
$\mathbf r=a\mathbf u+b\mathbf v$
Component equations
$a\mathbf u+b\mathbf v=\mathbf r\Longrightarrow\text{equate corresponding components}$
2D spanning condition
$\mathbf u\ne k\mathbf v$
Parallelogram diagonal
$\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{AD}$
Midpoint position
$\mathbf m=\frac{\mathbf a+\mathbf b}{2}$
Triangle centroid
$\mathbf g=\frac{\mathbf a+\mathbf b+\mathbf c}{3}$
Bearing components
$\mathbf d=\langle d\sin\beta,d\cos\beta\rangle$
Resultant journey
$\mathbf R=\mathbf d_1+\mathbf d_2+\cdots+\mathbf d_n$
Bearing from components
$\tan\beta=\frac{\text{east component}}{\text{north component}}$
Average velocity
$\mathbf v_{\mathrm{avg}}=\frac{\Delta\mathbf r}{\Delta t}$
Ground velocity
$\mathbf v_{\mathrm{ground}}=\mathbf v_{\mathrm{relative}}+\mathbf v_{\mathrm{medium}}$
Relative velocity
$\mathbf v_{\frac{A}{B}}=\mathbf v_A-\mathbf v_B$
Resultant force
$\mathbf R=\sum\limits_{i=1}^{n}\mathbf F_i$
Equilibrium
$\sum\limits_{i=1}^{n}\mathbf F_i=\mathbf0$
2D component balance
$\sum\limits_{i=1}^{n}F_{x,i}=0,\qquad\sum\limits_{i=1}^{n}F_{y,i}=0$
GDC magnitude input
$\operatorname{norm}(\langle a,b,c\rangle)$
GDC normalization
$\mathbf v\div\operatorname{norm}(\mathbf v)$
Quadrant-aware angle
$\theta=\operatorname{atan2}(v_y,v_x)$
Final audit
$\text{axes}\to\text{components}\to\text{operation}\to\text{magnitude}\to\text{direction}\to\text{units}$
Endpoint order
$\overrightarrow{AB}=B-A$
Parallel evidence
$\mathbf v=k\mathbf u\text{ in every component}$
Practice workflow
$\text{define axes}\to\text{write components}\to\text{operate}\to\text{find magnitude/direction}\to\text{interpret}$
Unit vector
$\widehat{\mathbf v}=\frac{\mathbf v}{|\mathbf v|}$
Relative motion
$\mathbf v_{\frac{A}{B}}=\mathbf v_A-\mathbf v_B$
Magnitude
$|\langle a,b,c\rangle|=\sqrt{a^2+b^2+c^2}$
Parallel test
$\mathbf v=k\mathbf u$
Topic 3.11Not started
Vector equation of a line
Represent straight paths in two and three dimensions using fixed position and direction vectors; convert to parametric form, classify line relationships, solve intersections and interpret practical route models.
180 min estimated
Topic 3.11 progress0%
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Learning objectives
Distinguish position vectors from direction vectors in two and three dimensions.
Write and interpret the vector equation of a line.
Convert accurately between vector and parametric forms.
Construct a line from a point and a nonzero direction vector.
Construct a line through two distinct points.
Determine exactly whether a point lies on a line.
Recognize equivalent equations of the same line.
Test whether two lines are parallel, distinct or coincident.
Find and verify intersection points in two dimensions.
Find and verify intersection points in three dimensions.
Recognize skew lines through algebraic inconsistency.
Use parameter restrictions for segments, rays and physical time.
Apply vector lines to navigation, route planning and constant-velocity motion.
Use calculator or GDC systems as a verification tool.
Present complete IB Mathematics AI HL solutions with notation, reasoning, context and units.
Key formulas
Line model
$\mathbf r=\mathbf a+\lambda\mathbf d,\qquad \lambda\in\mathbb R$
Displacement from the fixed point
$\mathbf r-\mathbf a=\lambda\mathbf d$
Position vector in 3D
$\overrightarrow{OP}=\begin{pmatrix}x\\[2pt]y\\[2pt]z\end{pmatrix}$
Vector between points
$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$
Position vector of P
$\overrightarrow{OP}=\begin{pmatrix}x_P\\y_P\\z_P\end{pmatrix}$
Nonzero direction requirement
$\mathbf d\ne\mathbf0$
Vector equation of a line
$\mathbf{r}=\mathbf{a}+t\mathbf{d},\qquad t\in\mathbb{R},\ \mathbf{d}\ne\mathbf{0}$
Vector line equation
$\mathbf r=\mathbf a+\lambda\mathbf d$
Unit-direction form
$\mathbf r=\mathbf a+s\widehat{\mathbf d}$
Parametric equations in 3D
$x=x_0+lt,\qquad y=y_0+mt,\qquad z=z_0+nt$
2D parametric form
$x=x_0+\lambda p,\qquad y=y_0+\lambda q$
3D parametric form
$x=x_0+\lambda p,\quad y=y_0+\lambda q,\quad z=z_0+\lambda r$
Point-direction line
$\mathbf r=\mathbf r_0+\lambda\mathbf d$
Line through two points
$\mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{a}),\qquad t\in\mathbb{R}$
Line through A and B
$\mathbf r=\mathbf a+\lambda(\mathbf b-\mathbf a)$
Point-on-line criterion
$\mathbf{q}-\mathbf{a}=t\mathbf{d}\text{ for one }t\in\mathbb{R}$
Membership condition
$\mathbf p-\mathbf a=\lambda\mathbf d\text{ for one common }\lambda$
Symmetric form
$\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}$
Component expansion
$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}x_0\\y_0\\z_0\end{pmatrix}+\lambda\begin{pmatrix}p\\q\\r\end{pmatrix}$
Same-line test
$\mathbf{d}_2=k\mathbf{d}_1\ (k\ne0)\quad\text{and}\quad \mathbf{a}_2-\mathbf{a}_1=\lambda\mathbf{d}_1$
Same-line test
$\mathbf d_2=k\mathbf d_1\quad\text{and}\quad\mathbf a_2-\mathbf a_1=m\mathbf d_1$
Line segment
$\mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{a}),\qquad 0\le t\le1$
Two-point parameterization
$\mathbf r(t)=(1-t)\mathbf a+t\mathbf b$
xy-plane condition
$z=0$
yz-plane condition
$x=0$
xz-plane condition
$y=0$
Constant-velocity line
$\mathbf{r}(t)=\mathbf{r}_0+t\mathbf{v}$
Constant-velocity motion
$\mathbf r(t)=\mathbf r_0+t\mathbf v$
Parallel-direction test
$\mathbf e=k\mathbf d,\qquad k\ne0$
Coincidence test
$\mathbf e=k\mathbf d\quad\text{and}\quad\mathbf b-\mathbf a=m\mathbf d$
Intersection equation
$\mathbf a+\lambda\mathbf d=\mathbf b+\mu\mathbf e$
2D component system
$a_x+\lambda d_x=b_x+\mu e_x,\qquad a_y+\lambda d_y=b_y+\mu e_y$
3D intersection system
$a_i+\lambda d_i=b_i+\mu e_i,\qquad i=1,2,3$
Verification
$\mathbf a+\lambda_0\mathbf d=\mathbf b+\mu_0\mathbf e$
Skew classification
$\mathbf e\ne k\mathbf d\quad\text{and}\quad\mathbf a+\lambda\mathbf d\ne\mathbf b+\mu\mathbf e\text{ for all }\lambda,\mu$
Segment AB
$\mathbf r=(1-t)\mathbf a+t\mathbf b,\qquad 0\le t\le1$
Ray
$\mathbf r=\mathbf a+t\mathbf d,\qquad t\ge0$
Bearing velocity
$\mathbf v=\begin{pmatrix}v\sin\beta\\v\cos\beta\end{pmatrix}$
Route model
$\mathbf r(t)=\mathbf r_0+t\mathbf v$
Linear-system form
$\begin{pmatrix}p&-u\\q&-v\end{pmatrix}\begin{pmatrix}\lambda\\\mu\end{pmatrix}=\begin{pmatrix}b_x-a_x\\b_y-a_y\end{pmatrix}$
Path-intersection model
$\mathbf a+\lambda\mathbf d=\mathbf b+\mu\mathbf e$
Reliable final check
$\mathbf a+\lambda\mathbf d\stackrel{?}{=}\mathbf b+\mu\mathbf e$
Core vector form
$\mathbf{r}=\mathbf{a}+t\mathbf{d}$
Core parametric form
$x=x_0+lt,\quad y=y_0+mt,\quad z=z_0+nt$
Core two-point form
$\mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{a})$
Core point test
$\mathbf{q}-\mathbf{a}=t\mathbf{d}$
Parallel test
$\mathbf e=k\mathbf d$
Intersection test
$\mathbf a+\lambda\mathbf d=\mathbf b+\mu\mathbf e$
Segment restriction
$\mathbf r=(1-t)\mathbf a+t\mathbf b,\quad 0\le t\le1$
Topic 3.12Not started
Vector kinematics
Model motion in two and three dimensions using time-dependent position, velocity and acceleration vectors; integrate and differentiate componentwise; analyze trajectories, relative motion, collisions and closest approach.
190 min estimated
Topic 3.12 progress0%
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Learning objectives
Interpret position vectors as functions of time in two and three dimensions.
Calculate displacement and average velocity over a time interval.
Differentiate position to obtain velocity and speed.
Differentiate velocity to obtain acceleration.
Integrate velocity to recover position using vector initial conditions.
Integrate acceleration to recover velocity and position.
Model constant-velocity and constant-acceleration motion.
Analyze planar and spatial particle trajectories.
Eliminate time to obtain a Cartesian trajectory equation.
Use velocity as a tangent direction and interpret changing speed.
Apply vector methods to projectile motion.
Calculate relative position, relative velocity and separation.
Determine closest approach by minimizing squared separation.
Distinguish geometric path intersections from collisions.
Solve and verify collision conditions in two and three dimensions.
Use calculator and GDC methods to support numerical motion analysis.
Evaluate modelling assumptions, domains, precision and units in context.
Present complete IB Mathematics AI HL solutions with clear vector reasoning.
Key formulas
2D position
$\mathbf r(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}$
3D position
$\mathbf r(t)=\begin{pmatrix}x(t)\\y(t)\\z(t)\end{pmatrix}$
Displacement
$\Delta\mathbf r=\mathbf r(t_2)-\mathbf r(t_1)$
Displacement magnitude
$|\Delta\mathbf r|=|\mathbf r(t_2)-\mathbf r(t_1)|$
Average velocity
$\mathbf v_{\mathrm{avg}}=\frac{\mathbf r(t_2)-\mathbf r(t_1)}{t_2-t_1}$
Velocity
$\mathbf v(t)=\frac{d\mathbf r}{dt}$
Component form
$\mathbf v(t)=\begin{pmatrix}x'(t)\\y'(t)\\z'(t)\end{pmatrix}$
2D speed
$|\mathbf v|=\sqrt{v_x^2+v_y^2}$
3D speed
$|\mathbf v|=\sqrt{v_x^2+v_y^2+v_z^2}$
Acceleration
$\mathbf a(t)=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}$
Component form
$\mathbf a(t)=\begin{pmatrix}x''(t)\\y''(t)\\z''(t)\end{pmatrix}$
Vector derivative
$\frac{d}{dt}\langle f,g,h\rangle=\langle f',g',h'\rangle$
Motion chain
$\mathbf r\xrightarrow{\frac{\,d}{dt\,}}\mathbf v\xrightarrow{\frac{\,d}{dt\,}}\mathbf a$
Indefinite integration
$\mathbf r(t)=\int\mathbf v(t)\,dt+\mathbf C$
Initial-value form
$\mathbf r(t)=\mathbf r(t_0)+\int_{t_0}^{t}\mathbf v(u)\,du$
Velocity from acceleration
$\mathbf v(t)=\mathbf v(t_0)+\int_{t_0}^{t}\mathbf a(u)\,du$
Position after two integrations
$\mathbf r(t)=\int\left(\int\mathbf a(t)\,dt+\mathbf C_1\right)dt+\mathbf C_2$
Initial position
$\mathbf r(t_0)=\mathbf r_0$
Initial velocity
$\mathbf v(t_0)=\mathbf v_0$
Constant-velocity position
$\mathbf r(t)=\mathbf r_0+t\mathbf v$
Zero acceleration
$\mathbf a(t)=\mathbf0$
Velocity
$\mathbf v(t)=\mathbf v_0+t\mathbf a$
Position
$\mathbf r(t)=\mathbf r_0+t\mathbf v_0+\frac12t^2\mathbf a$
Planar state
$\mathbf r=\langle x,y\rangle,\quad\mathbf v=\langle x',y'\rangle,\quad\mathbf a=\langle x'',y''\rangle$
Spatial motion
$\mathbf r(t)=\langle x(t),y(t),z(t)\rangle$
Spatial speed
$|\mathbf v|=\sqrt{x'(t)^2+y'(t)^2+z'(t)^2}$
Trajectory elimination
$t=g(x)\quad\Longrightarrow\quad y=f(g(x))$
Unit tangent
$\mathbf T(t)=\frac{\mathbf v(t)}{|\mathbf v(t)|}$
Speed derivative
$\frac{d}{dt}|\mathbf v|=\frac{\mathbf v\cdot\mathbf a}{|\mathbf v|}$
Gravity acceleration
$\mathbf a=\begin{pmatrix}0\\-g\end{pmatrix}$
Projectile position
$\mathbf r(t)=\mathbf r_0+t\mathbf v_0+\frac12t^2\begin{pmatrix}0\\-g\end{pmatrix}$
Relative position
$\mathbf r_{\frac{A}{B}}=\mathbf r_A-\mathbf r_B$
Relative velocity
$\mathbf v_{\frac{A}{B}}=\mathbf v_A-\mathbf v_B$
Separation
$d(t)=|\mathbf r_A(t)-\mathbf r_B(t)|$
Squared separation
$D(t)=|\mathbf r_A-\mathbf r_B|^2=(\mathbf r_A-\mathbf r_B)\cdot(\mathbf r_A-\mathbf r_B)$
Stationary condition
$D'(t)=0$
Collision condition
$\mathbf r_A(t)=\mathbf r_B(t)$
Path-intersection condition
$\mathbf r_A(u)=\mathbf r_B(v)$
Bearing components
$\mathbf v=\begin{pmatrix}V\sin\beta\\V\cos\beta\end{pmatrix}$
Ground velocity
$\mathbf v_{\mathrm{ground}}=\mathbf v_{\mathrm{\frac{object}{medium}}}+\mathbf v_{\mathrm{\frac{medium}{ground}}}$
Component displacement
$\Delta x=\int_{t_1}^{t_2}v_x(t)\,dt$
Component acceleration
$a_x(t)=\frac{dv_x}{dt}$
Numerical collision functions
$f_i(t)=r_{A,i}(t)-r_{B,i}(t)$
GDC minimum target
$D(t)=|\mathbf r_A(t)-\mathbf r_B(t)|^2$
Dimensional chain
$[\mathbf r]=L,\qquad[\mathbf v]=LT^{-1},\qquad[\mathbf a]=LT^{-2}$
Collision checklist
$\mathbf r_A(t)=\mathbf r_B(t)\text{ in every component and within the domain}$
Velocity
$\mathbf v=\frac{d\mathbf r}{dt}$
Speed
$\text{speed}=|\mathbf v|$
Acceleration
$\mathbf a=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}$
Position from velocity
$\mathbf r(t)=\mathbf r(t_0)+\int_{t_0}^{t}\mathbf v(u)\,du$
Constant acceleration
$\mathbf r=\mathbf r_0+t\mathbf v_0+\frac12t^2\mathbf a$
Collision
$\mathbf r_A(t)=\mathbf r_B(t)$
Topic 3.13Not started
Scalar and vector products
Products of vectors measure angular alignment and perpendicular area/direction. The dot product is widely used for angles; the cross product gives a vector perpendicular to two 3D vectors.
180 min estimated
Topic 3.13 progress0%
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Learning objectives
Calculate scalar products in two and three dimensions.
Interpret the scalar product using vector magnitudes and the included angle.
Find angles between vectors and identify acute, right and obtuse relationships.
Test vectors for perpendicularity using a zero scalar product.
Resolve one vector into components parallel and perpendicular to another.
Apply scalar products to work, direction and three-dimensional geometry.
Compute vector products accurately using components or determinants.
Use the right-hand rule and explain why cross-product order matters.
Find a vector perpendicular to two given non-parallel vectors.
Interpret the magnitude of a vector product geometrically.
Find areas of parallelograms and triangles using vector products.
Test vectors for parallelism using a zero vector product.
Find normals and equations of planes from points and direction vectors.
Use scalar triple products for volumes and coplanarity.
Use GDC methods to verify vector calculations while presenting complete reasoning.
Solve multi-step IB AI HL problems involving scalar and vector products.
Key formulas
Scalar product output
$\mathbf a\cdot\mathbf b\in\mathbb R$
Vector product output
$\mathbf a\times\mathbf b\in\mathbb R^3$
Geometric dot product
$\mathbf a\cdot\mathbf b=\lVert\mathbf a\rVert\lVert\mathbf b\rVert\cos\theta$
Self-product
$\mathbf a\cdot\mathbf a=\lVert\mathbf a\rVert^2$
Commutative law
$\mathbf a\cdot\mathbf b=\mathbf b\cdot\mathbf a$
2D component form
$\langle a_1,a_2\rangle\cdot\langle b_1,b_2\rangle=a_1b_1+a_2b_2$
3D component form
$\langle a_1,a_2,a_3\rangle\cdot\langle b_1,b_2,b_3\rangle=a_1b_1+a_2b_2+a_3b_3$
Angle between vectors
$\theta=\cos^{-1}\!\left(\frac{\mathbf a\cdot\mathbf b}{\lVert\mathbf a\rVert\lVert\mathbf b\rVert}\right)$
Perpendicular-vector test
$\mathbf a\perp\mathbf b\iff\mathbf a\cdot\mathbf b=0\quad(\mathbf a,\mathbf b\ne\mathbf0)$
Scalar projection
$\operatorname{comp}_{\mathbf a}\mathbf b=\frac{\mathbf a\cdot\mathbf b}{\lVert\mathbf a\rVert}$
Vector projection
$\operatorname{proj}_{\mathbf a}\mathbf b=\frac{\mathbf a\cdot\mathbf b}{\mathbf a\cdot\mathbf a}\mathbf a$
Perpendicular component
$\mathbf b_{\perp}=\mathbf b-\operatorname{proj}_{\mathbf a}\mathbf b$
Constant-force work
$W=\mathbf F\cdot\mathbf d=\lVert\mathbf F\rVert\lVert\mathbf d\rVert\cos\theta$
Cross-product magnitude
$\lVert\mathbf a\times\mathbf b\rVert=\lVert\mathbf a\rVert\lVert\mathbf b\rVert\sin\theta$
Order reversal
$\mathbf b\times\mathbf a=-(\mathbf a\times\mathbf b)$
Self cross product
$\mathbf a\times\mathbf a=\mathbf0$
Component formula
$\mathbf a\times\mathbf b=\langle a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1\rangle$
Determinant form
$\mathbf a\times\mathbf b=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}$
Distributive law
$\mathbf a\times(\mathbf b+\mathbf c)=\mathbf a\times\mathbf b+\mathbf a\times\mathbf c$
Scalar factors
$(s\mathbf a)\times(t\mathbf b)=st(\mathbf a\times\mathbf b)$
Normal from two directions
$\mathbf n=\mathbf u\times\mathbf v$
Normal checks
$\mathbf n\cdot\mathbf u=0,\qquad\mathbf n\cdot\mathbf v=0$
Parallelogram interpretation
$A_{\mathrm{parallelogram}}=\lVert\mathbf a\times\mathbf b\rVert$
Base-height interpretation
$\lVert\mathbf a\times\mathbf b\rVert=\lVert\mathbf a\rVert(\lVert\mathbf b\rVert\sin\theta)$
Parallelogram area
$A=\lVert\mathbf u\times\mathbf v\rVert$
Triangle area
$A_{\triangle ABC}=\frac12\left\lVert\overrightarrow{AB}\times\overrightarrow{AC}\right\rVert$
Parallel-vector test
$\mathbf a\parallel\mathbf b\iff\mathbf a\times\mathbf b=\mathbf0\quad(\mathbf a,\mathbf b\ne\mathbf0)$
Collinearity test
$A,B,C\text{ collinear}\iff\overrightarrow{AB}\times\overrightarrow{AC}=\mathbf0$
Point-normal plane
$\mathbf n\cdot(\mathbf r-\mathbf p)=0$
Cartesian plane
$ax+by+cz=d,\qquad\mathbf n=\langle a,b,c\rangle$
Angle between planes
$\theta=\cos^{-1}\!\left(\frac{|\mathbf n_1\cdot\mathbf n_2|}{\lVert\mathbf n_1\rVert\lVert\mathbf n_2\rVert}\right)$
Angle between line and plane
$\phi=\sin^{-1}\!\left(\frac{|\mathbf d\cdot\mathbf n|}{\lVert\mathbf d\rVert\lVert\mathbf n\rVert}\right)$
Parallelepiped volume
$V=|\mathbf a\cdot(\mathbf b\times\mathbf c)|$
Coplanarity test
$\mathbf a\cdot(\mathbf b\times\mathbf c)=0$
Tetrahedron volume
$V=\frac16|\mathbf a\cdot(\mathbf b\times\mathbf c)|$
Point-plane distance
$d=\frac{|\mathbf n\cdot(\mathbf p-\mathbf a)|}{\lVert\mathbf n\rVert}$
Cartesian point-plane distance
$d=\frac{|ax_0+by_0+cz_0-d_0|}{\sqrt{a^2+b^2+c^2}}$
Unit normal
$\widehat{\mathbf n}=\frac{\mathbf a\times\mathbf b}{\lVert\mathbf a\times\mathbf b\rVert}$
GDC angle input
$\cos^{-1}\!\left(\frac{\operatorname{dot}(\mathbf a,\mathbf b)}{\operatorname{norm}(\mathbf a)\operatorname{norm}(\mathbf b)}\right)$
Product decision
$\text{angle/projection}\to\cdot,\qquad\text{normal/area}\to\times$
Core angle formula
$\cos\theta=\frac{\mathbf a\cdot\mathbf b}{\lVert\mathbf a\rVert\lVert\mathbf b\rVert}$
Core area formula
$A_{\triangle}=\frac12\lVert\mathbf a\times\mathbf b\rVert$
Dot product
$\mathbf a\cdot\mathbf b=\sum\limits_{i=1}^{n}a_ib_i=\lVert\mathbf a\rVert\lVert\mathbf b\rVert\cos\theta$
Cross product
$\mathbf a\times\mathbf b=\langle a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1\rangle$
Perpendicular test
$\mathbf a\cdot\mathbf b=0$
Parallel test
$\mathbf a\times\mathbf b=\mathbf0$
Areas
$A_{\parallel}=\lVert\mathbf a\times\mathbf b\rVert,\qquad A_{\triangle}=\frac12\lVert\mathbf a\times\mathbf b\rVert$
Volume
$V=|\mathbf a\cdot(\mathbf b\times\mathbf c)|$
Topic 3.14Not started
Graph theory: fundamental concepts
Graph theory models networks by representing objects as vertices and relationships as edges. The mathematics is structural rather than geometric: the exact drawn shape of the network is often irrelevant.
185 min estimated
Topic 3.14 progress0%
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Learning objectives
Model real situations using graphs and networks.
Identify vertices, edges, order and size.
Distinguish directed from undirected graphs.
Distinguish weighted from unweighted graphs.
Calculate degree, in-degree and out-degree correctly.
Apply the Handshaking Lemma and parity consequences.
Determine connectedness, components and bridges.
Distinguish walks, trails, paths, circuits and cycles.
Recognize simple graphs, multigraphs and complete graphs.
Use formulas for the degree and size of complete graphs.
Identify subgraphs and induced subgraphs.
Recognize trees and fundamental tree properties.
Determine whether an undirected graph has an Euler trail or Euler circuit.
Apply directed Euler degree conditions where relevant.
Interpret graph-theory results in real-world network contexts.
Present complete IB AI HL reasoning using precise graph vocabulary.
Key formulas
Graph notation
$G=(V,E)$
Vertex and edge counts
$n=|V|,\qquad m=|E|$
Order
$\operatorname{ord}(G)=|V|$
Size
$\operatorname{size}(G)=|E|$
Directed arc
$(u,v)\equiv u\to v$
Undirected edge symmetry
$\{u,v\}=\{v,u\}$
Additive route weight
$W(P)=w_1+w_2+\cdots+w_k$
Undirected degree
$\deg(v)=\text{number of incident edge ends at }v$
Loop contribution
$\text{one loop contributes }2\text{ to }\deg(v)$
Directed degree notation
$\deg^-(v)=\text{in-degree},\qquad\deg^+(v)=\text{out-degree}$
Directed degree sums
$\sum\limits_{i=1}^{n}\deg^-(v_i)=\sum\limits_{i=1}^{n}\deg^+(v_i)=m$
Handshaking Lemma
$\sum\limits_{i=1}^{n}\deg(v_i)=2m$
Edge count from degrees
$m=\frac12\sum\limits_{i=1}^{n}\deg(v_i)$
Connectedness statement
$\forall u,v\in V,\ \exists\text{ a path from }u\text{ to }v$
Bridge test
$e\text{ is a bridge}\iff c(G-e)>c(G)$
Walk length
$\ell(W)=\text{number of traversed edges}$
Closed route condition
$v_0=v_k$
Containment of route types
$\text{cycle}\Rightarrow\text{circuit}\Rightarrow\text{trail}\Rightarrow\text{walk}$
Route checklist
$\text{adjacency}\to\text{edge repetition}\to\text{vertex repetition}\to\text{closure}$
Simple graph condition
$\text{no loops and at most one edge per unordered vertex pair}$
Complete-graph degree
$\deg(v)=n-1\text{ in }K_n$
Complete-graph size
$|E(K_n)|=\frac{n(n-1)}{2}$
Subgraph condition
$V(H)\subseteq V(G),\qquad E(H)\subseteq E(G)$
Spanning condition
$V(H)=V(G)$
Tree size
$m=n-1$
Tree path property
$\text{exactly one path between every pair of vertices}$
Euler trail requirement
$\text{every edge used exactly once}$
Euler circuit closure
$\text{Euler trail with start}=\text{finish}$
Euler circuit criterion
$0\text{ odd-degree vertices}\iff\text{Euler circuit}$
Open Euler trail criterion
$2\text{ odd-degree vertices}\iff\text{open Euler trail}$
Euler construction principle
$\text{use a bridge only when no unused non-bridge choice remains}$
Directed Euler circuit balance
$\deg^-(v)=\deg^+(v)\text{ for every relevant }v$
Directed open-trail start
$\deg^+(s)=\deg^-(s)+1$
Directed open-trail end
$\deg^-(t)=\deg^+(t)+1$
Adjacency matrix entry
$a_{ij}=\begin{cases}1,&v_i\text{ adjacent to }v_j\\0,&\text{otherwise}\end{cases}$
Undirected symmetry
$A=A^{\mathsf T}$
Basic weighted objective
$\text{total route cost}=\text{sum of traversed edge costs}$
Matrix degree check
$\deg(v_i)=\text{row sum of }A\text{ for a simple undirected graph}$
Undirected Euler decision
$\text{connected non-isolated part}+\text{odd-degree count}$
Degree sum
$\sum\limits_{i=1}^{n}\deg(v_i)=2m$
Complete graph
$|E(K_n)|=\frac{n(n-1)}2$
Euler parity
$0\text{ odd}\to\text{circuit},\quad2\text{ odd}\to\text{open trail}$
Graph counts
$n=|V|,\qquad m=|E|$
Complete graph size
$|E(K_n)|=\frac{n(n-1)}2$
Euler circuit
$\text{connected edge-part and all degrees even}$
Open Euler trail
$\text{connected edge-part and exactly two odd degrees}$
Topic 3.15Not started
Adjacency matrices, weighted tables and walks
Represent networks with adjacency and weighted matrices, reconstruct graphs from tables, and use matrix powers to count and interpret walks.
235 min estimated
Topic 3.15 progress0%
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Learning objectives
Define an adjacency matrix using a declared vertex order and row-to-column convention.
Construct adjacency matrices for directed and undirected graphs.
Reconstruct labelled graphs accurately from adjacency matrices.
Recognize symmetry, diagonal entries and row or column sums as graph information.
Represent loops and parallel edges under a stated matrix convention.
Construct and interpret weighted adjacency tables or matrices.
Distinguish zero weight from absence of an edge.
Calculate route weights from a weighted network.
Multiply matrices and compute powers using a GDC.
Explain why the entry of a matrix power counts walks of a specified length.
Interpret individual entries, row sums, column sums and diagonal entries of matrix powers.
Use powers of an adjacency matrix to test reachability within a fixed number of steps.
Distinguish walks counted by matrix powers from trails and paths.
Use trace formulas to count edges, returns and triangles in simple undirected graphs.
Distinguish adjacency, weighted and transition matrices.
Apply matrix representations to transport, communication, social and movement networks.
Use calculator matrix features efficiently and verify outputs independently.
Communicate IB AI HL solutions with vertex order, convention and contextual interpretation.
Key formulas
Matrix entry
$a_{ij}=\text{information about the connection }v_i\to v_j$
Simple adjacency entry
$a_{ij}=\begin{cases}1,&\text{if }v_i\to v_j\text{ is an edge},\\0,&\text{otherwise}\end{cases}$
Matrix order
$|V|=n\quad\Longrightarrow\quad A\in\mathbb R^{n\times n}$
Undirected symmetry
$a_{ij}=a_{ji}\quad\text{and hence}\quad A=A^T$
Simple graph diagonal
$a_{ii}=0$
Edge count
$m=\frac12\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}a_{ij}$
Directed entry convention
$a_{ij}=1\iff v_i\to v_j$
Transpose reverses arrows
$A^T\text{ represents the graph with every arrow reversed}$
Arc set
$E=\{(v_i,v_j):a_{ij}\ne0\}$
Multiplicity entry
$a_{ij}=\text{number of edges from }v_i\text{ to }v_j$
Degree with diagonal loop count
$d(v_i)=\sum\limits_{j=1}^{n}a_{ij}+a_{ii}$
Undirected test
$A=A^T$
Isolation in an undirected graph
$\sum\limits_{j=1}^{n}a_{ij}=0$
Undirected degree
$d(v_i)=\sum\limits_{j=1}^{n}a_{ij}$
Out-degree
$d^+(v_i)=\sum\limits_{j=1}^{n}a_{ij}$
In-degree
$d^-(v_j)=\sum\limits_{i=1}^{n}a_{ij}$
Directed degree balance
$\sum\limits_{i=1}^{n}d^+(v_i)=\sum\limits_{i=1}^{n}d^-(v_i)=|E|$
Weighted entry
$w_{ij}=\text{weight of the edge }v_i\to v_j$
Undirected weighted symmetry
$w_{ij}=w_{ji}$
Common distance-table convention
$w_{ii}=0,\qquad w_{ij}=\infty\text{ if no direct edge exists}$
Walk weight
$W(v_0v_1\cdots v_k)=\sum\limits_{r=1}^{k}w_{v_{r-1}v_r}$
Matrix product entry
$(BC)_{ij}=\sum\limits_{r=1}^{n}b_{ir}c_{rj}$
Adjacency square entry
$(A^2)_{ij}=\sum\limits_{r=1}^{n}a_{ir}a_{rj}$
Two-step interpretation
$(A^2)_{ij}=\#\{\text{walks of length }2\text{ from }v_i\text{ to }v_j\}$
Simple undirected diagonal
$(A^2)_{ii}=d(v_i)$
Walk-count theorem
$(A^k)_{ij}=\#\{\text{walks of length }k\text{ from }v_i\text{ to }v_j\}$
Power recursion
$A^{k+1}=A^kA$
Walks starting at (v_i)
$\sum\limits_{j=1}^{n}(A^k)_{ij}$
Walks ending at (v_j)
$\sum\limits_{i=1}^{n}(A^k)_{ij}$
All k-step walks
$\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}(A^k)_{ij}$
Trace
$\operatorname{tr}(A^k)=\sum\limits_{i=1}^{n}(A^k)_{ii}$
Based closed walks
$(A^k)_{ii}=\#\{\text{closed k-walks based at }v_i\}$
Edges from trace
$m=\frac12\operatorname{tr}(A^2)$
Triangles from trace
$N_{\triangle}=\frac16\operatorname{tr}(A^3)$
Reachability up to L
$R_L=A+A^2+\cdots+A^L$
Finite reachability test
$(A+A^2+\cdots+A^{n-1})_{ij}>0$
Containment of route types
$\text{path}\subseteq\text{trail}\subseteq\text{walk}$
Weighted square entry
$(W^2)_{ij}=\sum\limits_{r=1}^{n}w_{ir}w_{rj}$
Maximum-degree hub set
$H=\{v_i:d(v_i)=\max_{1\le r\le n}d(v_r)\}$
Row-stochastic condition
$p_{ij}\ge0,\qquad\sum\limits_{j=1}^{n}p_{ij}=1$
Two-step transition probability
$(P^2)_{ij}=\sum\limits_{r=1}^{n}p_{ir}p_{rj}$
Power command meaning
$A^k=\underbrace{A\cdot A\cdots A}_{k\text{ factors}}$
Core walk formula
$(A^k)_{ij}=\text{number of length-}k\text{ walks from }v_i\text{ to }v_j$
Triangle count
$N_{\triangle}=\frac{\operatorname{tr}(A^3)}{6}$
Adjacency
$a_{ij}=1\iff v_i\to v_j$
Walks
$(A^k)_{ij}=\#\text{ length-}k\text{ walks }v_i\to v_j$
Row total
$\sum\limits_{j=1}^{n}(A^k)_{ij}$
Closed walks
$\operatorname{tr}(A^k)=\sum\limits_{i=1}^{n}(A^k)_{ii}$
Topic 3.16Not started
Network routes, spanning trees and optimization algorithms
Optimize weighted networks using shortest-path, spanning-tree, route-inspection and travelling-salesperson methods, with complete algorithm traces and contextual interpretation.
260 min estimated
Topic 3.16 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Interpret vertices, edges, directions and weights in applied networks.
Distinguish walks, trails, paths, circuits and cycles in route problems.
Calculate and compare route weights with appropriate units.
Formulate a shortest-path problem with a defined source and destination.
Apply Dijkstra’s algorithm stage by stage to a non-negative weighted network.
Reconstruct a shortest route from permanent labels or predecessor records.
Recognize when Dijkstra’s algorithm is invalid or incomplete.
Define trees, spanning trees and minimum spanning trees.
Use the n−1 edge property to check a spanning tree.
Apply Kruskal’s algorithm with explicit cycle checks.
Apply Prim’s algorithm with explicit cut-edge choices.
Compare Kruskal’s and Prim’s algorithms and interpret non-unique MSTs.
Solve route-inspection or Chinese postman problems on connected weighted graphs.
Use odd-vertex pairing to minimize repeated edge weight.
Distinguish Eulerian and Hamiltonian optimization questions.
Use nearest-neighbour tours to obtain TSP upper bounds.
Use the deleted-vertex method to obtain TSP lower bounds.
Combine lower and upper bounds to evaluate a proposed tour.
Select a suitable algorithm for a transport, telecommunications or delivery problem.
Use technology to verify arithmetic without hiding algorithm decisions.
Evaluate assumptions, constraints and limitations of network models.
Communicate complete IB AI HL solutions with tables, choices and interpretations.
Key formulas
Optimization template
$\min\{\text{objective value}:\text{all constraints are satisfied}$
Route weight
$w(P)=\sum\limits_{r=1}^{m}w(e_r)$
Route-type containment
$\text{path}\subseteq\text{trail}\subseteq\text{walk}$
Feasible optimum
$w^*=\min_{P\in\mathcal F}w(P)$
Shortest distance
$d(s,t)=\min_{P:s\leadsto t}\sum\limits_{r=1}^{|P|}w(e_r)$
Relaxation rule
$L(v)\leftarrow\min\{L(v),L(u)+w(u,v)\}$
Initialization
$L(s)=0,\qquad L(v)=\infty\text{ for }v\ne s$
Final distance
$d(S,T)=13$
Visual shortest distance
$d(A,D)=6$
Validity condition
$w(e)\ge 0\quad\text{for every edge }e$
Tree edge count
$|E|=|V|-1$
Unique path property
$\forall u,v\in V,\quad\text{exactly one }u\text{–}v\text{ path exists}$
Spanning-tree size
$|E(T)|=|V(G)|-1$
MST objective
$w(T^*)=\min_{T\text{ spanning tree}}\sum\limits_{r=1}^{n-1}w(e_r)$
Kruskal stopping rule
$|E_{\text{accepted}}|=n-1$
Kruskal total
$w(T)=1+2+5=8$
Prim choice
$e^*=\arg\min\{w(e):e\text{ has exactly one endpoint in }V(T)\}$
Prim total
$w(T)=2+1+5=8$
Common result
$w(T_{\text{Kruskal}})=w(T_{\text{Prim}})=w(T^*)$
Sufficient uniqueness condition
$\text{all edge weights distinct}\Longrightarrow\text{unique MST}$
Euler circuit condition
$d(v)\equiv0\pmod 2\quad\text{for every non-isolated }v$
Open Euler trail condition
$|\{v:d(v)\text{ odd}\}|=2$
Postman optimum
$W_{\text{CPP}}=\sum\limits_{r=1}^{m}w(e_r)+\min(\text{total odd-pairing distance})$
Worked postman total
$W_{\text{CPP}}=15+1=16$
TSP objective
$W^*_{\text{TSP}}=\min_{H\text{ Hamiltonian cycle}}\sum\limits_{r=1}^{n}w(e_r)$
Upper-bound statement
$W^*_{\text{TSP}}\le W_{\text{NN}}$
Deleted-vertex lower bound
$L_v=w(\operatorname{MST}(G-v))+\text{two smallest weights incident with }v$
Bound interval
$L\le W^*_{\text{TSP}}\le U$
Absolute bound gap
$\text{gap}=U-L$
Relative gap convention
$\text{relative gap}=\frac{U-L}{L}\times100\%$
Shortest-path update
$L(v)\leftarrow\min\{L(v),L(u)+w(u,v)\}$
MST edge count
$|E(T)|=n-1$
TSP bounds
$L\le W^*_{\text{TSP}}\le U$
Dijkstra
$L(v)\leftarrow\min\{L(v),L(u)+w(u,v)\}$
Tree size
$|E|=|V|-1$
Postman
$W_{\text{CPP}}=\sum\limits_{r=1}^{m}w(e_r)+\text{minimum odd-pairing cost}$
Bounds
$L\le W^*_{\text{TSP}}\le U$
Unit 4
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Statistics and Probability
Statistics and Probability is the largest current AI HL content area. The shared course develops data presentation, regression, distributions, Spearman correlation and basic hypothesis testing. HL then adds study design, nonlinear regression, transformed random variables, the Central Limit Theorem, confidence intervals, Poisson models, advanced hypothesis testing and Markov chains.
19 topics
Unit 4 progress0%
Topics in this unit
Topic 4.1Not started
Population, samples and sampling
Statistical conclusions are only as trustworthy as the data collection process. Students should judge source quality, bias and representativeness before calculation.
240 min estimated
Topic 4.1 progress0%
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Learning objectives
Distinguish population and sample.
Distinguish discrete and continuous data.
Recognize random and non-random sampling approaches.
Evaluate possible sampling bias.
Interpret outliers and question data reliability.
Comment on whether conclusions can reasonably be generalized.
Key formulas
Sample proportion
$\hat p=\frac{x}{n}$
Sampling fraction
$f=\frac{n}{N}$
Systematic sampling interval
$k=\frac{N}{n}$
Selected positions
$r,r+k,r+2k,\ldots,r+(n-1)k$
Proportional stratum allocation
$n_h=n\frac{N_h}{N}$
Response rate
$\text{response rate}=\frac{\text{usable responses}}{\text{people selected}}\times100\%$
Sampling calculation checklist
$\text{define population}\to\text{choose frame}\to\text{select method}\to\text{evaluate limitations}$
Core sampling formulas
$\hat p=\frac{x}{n},\qquad k=\frac{N}{n},\qquad n_h=n\frac{N_h}{N}$
Systematic interval
$k=\frac{N}{n}$
Stratified allocation
$n_h=n\frac{N_h}{N}$
Topic 4.2Not started
Presentation of data
Graphs should reveal structure rather than decorate a solution. AI students are expected to read and construct common univariate displays.
250 min estimated
Topic 4.2 progress0%
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Learning objectives
Use frequency tables for discrete and continuous data.
Construct and interpret histograms.
Use cumulative-frequency graphs.
Determine median, quartiles and percentiles.
Use box-and-whisker plots to compare distributions.
Calculate range and interquartile range.
Key formulas
Relative frequency
$\text{relative frequency}=\frac{f}{n}$
Percentage frequency
$\text{percentage frequency}=\frac{f}{n}\times100\%$
Cumulative frequency
$F_i=\sum\limits_{j=1}^{i}f_j$
Class width
$w=U-L$
Class midpoint
$m=\frac{L+U}{2}$
Frequency density
$\text{frequency density}=\frac{f}{w}$
Histogram area
$f=(\text{frequency density})(\text{class width})$
Frequency-polygon point
$(m_i,f_i),\quad m_i=\frac{L_i+U_i}{2}$
Frequency between two values
$f(a<x\le b)\approx F(b)-F(a)$
Quartile levels
$Q_1:\frac{N}{4},\qquad Q_2:\frac{N}{2},\qquad Q_3:\frac{3N}{4}$
Percentile level
$\text{CF level}=\frac{p}{100}N$
Interquartile range
$\operatorname{IQR}=Q_3-Q_1$
Range
$\text{range}=\max-\min$
Outlier fences
$Q_1-1.5\operatorname{IQR},\qquad Q_3+1.5\operatorname{IQR}$
Difference in medians
$\Delta\widetilde{x}=\widetilde{x}_A-\widetilde{x}_B$
Display workflow
$\text{classify data}\to\text{calculate}\to\text{construct}\to\text{interpret}$
Core presentation formulas
$\frac{f}{n},\qquad \frac{f}{w},\qquad \operatorname{IQR}=Q_3-Q_1$
Relative frequency
$\frac{f}{n}$
Frequency density
$\frac{f}{w}$
Quartile levels
$\frac{N}{4},\quad\frac{N}{2},\quad\frac{3N}{4}$
Topic 4.3Not started
Measures of central tendency and spread
Different summaries answer different questions. Students should know when the mean is sensitive to extreme values and when the median/IQR are more robust.
250 min estimated
Topic 4.3 progress0%
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Learning objectives
Calculate mean, median and mode.
Estimate mean from grouped data.
Identify modal class.
Use variance and standard deviation.
Explain the effect of adding/multiplying all data values by a constant.
Compare data sets using both center and spread.
Key formulas
Sample mean
$\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_i$
Zero-sum deviations
$\sum\limits_{i=1}^{n}(x_i-\bar{x})=0$
Odd-n median position
$\frac{n+1}{2}$
Even-n median
$\widetilde{x}=\frac{x_{(\frac{n}{2})}+x_{(\frac{n}{2}+1)}}{2}$
Quartile cumulative levels
$\frac{N}{4},\qquad\frac{N}{2},\qquad\frac{3N}{4}$
Percentile cumulative level
$\frac{p}{100}N$
Range
$\text{range}=x_{\max}-x_{\min}$
Interquartile range
$\operatorname{IQR}=Q_3-Q_1$
Population variance
$\sigma^2=\frac{1}{N}\sum\limits_{i=1}^{N}(x_i-\mu)^2$
Population computational form
$\sigma^2=\frac{1}{N}\sum\limits_{i=1}^{N}x_i^2-\mu^2$
Sample variance
$s^2=\frac{1}{n-1}\sum\limits_{i=1}^{n}(x_i-\bar{x})^2$
Standard deviations
$\sigma=\sqrt{\sigma^2},\qquad s=\sqrt{s^2}$
Frequency mean
$\bar{x}=\frac{\sum\limits_{i=1}^{k}f_ix_i}{\sum\limits_{i=1}^{k}f_i}$
Grouped mean estimate
$\bar{x}\approx\frac{\sum\limits_{i=1}^{k}f_im_i}{\sum\limits_{i=1}^{k}f_i}$
Grouped population variance estimate
$\sigma^2\approx\frac{\sum\limits_{i=1}^{k}f_im_i^2}{\sum\limits_{i=1}^{k}f_i}-\bar{x}^{\,2}$
Transformed mean
$\bar{y}=a\bar{x}+b$
Transformed variance
$\operatorname{Var}(Y)=a^2\operatorname{Var}(X)$
Transformed standard deviation
$\sigma_Y=|a|\sigma_X$
Transformed IQR
$\operatorname{IQR}_Y=|a|\operatorname{IQR}_X$
Common potential-outlier fences
$Q_1-1.5\operatorname{IQR},\qquad Q_3+1.5\operatorname{IQR}$
Exam workflow
$\text{identify data role}\to\text{calculate}\to\text{interpret}\to\text{check units}$
Core center-and-spread formulas
$\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_i,\qquad \operatorname{IQR}=Q_3-Q_1,\qquad \sigma=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}(x_i-\mu)^2}$
Mean
$\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_i$
IQR
$\operatorname{IQR}=Q_3-Q_1$
Topic 4.4Not started
Linear correlation and regression
Linear regression describes association between two quantitative variables. The fitted equation supports prediction, but interpretation and limitations remain essential.
250 min estimated
Topic 4.4 progress0%
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Learning objectives
Construct and interpret a scatter diagram.
Use Pearson correlation coefficient r.
Find the regression line of y on x using technology.
Interpret slope and intercept where meaningful.
Use the model for prediction.
Distinguish association from causation and interpolation from extrapolation.
Key formulas
Pearson’s product-moment correlation coefficient
$r=\frac{\displaystyle\sum\limits_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\displaystyle\left[\sum\limits_{i=1}^{n}(x_i-\bar{x})^2\right]\left[\sum\limits_{i=1}^{n}(y_i-\bar{y})^2\right]}}$
Range of r
$-1\le r\le1$
Strength comparison
$|r_1|>|r_2|\quad\Longrightarrow\quad r_1\text{ represents the stronger linear association}$
Coefficient of determination
$0\le r^2\le1$
Mean point
$(\bar{x},\bar{y})=\left(\frac{1}{n}\sum\limits_{i=1}^{n}x_i,\frac{1}{n}\sum\limits_{i=1}^{n}y_i\right)$
Gradient from two points on a line
$m=\frac{y_2-y_1}{x_2-x_1}$
Regression line of y on x
$\hat{y}=ax+b$
Regression coefficients
$a=r\frac{s_y}{s_x},\qquad b=\bar{y}-a\bar{x}$
Residual
$e=y-\hat{y}$
Correlation and regression toolkit
$-1\le r\le1,\qquad \hat{y}=ax+b,\qquad e=y-\hat{y}$
Pearson correlation coefficient
$r=\frac{\displaystyle\sum\limits_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\displaystyle\left[\sum\limits_{i=1}^{n}(x_i-\bar{x})^2\right]\left[\sum\limits_{i=1}^{n}(y_i-\bar{y})^2\right]}}$
Regression and residual
$\hat{y}=ax+b,\qquad e=y-\hat{y}$
Topic 4.5Not started
Probability: fundamental concepts
Probability formalizes uncertainty using events in a sample space.
250 min estimated
Topic 4.5 progress0%
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Learning objectives
Identify trials, outcomes, sample spaces and events.
Use equally likely outcomes when appropriate.
Use relative frequency as an empirical estimate.
Apply complements.
Calculate expected numbers of occurrences from repeated trials.
Key formulas
Counting outcomes in successive choices
$n(U)=n_1n_2\cdots n_k$
Event containment
$A\subseteq U$
Probability bounds
$0\le P(A)\le1$
Impossible and certain events
$P(\varnothing)=0,\qquad P(U)=1$
Equally likely outcomes
$P(A)=\frac{n(A)}{n(U)}$
Complement rule
$P(A')=1-P(A)$
Complement total
$P(A)+P(A')=1$
Mutually exclusive condition
$A\cap B=\varnothing\qquad\Longleftrightarrow\qquad P(A\cap B)=0$
Addition rule for mutually exclusive events
$P(A\cup B)=P(A)+P(B)$
Relative frequency
$\widehat{P}(A)=\frac{f_A}{N}$
Expected number of occurrences
$E(\text{number of occurrences of }A)=NP(A)$
General addition rule
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Multiplication rule
$P(A\cap B)=P(A)P(B\mid A)$
Independent-event multiplication rule
$P(A\cap B)=P(A)P(B)$
Core probability rules
$P(U)=1,\qquad P(\varnothing)=0,\qquad 0\le P(A)\le1$
Foundational probability toolkit
$P(A)=\frac{n(A)}{n(U)},\qquad P(A')=1-P(A),\qquad \widehat{P}(A)=\frac{f_A}{N}$
Probability scale and equally likely outcomes
$0\le P(A)\le1,\qquad P(A)=\frac{n(A)}{n(U)}$
Complement, relative frequency and expectation
$P(A')=1-P(A),\qquad \widehat{P}(A)=\frac{f_A}{N},\qquad E(\text{count})=NP(A)$
Topic 4.6Not started
Probability diagrams, combined events and conditional probability
Venn diagrams, tree diagrams and tables organize compound probability. Students should select the representation that makes the dependence structure clear.
250 min estimated
Topic 4.6 progress0%
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Learning objectives
Use Venn diagrams, trees and outcome tables.
Apply the addition rule.
Recognize mutually exclusive events.
Calculate conditional probabilities.
Recognize and use independence.
Translate words such as “given”, “at least” and “exactly” into correct events.
Key formulas
Two-set region counts
$n(A\text{ only})=n(A)-n(A\cap B),\qquad n((A\cup B)')=n(U)-n(A\cup B)$
De Morgan’s laws
$(A\cup B)'=A'\cap B',\qquad (A\cap B)'=A'\cup B'$
General addition rule
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Mutually exclusive condition
$A\cap B=\varnothing\qquad\Longleftrightarrow\qquad P(A\cap B)=0$
Disjoint addition
$P(A\cup B)=P(A)+P(B)$
Tree-diagram strategy
$P(E)=\sum\limits_{j=1}^{m}\prod_{k=1}^{s_j}p_{jk}$
Conditional probability
$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(B)>0$
Conditional probability from counts
$P(A\mid B)=\frac{n(A\cap B)}{n(B)}$
General multiplication rule
$P(A\cap B)=P(A)P(B\mid A)=P(B)P(A\mid B)$
Three-stage path
$P(A\cap B\cap C)=P(A)P(B\mid A)P(C\mid A\cap B)$
Independent-events product condition
$P(A\cap B)=P(A)P(B)$
Equivalent conditional condition
$P(A\mid B)=P(A),\qquad P(B)>0$
Combined-events toolkit
$P(A\cup B)=P(A)+P(B)-P(A\cap B),\qquad P(A\mid B)=\frac{P(A\cap B)}{P(B)}$
Addition and complement
$P(A\cup B)=P(A)+P(B)-P(A\cap B),\qquad P(A')=1-P(A)$
Conditional and multiplication rules
$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(A\cap B)=P(A)P(B\mid A)$
Independence test
$A\text{ and }B\text{ independent}\quad\Longleftrightarrow\quad P(A\cap B)=P(A)P(B)$
Topic 4.7Not started
Discrete random variables
A discrete random variable assigns numerical values to outcomes, allowing probability distributions and expected value to be analyzed.
250 min estimated
Topic 4.7 progress0%
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Learning objectives
Construct or interpret a discrete probability distribution.
Check that probabilities are valid and sum to one.
Calculate expected value.
Use expected value as a long-run average rather than a guaranteed single outcome.
Apply distributions to realistic decisions.
Key formulas
Valid probability values
$0\le P(X=x_i)\le1\quad\text{for }i=1,2,\ldots,n$
Total probability
$\sum\limits_{i=1}^{n}P(X=x_i)=1$
Normalization equation
$\sum\limits_{i=1}^{n}p_i=1$
Expected value
$E(X)=\mu=\sum\limits_{i=1}^{n}x_iP(X=x_i)$
Fair-game condition
$E(X)=0\quad\text{when }X\text{ is the player's net gain}$
Second moment
$E(X^2)=\sum\limits_{i=1}^{n}x_i^2P(X=x_i)$
Variance
$\operatorname{Var}(X)=\sum\limits_{i=1}^{n}(x_i-\mu)^2P(X=x_i)=E(X^2)-[E(X)]^2$
Standard deviation
$\sigma_X=\sqrt{\operatorname{Var}(X)}$
Expectation under a linear transformation
$E(aX+b)=aE(X)+b$
Variance under a linear transformation
$\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$
Standard deviation under a linear transformation
$\sigma_{aX+b}=|a|\sigma_X$
Probability total
$\sum\limits_{i=1}^{n}P(X=x_i)=1$
Expected value
$E(X)=\sum\limits_{i=1}^{n}x_iP(X=x_i)$
Variance shortcut
$\operatorname{Var}(X)=E(X^2)-[E(X)]^2$
Fair-game condition
$E(X)=0\quad\text{for signed net gain }X$
Topic 4.8Not started
Binomial distribution
The binomial model applies to a fixed number of independent Bernoulli trials with constant success probability.
250 min estimated
Topic 4.8 progress0%
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Learning objectives
Check the assumptions before selecting a binomial model.
Use X ~ B(n,p).
Calculate exact, cumulative and tail probabilities using technology.
Use the mean and variance.
Interpret the probability in context.
Key formulas
Binomial notation
$X\sim B(n,p)$
Failure probability
$q=1-p$
Binomial support
$x\in\{0,1,2,\ldots,n\}$
Exact binomial probability
$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$
Combination
$\binom{n}{x}=\frac{n!}{x!(n-x)!}$
GDC exact-probability input
$\operatorname{binompdf}(n,p,x)=P(X=x)$
GDC cumulative-probability input
$\operatorname{binomcdf}(n,p,x)=P(X\le x)$
At least r successes
$P(X\ge r)=1-P(X\le r-1)$
More than r successes
$P(X>r)=1-P(X\le r)$
Inclusive interval
$P(a\le X\le b)=P(X\le b)-P(X\le a-1)$
Binomial mean
$E(X)=\mu=np$
Binomial variance
$\operatorname{Var}(X)=\sigma^2=np(1-p)=npq$
Binomial standard deviation
$\sigma=\sqrt{np(1-p)}$
Binomial model
$X\sim B(n,p)$
Exact probability
$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$
At least r
$P(X\ge r)=1-P(X\le r-1)$
Mean
$E(X)=np$
Variance
$\operatorname{Var}(X)=np(1-p)$
Standard deviation
$\sigma=\sqrt{np(1-p)}$
Topic 4.9Not started
Normal distribution
The normal distribution models many continuous variables and supports probability and inverse-probability calculations.
250 min estimated
Topic 4.9 progress0%
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Learning objectives
Recognize the symmetric bell-shaped model.
Use X ~ N(mu, $sigma^{2}$ ).
Calculate interval and tail probabilities.
Use inverse normal calculations to find cut-off values.
Distinguish standard deviation from variance when entering technology.
Sketch or annotate the probability region.
Key formulas
Total probability
$P(-\infty<X<\infty)=1$
Center of a normal distribution
$\text{mean}=\text{median}=\text{mode}=\mu$
Area on either side of the mean
$P(X<\mu)=P(X>\mu)=\frac{1}{2}$
Normal-distribution notation
$X\sim N(\mu,\sigma^2)$
Expected value
$E(X)=\mu$
Variance
$\operatorname{Var}(X)=\sigma^2$
Standard deviation
$\operatorname{SD}(X)=\sigma=\sqrt{\operatorname{Var}(X)}$
Within one standard deviation
$P(\mu-\sigma<X<\mu+\sigma)\approx0.68$
Within two standard deviations
$P(\mu-2\sigma<X<\mu+2\sigma)\approx0.95$
Within three standard deviations
$P(\mu-3\sigma<X<\mu+3\sigma)\approx0.997$
Complement rule
$P(X>a)=1-P(X\le a)$
Interval from cumulative probabilities
$P(a<X<b)=P(X<b)-P(X\le a)$
Probability at one point
$P(X=a)=0$
Mirror-tail symmetry
$P(X<\mu-d)=P(X>\mu+d)$
Each tail outside a central area
$P(X<L)=P(X>U)=\frac{1-c}{2}$
Interval probability on a GDC
$P(a<X<b)=\operatorname{normalcdf}(a,b,\mu,\sigma)$
Lower-tail probability
$P(X<b)=\operatorname{normalcdf}(-\infty,b,\mu,\sigma)$
Upper-tail probability
$P(X>a)=\operatorname{normalcdf}(a,\infty,\mu,\sigma)$
Quantile definition
$P(X\le x_q)=q$
Inverse-normal command
$x_q=\operatorname{invNorm}(q,\mu,\sigma)$
Convert an upper-tail area
$P(X>x)=p\quad\Longrightarrow\quad P(X\le x)=1-p$
Highest p cutoff
$P(X>k)=p\quad\Longrightarrow\quad k=\operatorname{invNorm}(1-p,\mu,\sigma)$
Central interval tail area
$\text{each tail}=\frac{1-c}{2}$
Central interval boundaries
$L=\operatorname{invNorm}\!\left(\frac{1-c}{2},\mu,\sigma\right),\quad U=\operatorname{invNorm}\!\left(\frac{1+c}{2},\mu,\sigma\right)$
Normal density
$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$
Inflection points
$x=\mu-\sigma\quad\text{and}\quad x=\mu+\sigma$
Expected count
$\text{expected count}=nP(\text{event})$
Normal model
$X\sim N(\mu,\sigma^2)$
Total area
$P(-\infty<X<\infty)=1$
Symmetry
$P(X<\mu-d)=P(X>\mu+d)$
Normal CDF
$P(a<X<b)=\operatorname{normalcdf}(a,b,\mu,\sigma)$
Inverse normal
$P(X\le x_q)=q\quad\Longrightarrow\quad x_q=\operatorname{invNorm}(q,\mu,\sigma)$
Topic 4.10Not started
Spearman rank correlation
Spearman correlation measures monotonic association using ranks and can be more appropriate than Pearson correlation when data are ordinal or strongly affected by nonlinearity/outliers.
260 min estimated
Topic 4.10 progress0%
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Learning objectives
Rank paired data consistently, including ties as instructed.
Calculate or obtain Spearman rank correlation.
Interpret sign and strength.
Compare when Pearson or Spearman is more suitable.
Discuss the effect of outliers and monotonic but non-linear patterns.
Key formulas
Untied rank set
$R_i\in\{1,2,\ldots,n\}$
Average rank for a tie
$\overline{R}=\frac{r+(r+1)+\cdots+s}{s-r+1}$
Rank difference
$d_i=R_{x,i}-R_{y,i}$
Sum of squared rank differences
$\sum\limits_{i=1}^{n}d_i^2=d_1^2+d_2^2+\cdots+d_n^2$
Spearman shortcut without ties
$r_s=1-\frac{6\sum\limits_{i=1}^{n}d_i^2}{n(n^2-1)}$
Spearman as correlation of ranks
$r_s=r(R_x,R_y)$
Spearman range
$-1\le r_s\le1$
No-ties Spearman formula
$r_s=1-\frac{6\sum\limits_{i=1}^{n}d_i^2}{n(n^2-1)}$
Tied-rank method
$r_s=r(R_x,R_y)$
Coefficient range
$-1\le r_s\le1$
Topic 4.11Not started
Hypothesis testing: chi-square and two-sample t procedures
The shared AI course introduces formal testing through null/alternative hypotheses, significance levels and p-values.
300 min estimated
Topic 4.11 progress0%
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Learning objectives
State H0 and H1 in terms of the context.
Choose one-tailed or two-tailed alternatives appropriately.
Use chi-square tests for independence and goodness of fit.
Use the required t-test for comparing two population means.
Compare p-value with significance level.
Write a conclusion about evidence in the original context rather than “accepting” H0.
Key formulas
p-value decision rule
$p\le\alpha\Rightarrow\text{reject }H_0;\qquad p>\alpha\Rightarrow\text{do not reject }H_0$
Critical-value decision for chi-square
$\chi^2\ge\chi^2_{\mathrm{crit}}\Rightarrow\text{reject }H_0$
Type I error probability
$P(\text{Type I error})=\alpha$
Power of a test
$\text{power}=1-\beta$
Expected count from a stated probability
$E_i=Np_i$
One chi-square contribution
$\frac{(O-E)^2}{E}$
Chi-square statistic
$\chi^2=\sum\limits_{i=1}^{k}\frac{(O_i-E_i)^2}{E_i}$
Goodness-of-fit degrees of freedom
$df=k-1$
Expected frequency for independence
$E_{ij}=\frac{(\text{row total}_i)(\text{column total}_j)}{N}$
Independence-test degrees of freedom
$df=(r-1)(c-1)$
Null hypothesis for two means
$H_0:\mu_1-\mu_2=0$
Two-tailed alternative
$H_1:\mu_1-\mu_2\ne0$
Right-tailed alternative
$H_1:\mu_1-\mu_2>0$
Left-tailed alternative
$H_1:\mu_1-\mu_2<0$
Pooled variance
$s_p^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$
Pooled two-sample t statistic
$t=\frac{(\bar{x}_1-\bar{x}_2)-\Delta_0}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$
Pooled t-test degrees of freedom
$df=n_1+n_2-2$
Goodness-of-fit GDC inputs
$\text{observed list},\ \text{expected list},\ df=k-1$
Independence GDC input
$\text{observed contingency matrix}$
Two-sample t GDC inputs
$\bar{x}_1,s_1,n_1,\bar{x}_2,s_2,n_2,\Delta_0=0$
Universal decision rule
$p\le\alpha\Rightarrow\text{reject }H_0;\qquad p>\alpha\Rightarrow\text{do not reject }H_0$
Goodness-of-fit expected count
$E_i=Np_i$
Independence expected count
$E_{ij}=\frac{(\text{row total}_i)(\text{column total}_j)}{N}$
Independence degrees of freedom
$df=(r-1)(c-1)$
Topic 4.12Not started
Data collection and study design
Plan, conduct and critique surveys, observational studies and randomized experiments so that conclusions match the evidence collected.
235 min estimated
Topic 4.12 progress0%
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Learning objectives
Distinguish a population, target population, sampling frame, census and sample.
Distinguish a population parameter from a sample statistic.
Choose and justify simple random, systematic, stratified, cluster or multistage sampling.
Recognize when convenience, voluntary-response and quota samples are biased.
Separate sampling error from non-sampling error.
Distinguish an observational study from a randomized experiment.
Identify experimental units, factors, levels, treatments and response variables.
Explain the roles of control groups, placebos and blinding.
Distinguish random sampling from random assignment and state what each permits.
Explain how randomization reduces systematic allocation bias.
Explain replication and recognize pseudoreplication.
Use blocking or matched pairs to control known sources of variability.
Identify lurking and confounding variables.
Identify selection, undercoverage, non-response, response and measurement bias.
Distinguish reliability, validity, accuracy and precision.
Write neutral survey questions with clear operational definitions.
Design a feasible and ethical statistical investigation.
Choose suitable variables and a data structure before collection.
Evaluate whether a conclusion supports association, causation or generalization.
Use technology for random selection, random assignment and transparent data checking.
Communicate limitations without claiming that a large sample fixes a biased design.
Answer IB AI HL study-design questions with design-specific evidence and contextual conclusions.
Key formulas
Study-design principle
$\text{strength of conclusion}\leq\text{strength of design}$
Sampling fraction
$f=\frac{n}{N}$
Sample mean
$\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_i$
Sample proportion
$\hat p=\frac{x}{n}$
Coverage rate
$\text{coverage rate}=\frac{\text{eligible units on frame}}{\text{eligible units in target population}}$
Equal inclusion probability in an SRS
$\Pr(\text{a particular unit is selected})=\frac{n}{N}$
Systematic interval
$k\approx\frac{N}{n}$
Random start
$j\in\{1,2,\ldots,k\}$
Proportional stratum size
$n_h=n\frac{N_h}{N}$
Weighted stratified mean
$\hat\mu_{\mathrm{strat}}=\sum\limits_{h=1}^{H}\frac{N_h}{N}\bar{x}_h$
Overall inclusion probability
$\pi_i=\prod_{s=1}^{S}\Pr(\text{selected at stage }s\mid\text{previous stages})$
Unknown inclusion probability
$\Pr(\text{selection of unit }i)\text{ is not known}$
Typical sampling variability pattern
$\mathrm{SE}\propto\frac{1}{\sqrt{n}}$
Estimator bias
$\operatorname{Bias}(\hat\theta)=E(\hat\theta)-\theta$
Observed association
$\text{association}=\text{possible effect}+\text{confounding}+\text{chance}+\text{bias}$
Estimated treatment effect
$\widehat\tau=\bar{x}_{T}-\bar{x}_{C}$
Number of treatments in a full factorial design
$t=\prod_{j=1}^{q}L_j$
Net treatment contrast
$\text{net effect}=\bar{x}_{\text{active}}-\bar{x}_{\text{control}}$
Inference map
$\text{random sample}\Rightarrow\text{generalize},\qquad\text{random assignment}\Rightarrow\text{causal comparison}$
Equal-sized allocation count
$\text{number of allocations}=\binom{n}{n_T}$
Independent replication count
$n_{\mathrm{rep}}=\text{number of independently assigned experimental units}$
Matched difference
$d_i=x_{i,A}-x_{i,B}$
Mean matched difference
$\bar d=\frac{1}{n}\sum\limits_{i=1}^{n}d_i$
Confounder conditions
$C\leftrightarrow X\quad\text{and}\quad C\to Y$
Response rate
$\text{response rate}=\frac{\text{completed eligible responses}}{\text{eligible people contacted}}\times100\%$
Attrition rate
$\text{attrition rate}=\frac{n_{\mathrm{enrolled}}-n_{\mathrm{completed}}}{n_{\mathrm{enrolled}}}\times100\%$
Measurement model
$X_{\mathrm{observed}}=X_{\mathrm{true}}+\text{systematic error}+\text{random error}$
Key distinction
$\text{reliability}\ne\text{validity}$
Category requirement
$\text{response categories}=\text{mutually exclusive}+\text{collectively exhaustive}$
Average of repeated readings
$\bar{x}_{\mathrm{unit}}=\frac{1}{m}\sum\limits_{j=1}^{m}x_j$
Data minimization principle
$\text{collected data}\subseteq\text{data necessary for the stated purpose}$
Expected cell count
$E_{ij}=\frac{(\text{row }i\text{ total})(\text{column }j\text{ total})}{n}$
Chi-square degrees of freedom
$\mathrm{df}=(r-1)(c-1)$
Design chain
$\text{question}\to\text{population}\to\text{design}\to\text{measurement}\to\text{analysis}\to\text{conclusion}$
Conclusion scope
$\text{scope}=\text{population represented}\cap\text{conditions studied}$
Missing-data rate
$\text{missing rate}=\frac{n_{\mathrm{missing}}}{n_{\mathrm{expected}}}\times100\%$
Random sort key
$u_i\sim U(0,1)$
Correction rule
$\text{design feature}\to\text{purpose}\to\text{permitted conclusion}$
IB response pattern
$\text{identify}\to\text{apply to context}\to\text{justify}\to\text{conclude}$
Proportional allocation
$n_h=n\frac{N_h}{N}$
Response rate
$R=\frac{n_{\mathrm{complete}}}{n_{\mathrm{eligible\ contacted}}}\times100\%$
Stratified allocation
$n_h=n\frac{N_h}{N}$
Treatment contrast
$\widehat\tau=\bar{x}_{T}-\bar{x}_{C}$
Topic 4.13Not started
Nonlinear regression and goodness of fit
Fit, compare and critique linear and nonlinear regression models using parameters, residuals, residual plots, goodness-of-fit measures and contextual reasoning.
245 min estimated
Topic 4.13 progress0%
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Learning objectives
Review the least-squares linear regression model and interpret its parameters.
Explain why a curved relationship may require nonlinear regression.
Fit and interpret quadratic and higher-degree polynomial models.
Fit and interpret exponential growth or decay models.
Fit and interpret power and logarithmic models.
Recognize when logistic, sinusoidal or other contextual models are appropriate.
Choose candidate models from scatter-plot shape and contextual theory.
Interpret regression parameters with units and domain restrictions.
Calculate and interpret a residual as observed minus predicted.
Construct and read residual plots.
Recognize curvature, unequal spread, clustering and dependence in residual plots.
Identify outliers, high-leverage points and influential observations.
Calculate and interpret the sum of squared errors and root mean squared error.
Calculate and interpret the coefficient of determination.
Explain why a high coefficient of determination does not prove model validity.
Use adjusted goodness-of-fit measures cautiously when model complexity differs.
Compare competing models using the same response scale and data set.
Explain how transformed regression changes the fitting criterion.
Distinguish interpolation from extrapolation.
Evaluate model reliability using data quality, domain, residuals and context.
Use validation or holdout data to assess predictive performance.
Use a GDC or spreadsheet to fit models, store predictions and graph residuals.
Avoid overfitting and false precision in reported regression models.
Answer IB AI HL regression questions with calculations, diagnostics and contextual conclusions.
Key formulas
Model-quality principle
$\text{useful fit}=\text{shape}+\text{residuals}+\text{context}+\text{valid domain}$
Paired observation
$(x_i,y_i),\qquad i=1,2,\ldots,n$
Linear regression model
$\hat y=a+bx$
Gradient meaning
$b=\frac{\text{change in predicted }y}{\text{one-unit increase in }x}$
Residual
$e_i=y_i-\hat y_i$
Least-squares objective
$\mathrm{SSE}=\sum\limits_{i=1}^{n}(y_i-\hat y_i)^2=\sum\limits_{i=1}^{n}e_i^2$
Nonlinear mean relation
$\hat y=f(x;\theta_1,\theta_2,\ldots,\theta_p)$
Quadratic model
$\hat y=ax^2+bx+c$
Vertex abscissa
$x_{\mathrm v}=-\frac{b}{2a}$
Degree-$d$ polynomial
$\hat y=a_dx^d+a_{d-1}x^{d-1}+\cdots+a_1x+a_0$
Parameter count
$p=d+1$
Exponential model
$\hat y=ab^x=ae^{kx}$
Parameter conversion
$b=e^k,\qquad k=\ln b$
Percentage change
$100(b-1)\%$
Power model
$\hat y=ax^b$
Log-linear form
$\ln\hat y=\ln a+b\ln x$
Elasticity
$\frac{d\ln y}{d\ln x}=b$
Logarithmic model
$\hat y=a+b\ln x,\qquad x>0$
Multiplicative-input effect
$\hat y(cx)-\hat y(x)=b\ln c$
Logistic growth model
$\hat y=\frac{L}{1+Ae^{-kx}}$
Inflection location
$x_{\mathrm{inf}}=\frac{\ln A}{k},\qquad \hat y=\frac{L}{2}$
Sinusoidal model
$\hat y=D+A\sin(B(x-C))$
Period
$T=\frac{2\pi}{|B|}$
Selection rule
$\text{candidate}\xrightarrow{\text{fit}}\text{diagnostics}\xrightarrow{\text{context}}\text{final model}$
Continuous-to-discrete rate
$r_{\%}=100(e^k-1)\%$
Quadratic curvature
$\frac{d^2\hat y}{dx^2}=2a$
Residual
$e_i=y_i-f(x_i;\hat\theta)$
Percentage residual
$100\frac{y_i-\hat y_i}{y_i}\%$
Residual-plot coordinates
$(x_i,e_i)\quad\text{or}\quad(\hat y_i,e_i)$
Adequate mean condition
$E(e\mid x)\approx0$
Constant-spread ideal
$\operatorname{Var}(e\mid x)\approx\sigma^2$
Residual size
$|e_i|=|y_i-\hat y_i|$
Sensitivity change
$\Delta\hat\theta=\hat\theta_{\mathrm{all}}-\hat\theta_{(-i)}$
Sum of squared errors
$\mathrm{SSE}=\sum\limits_{i=1}^{n}e_i^2$
Root mean squared error
$\mathrm{RMSE}=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}e_i^2}$
Residual standard error
$s_e=\sqrt{\frac{\mathrm{SSE}}{n-p}}$
Total sum of squares
$\mathrm{SST}=\sum\limits_{i=1}^{n}(y_i-\bar y)^2$
Coefficient of determination
$R^2=1-\frac{\mathrm{SSE}}{\mathrm{SST}}$
Adjusted coefficient of determination
$R^2_{\mathrm{adj}}=1-\frac{\frac{\mathrm{SSE}}{(n-p)}}{\frac{\mathrm{SST}}{(n-1)}}$
Fair comparison requirement
$\{(x_i,y_i)\}_{i=1}^{n}\text{ and response scale must match}$
Exponential linearization
$\ln y=\ln a+x\ln b$
Power linearization
$\ln y=\ln a+b\ln x$
Interpolation condition
$x_{\min}\leq x_0\leq x_{\max}$
Extrapolation condition
$x_0<x_{\min}\quad\text{or}\quad x_0>x_{\max}$
Reliability checklist
$\text{data quality}+\text{diagnostics}+\text{domain}+\text{stability}$
Validation RMSE
$\mathrm{RMSE}_{\mathrm{val}}=\sqrt{\frac{1}{m}\sum\limits_{j=1}^{m}(y_j-\hat y_j)^2}$
Technology residual list
$L_e=L_y-f(L_x)$
Prediction
$\hat y_0=f(x_0)$
Safe model-choice rule
$\text{choose}\ne\arg\max R^2\text{ alone}$
IB comparison structure
$\text{equation/fit}+\text{residuals}+\text{context}+\text{conclusion}$
Residual
$e=y-\hat y$
SSE
$\mathrm{SSE}=\sum\limits_{i=1}^{n}e_i^2$
Linear
$\hat y=a+bx$
Quadratic
$\hat y=ax^2+bx+c$
Exponential
$\hat y=ab^x$
Power
$\hat y=ax^b$
Logarithmic
$\hat y=a+b\ln x$
Logistic
$\hat y=\frac{L}{1+Ae^{-kx}}$
Topic 4.14Not started
Linear transformations of random variables
Affine transformations and linear combinations change the centre and spread of random variables in predictable, useful ways.
185 min estimated
Topic 4.14 progress0%
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Learning objectives
Distinguish a random variable from an observed value and describe its probability distribution.
Calculate and interpret expected value, variance and standard deviation.
Derive and use the expectation rule for a linear transformation.
Derive and use the variance rule for a linear transformation.
Explain why a translation changes the mean but not the spread.
Explain why a scale factor changes variance by its square and standard deviation by its absolute value.
Interpret a negative scale factor as a reflection and rescaling.
Transform the support and probabilities of a discrete random variable.
Reverse a linear transformation when the transformed statistics are known.
Standardize a random variable and interpret a z-score.
Convert units while transforming means and measures of spread correctly.
Use linearity of expectation for sums, differences and weighted totals.
Recognize that linearity of expectation does not require independence.
Use covariance to find the variance of a linear combination.
Apply independence to simplify variances of sums and differences.
Explain why variances add when independent random variables are subtracted.
Find the mean, variance and standard deviation of an independent total.
Find the mean and standard error of an independent sample mean.
Transform normal random variables and state the resulting distribution.
Model costs, revenues, temperatures and measurements with affine transformations.
Use a GDC or spreadsheet to verify transformed distributions and summary statistics.
Communicate assumptions, units and independence conditions in IB AI HL solutions.
Detect common errors involving variance, standard deviation and covariance.
Solve multi-stage IB AI HL problems involving transformations and combinations of random variables.
Key formulas
Expectation
$E(aX+b)=aE(X)+b$
Variance
$\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$
Standard deviation
$\sigma_{aX+b}=|a|\sigma_X$
Discrete total probability
$\sum\limits_{i=1}^{n}P(X=x_i)=1$
Discrete expectation
$E(X)=\sum\limits_{i=1}^{n}x_iP(X=x_i)$
Expectation of a constant
$E(c)=c$
Variance definition
$\operatorname{Var}(X)=E\left[(X-\mu_X)^2\right]$
Variance identity
$\operatorname{Var}(X)=E(X^2)-[E(X)]^2$
Standard deviation
$\sigma_X=\sqrt{\operatorname{Var}(X)}$
Second moment
$E(X^2)=\sum\limits_{i=1}^{n}x_i^2P(X=x_i)$
Linearity for one variable
$E(aX+b)=aE(X)+b$
Linearity for several variables
$E\left(\sum\limits_{i=1}^{n}c_iX_i+d\right)=\sum\limits_{i=1}^{n}c_iE(X_i)+d$
Deviation transformation
$Y-E(Y)=a[X-E(X)]$
Variance transformation
$\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$
Standard-deviation transformation
$\sigma_{aX+b}=|a|\sigma_X$
Probability transfer
$P(Y=ax_i+b)=P(X=x_i),\quad a\ne0$
Inverse variable
$X=\frac{Y-b}{a}$
Inverse mean
$E(X)=\frac{E(Y)-b}{a}$
Inverse variance
$\operatorname{Var}(X)=\frac{\operatorname{Var}(Y)}{a^2}$
Inverse standard deviation
$\sigma_X=\frac{\sigma_Y}{|a|}$
z-score
$z=\frac{x-\mu_X}{\sigma_X}$
Standardized mean
$E(Z)=0$
Standardized variance
$\operatorname{Var}(Z)=1$
Two-variable expectation
$E(aX+bY+c)=aE(X)+bE(Y)+c$
Sum expectation
$E(X+Y)=E(X)+E(Y)$
Difference expectation
$E(X-Y)=E(X)-E(Y)$
Covariance definition
$\operatorname{Cov}(X,Y)=E[(X-\mu_X)(Y-\mu_Y)]$
General variance
$\operatorname{Var}(aX+bY)=a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Y)+2ab\operatorname{Cov}(X,Y)$
Variance of a sum
$\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y)$
Variance of a difference
$\operatorname{Var}(X-Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)-2\operatorname{Cov}(X,Y)$
Independent covariance
$X\perp Y\implies\operatorname{Cov}(X,Y)=0$
Independent sum variance
$\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)$
Independent difference variance
$\operatorname{Var}(X-Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)$
Weighted mean
$E(T)=w_1E(X)+w_2E(Y)$
Weighted variance
$\operatorname{Var}(T)=w_1^2\operatorname{Var}(X)+w_2^2\operatorname{Var}(Y)+2w_1w_2\operatorname{Cov}(X,Y)$
Independent total mean
$E(S_n)=n\mu$
Independent total variance
$\operatorname{Var}(S_n)=n\sigma^2$
Independent total standard deviation
$\sigma_{S_n}=\sqrt n\,\sigma$
Sample mean
$\bar X=\frac1n\sum\limits_{i=1}^{n}X_i$
Mean of sample mean
$E(\bar X)=\mu$
Variance of sample mean
$\operatorname{Var}(\bar X)=\frac{\sigma^2}{n}$
Standard error
$\sigma_{\bar X}=\frac{\sigma}{\sqrt n}$
Normal affine transformation
$X\sim N(\mu,\sigma^2)\Longrightarrow aX+b\sim N(a\mu+b,a^2\sigma^2)$
Normal standardization
$Z=\frac{X-\mu}{\sigma}\sim N(0,1)$
Independent normal combination
$aX+bY+c\sim N(a\mu_X+b\mu_Y+c,\ a^2\sigma_X^2+b^2\sigma_Y^2)$
Binomial mean
$E(X)=np$
Binomial variance
$\operatorname{Var}(X)=np(1-p)$
Transformed binomial mean
$E(Y)=anp+b$
Transformed binomial variance
$\operatorname{Var}(Y)=a^2np(1-p)$
Composed affine transformation
$Z=c(aX+b)+d=(ca)X+(cb+d)$
Constant expectation
$E(b)=b$
Constant variance
$\operatorname{Var}(b)=0$
Affine mean
$E(aX+b)=a\mu_X+b$
Affine variance
$\operatorname{Var}(aX+b)=a^2\sigma_X^2$
Affine standard deviation
$\sigma_{aX+b}=|a|\sigma_X$
General combination variance
$\operatorname{Var}(aX+bY)=a^2\sigma_X^2+b^2\sigma_Y^2+2ab\operatorname{Cov}(X,Y)$
Independent total
$E(S_n)=n\mu,\quad \operatorname{Var}(S_n)=n\sigma^2,\quad \sigma_{S_n}=\sqrt n\sigma$
Independent average
$E(\bar X)=\mu,\quad \operatorname{Var}(\bar X)=\frac{\sigma^2}{n},\quad \sigma_{\bar X}=\frac{\sigma}{\sqrt n}$
Standardization
$Z=\frac{X-\mu}{\sigma}$
Topic 4.15Not started
Normal variables, sums and the Central Limit Theorem
Exact normal combinations and the Central Limit Theorem connect population models, totals and sample means to practical probability calculations.
190 min estimated
Topic 4.15 progress0%
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Learning objectives
Describe a normal random variable using its mean and variance or standard deviation.
Standardize normal observations and interpret z-scores.
Calculate normal probabilities and quantiles using a GDC.
Transform a normal random variable under an affine rule.
Find the distribution of a sum of independent normal variables.
Find the distribution of a difference of independent normal variables.
Find the distribution of a general linear combination of independent normal variables.
Include covariance when random variables are not independent.
Find the mean, variance and standard deviation of an independent total.
Find the exact distribution of a sample mean from a normal population.
Distinguish the population distribution from a sampling distribution.
Distinguish a total from a sample mean and use the correct spread.
Explain the Central Limit Theorem in student-friendly language.
State the conditions and assumptions required for a CLT approximation.
Judge whether a sample size is adequate from the population shape and context.
Apply the CLT to approximate the distribution of a sample mean.
Apply the CLT to approximate the distribution of an independent sum.
Standardize sample means and sums using the correct standard error.
Find probability cutoffs using inverse-normal technology.
Apply continuity correction when approximating a discrete count by a normal model.
Use a finite-population correction where sampling without replacement makes it necessary.
Model quality control, demand, waiting times, polling and finance applications.
Use calculator/GDC functions accurately and report technology inputs.
Identify common mistakes involving variance, independence, normality and sample size.
Communicate assumptions, units and approximation language in IB AI HL solutions.
Solve multi-stage IB AI HL problems involving exact normal results and CLT approximations.
Key formulas
Mean of sample mean
$E(\bar X)=\mu$
Variance of sample mean
$\operatorname{Var}(\bar X)=\frac{\sigma^2}{n}$
Standard error
$\sigma_{\bar X}=\frac{\sigma}{\sqrt n}$
Normal notation
$X\sim N(\mu,\sigma^2)$
Variance-standard deviation link
$\operatorname{Var}(X)=\sigma^2$
Total density area
$P(-\infty<X<\infty)=1$
z-score
$Z=\frac{X-\mu}{\sigma}$
Standard normal
$Z\sim N(0,1)$
Reverse standardization
$X=\mu+Z\sigma$
Upper-tail complement
$P(X>a)=1-P(X\le a)$
Between probability
$P(a<X<b)=P(X<b)-P(X\le a)$
Affine normal rule
$aX+b\sim N(a\mu+b,a^2\sigma^2)$
Independent normal sum
$X+Y\sim N(\mu_X+\mu_Y,\sigma_X^2+\sigma_Y^2)$
Independent normal difference
$X-Y\sim N(\mu_X-\mu_Y,\sigma_X^2+\sigma_Y^2)$
Combination mean
$E(T)=\sum\limits_{i=1}^{k}a_i\mu_i+c$
Independent combination variance
$\operatorname{Var}(T)=\sum\limits_{i=1}^{k}a_i^2\sigma_i^2$
Independent normal combination
$T\sim N\left(\sum\limits_{i=1}^{k}a_i\mu_i+c,\sum\limits_{i=1}^{k}a_i^2\sigma_i^2\right)$
General two-variable variance
$\operatorname{Var}(aX+bY)=a^2\sigma_X^2+b^2\sigma_Y^2+2ab\operatorname{Cov}(X,Y)$
Covariance from correlation
$\operatorname{Cov}(X,Y)=\rho\sigma_X\sigma_Y$
Normal total distribution
$S_n\sim N(n\mu,n\sigma^2)$
Normal total SD
$\sigma_{S_n}=\sqrt n\,\sigma$
Exact normal sample mean
$\bar X\sim N\left(\mu,\frac{\sigma^2}{n}\right)$
Sample-mean standard error
$\sigma_{\bar X}=\frac{\sigma}{\sqrt n}$
Total
$S_n=\sum\limits_{i=1}^{n}X_i$
Average
$\bar X=\frac{S_n}{n}$
CLT standardized sum
$\frac{S_n-n\mu}{\sigma\sqrt n}\xrightarrow{d}N(0,1)$
CLT sample-mean approximation
$\bar X\approx N\left(\mu,\frac{\sigma^2}{n}\right)$
Standard-error ratio
$\frac{\operatorname{SE}_{n_2}}{\operatorname{SE}_{n_1}}=\sqrt{\frac{n_1}{n_2}}$
Approximate sample-mean distribution
$\bar X\approx N\left(\mu,\frac{\sigma^2}{n}\right)$
Sample-mean z-score
$Z=\frac{\bar X-\mu}{\frac{\sigma}{\sqrt n}}$
Approximate sum distribution
$S_n\approx N(n\mu,n\sigma^2)$
Sum z-score
$Z=\frac{S_n-n\mu}{\sigma\sqrt n}$
Generic statistic z-score
$Z=\frac{T-E(T)}{\sqrt{\operatorname{Var}(T)}}$
Mean z-score
$Z=\frac{\bar X-\mu}{\frac{\sigma}{\sqrt n}}$
Sample-mean percentile
$x_p=\mu+z_p\frac{\sigma}{\sqrt n}$
Bernoulli mean and variance
$E(X)=p,\quad \operatorname{Var}(X)=p(1-p)$
Sample-proportion approximation
$\hat p\approx N\left(p,\frac{p(1-p)}{n}\right)$
Binomial normal approximation
$X\sim B(n,p)\approx N(np,np(1-p))$
Finite-population standard error
$\operatorname{SE}(\bar X)=\frac{\sigma}{\sqrt n}\sqrt{\frac{N-n}{N-1}}$
Independent normal combination
$\sum\limits_{i=1}^{k}a_iX_i+c\sim N\left(\sum\limits_{i=1}^{k}a_i\mu_i+c,\sum\limits_{i=1}^{k}a_i^2\sigma_i^2\right)$
Independent total
$S_n\sim N(n\mu,n\sigma^2)\quad\text{exactly if X is normal, approximately by CLT otherwise}$
Sample mean
$\bar X\sim N\left(\mu,\frac{\sigma^2}{n}\right)\quad\text{exactly if X is normal, approximately by CLT otherwise}$
Mean standard error
$\operatorname{SE}(\bar X)=\frac{\sigma}{\sqrt n}$
Sum standard deviation
$\sigma_{S_n}=\sigma\sqrt n$
Mean z-score
$z=\frac{\bar x-\mu}{\frac{\sigma}{\sqrt n}}$
Sum z-score
$z=\frac{s-n\mu}{\sigma\sqrt n}$
Topic 4.16Not started
Confidence intervals
Confidence intervals turn sample evidence into an estimate of an unknown population mean while making sampling uncertainty, assumptions and precision explicit.
225 min estimated
Topic 4.16 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Distinguish a population parameter from a sample statistic.
Explain estimation as learning about an unknown parameter from a sample.
Distinguish point estimates from interval estimates.
Identify the sample mean as a point estimate of the population mean.
Explain the role of a sampling distribution in interval estimation.
Calculate and interpret the standard error of a sample mean.
Explain confidence level as long-run coverage of a method.
Interpret a confidence interval correctly in context.
Reject common probability and population-spread misinterpretations.
Identify the estimate, critical value, standard error and margin of error.
Construct a z confidence interval for a mean when population standard deviation is known.
Construct a t confidence interval for a mean when population standard deviation is unknown.
Select the appropriate z or t procedure.
Use degrees of freedom correctly in a one-sample t interval.
Check randomness, independence and distributional conditions.
Calculate and interpret margin of error and interval width.
Explain how sample size affects interval precision.
Explain how confidence level affects interval width.
Explain how population or sample variability affects interval width.
Plan a minimum sample size for a target margin of error.
Round a required sample size upward.
Apply a finite-population correction when the sampling fraction is large.
Use a GDC to obtain one-sample z and t intervals.
Interpret calculator output rather than reporting endpoints alone.
Relate a two-sided confidence interval to a matching hypothesis test.
Distinguish statistical significance from practical importance.
Recognize that a narrow interval can still be biased.
Solve multi-stage and real-world confidence-interval problems.
Communicate assumptions, units and context in IB-style solutions.
Audit a completed solution for method, arithmetic and interpretation errors.
Key formulas
Sample mean
$\bar x=\frac{1}{n}\sum\limits_{i=1}^{n}x_i$
Target parameter
$\mu=E(X)$
Generic point estimate
$\hat\theta$
Generic interval
$\hat\theta\pm E$
Mean of sample mean
$E(\bar X)=\mu$
Standard error, known sigma
$SE(\bar X)=\frac{\sigma}{\sqrt n}$
Estimated standard error
$\widehat{SE}(\bar X)=\frac{s}{\sqrt n}$
Confidence level
$C=1-\alpha$
Two-tail area
$\frac{\alpha}{2}=\frac{1-C}{2}$
Interval notation
$L<\mu<U$
Interval centre
$\bar x=\frac{L+U}{2}$
Interval template
$\text{estimate}\pm(\text{critical value})(\text{standard error})$
Margin of error
$E=(\text{critical value})(SE)$
Width
$U-L=2E$
z interval for mean
$\bar x\pm z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt n}$
z margin of error
$E=z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt n}$
Upper cutoff area
$P(Z\le z_{\frac{\alpha}{2}})=1-\frac{\alpha}{2}$
Key z values
$z^*_{90\%}\approx1.645,\quad z^*_{95\%}\approx1.960,\quad z^*_{99\%}\approx2.576$
t interval for mean
$\bar x\pm t_{\frac{\alpha}{2},n-1}\frac{s}{\sqrt n}$
t margin of error
$E=t_{\frac{\alpha}{2},n-1}\frac{s}{\sqrt n}$
Degrees of freedom
$df=n-1$
z standard error
$SE=\frac{\sigma}{\sqrt n}$
t estimated standard error
$\widehat{SE}=\frac{s}{\sqrt n}$
Sampling fraction
$f=\frac{n}{N}$
Usual independence check
$n\le0.10N$
Margin from endpoints
$E=\frac{U-L}{2}$
Endpoints
$L=\bar x-E,\quad U=\bar x+E$
Sample-size scaling
$E\propto\frac{1}{\sqrt n}$
Margin ratio
$\frac{E_2}{E_1}=\sqrt{\frac{n_1}{n_2}}$
Confidence–width link
$C\uparrow\ \Rightarrow\ z^*\text{ or }t^*\uparrow\ \Rightarrow\ E\uparrow$
z margin at fixed SE
$E=z^*SE$
Variability–width link
$E\propto\sigma\quad\text{or}\quad E\propto s$
Scale transformation
$CI(aX+b)=a\,CI(X)+b\quad(a>0)$
Required sample size
$n\ge\left(\frac{z^*\sigma}{E}\right)^2$
Planning rule
$n_{min}=\left\lceil\left(\frac{z^*\sigma}{E}\right)^2\right\rceil$
Finite-population standard error
$SE=\frac{\sigma}{\sqrt n}\sqrt{\frac{N-n}{N-1}}$
GDC z inputs
$(\bar x,\sigma,n,C)$
GDC z output
$(L,U)$
GDC t inputs
$(\bar x,s,n,C)$
Implicit degrees of freedom
$df=n-1$
Report format
$CI_C(\mu)=(L,U)$
Consistent rounding
$(L,U)\mapsto(\operatorname{round}(L),\operatorname{round}(U))$
Quality-control standard error
$SE=\frac{1.8}{\sqrt{81}}=0.2$
Quality-control interval
$249.4\pm1.96(0.2)$
Service-time SE
$SE=\frac{7.5}{\sqrt{20}}$
Service-time interval
$38.6\pm1.729\frac{7.5}{\sqrt{20}}$
Centre from bounds
$\bar x=\frac{L+U}{2}$
Recover sigma
$\sigma=\frac{E\sqrt n}{z^*}$
Matching levels
$C=1-\alpha$
Decision by interval
$\mu_0\notin CI\Rightarrow\text{reject }H_0$
Effect versus threshold
$CI\cap\{\text{practically important values}\}$
Null comparison
$0\in CI\ \text{or}\ 0\notin CI$
Design chain
$\text{question}\to\text{sample}\to\text{data}\to\text{interval}\to\text{decision}$
Total error idea
$\text{total uncertainty}=\text{sampling error}+\text{possible nonsampling error}$
Variance to SD
$\sigma=\sqrt{\sigma^2}$
Correct mean standard error
$SE=\frac{\sigma}{\sqrt n}\ \text{or}\ \frac{s}{\sqrt n}$
Solution audit
$\text{parameter}\to\text{procedure}\to\text{conditions}\to\text{interval}\to\text{interpretation}$
Width drivers
$\text{width}=2(\text{critical value})\frac{\text{spread}}{\sqrt n}$
Core z practice formula
$\bar x\pm z^*\frac{\sigma}{\sqrt n}$
Core t practice formula
$\bar x\pm t^*\frac{s}{\sqrt n}$
z mean interval
$\bar x\pm z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt n}$
t mean interval
$\bar x\pm t_{\frac{\alpha}{2},n-1}\frac{s}{\sqrt n}$
Sample size
$n\ge\left(\frac{z^*\sigma}{E}\right)^2$
Topic 4.17Not started
Poisson distribution
The Poisson distribution models independent event counts at an approximately constant average rate across time, distance, area, volume or another exposure measure.
220 min estimated
Topic 4.17 progress0%
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Learning objectives
Explain a Poisson random variable as a count in a fixed exposure interval.
Recognize suitable and unsuitable situations for a Poisson model.
State the independence and constant-rate assumptions.
Interpret the parameter lambda as an expected event count.
Distinguish an event rate from the Poisson mean for a specified exposure.
Scale lambda when time, distance, area, volume or exposure changes.
Use correct Poisson distribution notation.
Calculate exact Poisson probabilities from the probability mass function.
Use factorials and the discrete support correctly.
Calculate probabilities of zero, one and multiple events.
Calculate cumulative at-most probabilities.
Calculate at-least probabilities using complements.
Calculate inclusive and exclusive interval probabilities.
Use the equality of Poisson mean and variance.
Calculate the Poisson standard deviation.
Find an unknown parameter from a mean, variance or standard deviation.
Find an unknown parameter from an exact or cumulative probability.
Use GDC Poisson PDF and CDF functions accurately.
Find quantile-type cutoffs using a GDC table or search.
Combine independent Poisson variables by adding their parameters.
Apply Poisson thinning or splitting where relevant.
Explain the connection between binomial and Poisson models.
Use a Poisson approximation to a binomial distribution where appropriate.
Use a normal approximation to a Poisson distribution where appropriate.
Apply a continuity correction in normal approximations.
Connect Poisson counts with exponential waiting times where relevant.
Estimate an event rate from observed counts and total exposure.
Evaluate model fit using the mean–variance relationship and context.
Solve multi-stage real-world event-rate problems.
Communicate assumptions, units and probability statements in IB AI HL style.
Key formulas
Poisson support
$x\in\{0,1,2,\ldots\}$
Distribution notation
$X\sim\operatorname{Po}(\lambda)$
Small-exposure idea
$P(\text{two or more events in }\Delta t)=o(\Delta t)$
Constant-rate expectation
$E(X)=rE$
Expected count
$E(X)=\lambda$
Rate–exposure relation
$\lambda=rE$
Scale from one interval
$\lambda_2=\lambda_1\frac{E_2}{E_1}$
Rate times exposure
$\lambda=rE$
Poisson notation
$X\sim\operatorname{Po}(\lambda)$
Discrete inequality
$P(X<a)=P(X\le a-1)\quad(a\in\mathbb Z)$
Poisson PMF
$P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!}$
Factorial
$x!=x(x-1)\cdots2\cdot1$
Zero factorial
$0!=1$
Poisson recurrence
$P(X=x+1)=P(X=x)\frac{\lambda}{x+1}$
No events
$P(X=0)=e^{-\lambda}$
At least one event
$P(X\ge1)=1-e^{-\lambda}$
Poisson CDF
$P(X\le k)=\sum\limits_{x=0}^{k}\frac{e^{-\lambda}\lambda^x}{x!}$
Strict lower count
$P(X<k)=P(X\le k-1)$
At least k
$P(X\ge k)=1-P(X\le k-1)$
More than k
$P(X>k)=1-P(X\le k)$
Inclusive interval
$P(a\le X\le b)=F(b)-F(a-1)$
Strict-lower interval
$P(a<X\le b)=F(b)-F(a)$
Poisson mean
$E(X)=\lambda$
Poisson variance
$\operatorname{Var}(X)=\lambda$
Poisson standard deviation
$\sigma_X=\sqrt\lambda$
From mean
$\lambda=E(X)$
From variance
$\lambda=\operatorname{Var}(X)$
From SD
$\lambda=\sigma_X^2$
Lambda from zero probability
$\lambda=-\ln P(X=0)$
Consecutive probability ratio
$\frac{P(X=x+1)}{P(X=x)}=\frac{\lambda}{x+1}$
GDC PDF input
$\operatorname{poissonPdf}(\lambda,k)$
Exact event
$P(X=k)$
GDC CDF input
$\operatorname{poissonCdf}(\lambda,k)=P(X\le k)$
Quantile condition
$F(k-1)<p\le F(k)$
Independent Poisson sum
$X\sim\operatorname{Po}(a),\ Y\sim\operatorname{Po}(b)\Rightarrow X+Y\sim\operatorname{Po}(a+b)$
n independent equal intervals
$\sum\limits_{i=1}^{n}X_i\sim\operatorname{Po}(n\lambda)$
Thinned Poisson count
$Y\sim\operatorname{Po}(p\lambda)$
Conditional category count
$Y\mid X=n\sim\operatorname{Bin}(n,p)$
Poisson approximation
$\operatorname{Bin}(n,p)\approx\operatorname{Po}(np)$
Approximation parameter
$\lambda=np$
Normal approximation
$X\sim\operatorname{Po}(\lambda)\approx Y\sim\mathcal N(\lambda,\lambda)$
Continuity correction for at least k
$P(X\ge k)\approx P(Y>k-0.5)$
Poisson-process count
$N(t)\sim\operatorname{Po}(rt)$
Exponential survival
$P(T>t)=e^{-rt}$
First-event CDF
$P(T\le t)=1-e^{-rt}$
Equal-interval estimate
$\hat\lambda=\bar x$
Pooled rate estimate
$\hat r=\frac{\text{total events}}{\text{total exposure}}$
Predicted interval parameter
$\hat\lambda=\hat rE$
Poisson moment check
$\operatorname{Var}(X)=E(X)$
Dispersion ratio
$D=\frac{s^2}{\bar x}$
Call interval parameter
$\lambda=7.5\left(\frac{24}{60}\right)=3$
More than four
$P(X>4)=1-P(X\le4)$
Distance-scaled parameter
$\lambda=(2.4)(3.5)=8.4$
Distance-count spread
$\sigma_Y=\sqrt{8.4}$
Multi-stage parameter
$\lambda=\left(\frac{\text{mean count}}{\text{original exposure}}\right)(\text{new exposure})$
Exposure from no-event probability
$E=-\frac{\ln P(X=0)}{r}$
Correct at-least complement
$P(X\ge k)=1-F(k-1)$
Correct interval scaling
$\lambda_{new}=rE_{new}$
Exam workflow
$\text{context}\to r\to\lambda\to\text{event}\to\text{probability}$
Final probability form
$0\le P(\text{event})\le1$
Practice PMF
$P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!}$
Practice CDF complement
$P(X\ge k)=1-F(k-1)$
Practice scaling
$\lambda=rE$
Core model
$X\sim\operatorname{Po}(\lambda)$
Core PMF
$P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!}$
Core moments
$E(X)=\operatorname{Var}(X)=\lambda,\quad\sigma_X=\sqrt\lambda$
Topic 4.18Not started
Advanced hypothesis testing, errors and critical regions
Advanced hypothesis testing formalizes evidence through hypotheses, test statistics, p-values and critical regions while quantifying Type I and Type II decision risks.
235 min estimated
Topic 4.18 progress0%
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Learning objectives
State null and alternative hypotheses using population parameters.
Distinguish a parameter claim from a sample statistic.
Choose a left-tailed, right-tailed or two-tailed alternative from the research question.
Explain the significance level as a preselected Type I error bound or rate.
Interpret a p-value as a probability calculated under the null hypothesis.
Use p-values to make reject or fail-to-reject decisions.
Calculate and interpret standardized test statistics.
Find critical values for one-tailed and two-tailed tests.
Construct and use critical regions.
Explain the equivalence of p-value and critical-region methods.
Define Type I and Type II errors in general and in context.
Calculate the actual probability of a Type I error.
Calculate Type II error probabilities for specified alternatives.
Calculate and interpret test power.
Explain how sample size, significance level and effect size affect power.
Select an appropriate test for a mean, proportion, Poisson mean or correlation.
Select between one-sample, paired and independent two-sample procedures.
Apply z or t tests for population means in the required settings.
Apply tests for population proportions.
Apply exact or critical-region tests for a Poisson mean in the current syllabus.
Test whether a population correlation coefficient is zero in the required setting.
Use chi-square tests of goodness of fit and independence appropriately.
Use GDC test menus and interpret complete output.
Check randomness, independence, distribution and expected-frequency conditions.
Distinguish statistical significance from practical importance.
Write conclusions using evidence language rather than certainty language.
Recognize the limitations of multiple testing and post-hoc tail selection.
Solve multi-stage IB AI HL hypothesis-testing problems.
Audit a testing solution for direction, boundaries, assumptions and interpretation.
Communicate technology-supported inference clearly and accurately.
Key formulas
Testing framework
$\text{claim}\to\text{null model}\to\text{sample evidence}\to\text{decision}$
Evidence probability
$P(\text{result at least as extreme as observed}\mid H_0)$
Null hypothesis
$H_0:\theta=\theta_0$
Alternative forms
$H_1:\theta<\theta_0,\quad\theta>\theta_0,\quad\text{or}\quad\theta\ne\theta_0$
Left tail
$H_1:\theta<\theta_0$
Right tail
$H_1:\theta>\theta_0$
Two tails
$H_1:\theta\ne\theta_0$
Significance level
$\alpha=P(\text{reject }H_0\mid H_0\text{ true})$
Two-tail allocation
$\alpha=\frac{\alpha}{2}+\frac{\alpha}{2}$
Generic standardized statistic
$\text{test statistic}=\frac{\text{estimate}-\text{null value}}{\text{null standard error}}$
Two-sided extremeness
$|T|\ge |t_{obs}|$
Right-tail p-value
$p=P(T\ge t_{obs}\mid H_0)$
Left-tail p-value
$p=P(T\le t_{obs}\mid H_0)$
Two-tail p-value
$p=P(|T|\ge|t_{obs}|\mid H_0)$
Reject rule
$p\le\alpha\Rightarrow\text{reject }H_0$
Fail-to-reject rule
$p>\alpha\Rightarrow\text{fail to reject }H_0$
Right critical value
$P(T\ge c\mid H_0)=\alpha$
Two-tail critical values
$P(T\le-c\mid H_0)+P(T\ge c\mid H_0)=\alpha$
Equivalent rejection
$T\in\mathcal C\Longleftrightarrow p\le\alpha$
Critical region probability
$P_{H_0}(T\in\mathcal C)\le\alpha$
Type I event
$\text{reject }H_0\text{ when }H_0\text{ is true}$
Type I probability
$\alpha_{actual}=P(\text{reject }H_0\mid H_0\text{ true})$
Type II event
$\text{fail to reject }H_0\text{ when }H_1\text{ is true}$
Type II probability
$\beta(\theta_1)=P_{\theta_1}(\text{fail to reject }H_0)$
Power
$\operatorname{Power}(\theta_1)=1-\beta(\theta_1)$
Power event
$P_{\theta_1}(\text{reject }H_0)$
Discrete Type I probability
$\alpha_{actual}=P_{\theta_0}(X\in\mathcal C)$
Discrete Type II probability
$\beta(\theta_1)=P_{\theta_1}(X\notin\mathcal C)$
Selection keys
$\text{variable type}+\text{parameter}+\text{design}+\text{conditions}$
Paired reduction
$D_i=X_{after,i}-X_{before,i}$
One-sample z statistic
$z=\frac{\bar x-\mu_0}{\frac{\sigma}{\sqrt n}}$
Null sampling model
$\bar X\sim\mathcal N\left(\mu_0,\frac{\sigma^2}{n}\right)$
One-sample t statistic
$t=\frac{\bar x-\mu_0}{\frac{s}{\sqrt n}}$
Degrees of freedom
$df=n-1$
Sample proportion
$\hat p=\frac{x}{n}$
One-proportion z statistic
$z=\frac{\hat p-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$
Null expected counts
$np_0\quad\text{and}\quad n(1-p_0)$
Poisson null model
$X\mid H_0\sim\operatorname{Po}(\lambda_0)$
Upper-tail Poisson p-value
$p=P_{\lambda_0}(X\ge x_{obs})$
Lower-tail Poisson p-value
$p=P_{\lambda_0}(X\le x_{obs})$
Correlation t statistic
$t=r\sqrt{\frac{n-2}{1-r^2}}$
Correlation test degrees of freedom
$df=n-2$
Correlation hypotheses
$H_0:\rho=0$
Chi-square statistic
$\chi^2=\sum\limits_{i=1}^{k}\frac{(O_i-E_i)^2}{E_i}$
GOF degrees of freedom
$df=k-1-m$
Expected cell frequency
$E_{ij}=\frac{(\text{row total})(\text{column total})}{\text{grand total}}$
Independence degrees of freedom
$df=(r-1)(c-1)$
Paired t statistic
$t=\frac{\bar d-\mu_{D,0}}{\frac{s_d}{\sqrt n}}$
Difference in means
$H_0:\mu_1-\mu_2=\Delta_0$
GDC output set
$(\text{test statistic},p,df,\text{sample estimates})$
Decision audit
$p\mathrel{?}\alpha$
Reject conclusion
$p\le\alpha\Rightarrow\text{sufficient evidence for }H_1$
Non-reject conclusion
$p>\alpha\Rightarrow\text{insufficient evidence for }H_1$
Significance question
$p\le\alpha$
Importance question
$|\text{effect}|\ge\text{contextual threshold}$
Familywise false-positive chance
$1-(1-\alpha)^m$
Bonferroni per-test level
$\alpha_{per}=\frac{\alpha_{family}}{m}$
Discrete upper critical boundary
$P_{\theta_0}(X\ge c)\le\alpha<P_{\theta_0}(X\ge c-1)$
Actual level
$\alpha_{actual}=P_{\theta_0}(X\ge c)$
Critical sample-mean boundary
$c=\mu_0+z_{1-\alpha}\frac{\sigma}{\sqrt n}$
Right-test beta
$\beta(\mu_1)=P_{\mu_1}(\bar X<c)$
Power at alternative
$1-\beta(\mu_1)$
Correct conditioning
$p=P(\text{extreme data}\mid H_0)$
Incorrect reversal
$p\ne P(H_0\mid\text{data})$
Testing workflow
$H_0,H_1\to\text{test}\to\text{conditions}\to T,p\to\text{decision}\to\text{context}$
Error workflow
$\mathcal C\to P_{\theta_0}(\mathcal C)\text{ or }P_{\theta_1}(\mathcal C^c)$
Practice decision
$p\le\alpha\Rightarrow\text{reject }H_0$
Practice power
$1-\beta$
Practice chi-square df
$(r-1)(c-1)$
Core decision
$p\le\alpha\Longleftrightarrow T\in\mathcal C$
Core errors
$\alpha=P_{H_0}(\text{reject}),\quad\beta(\theta_1)=P_{\theta_1}(\text{fail to reject})$
Core power
$\operatorname{Power}(\theta_1)=1-\beta(\theta_1)$
Topic 4.19Not started
Markov chains and steady state
Markov chains model systems that move among states according to fixed transition probabilities. Matrix powers describe repeated steps, while a steady-state vector describes long-term proportions when appropriate.
240 min estimated
Topic 4.19 progress0%
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Learning objectives
Explain a stochastic process as a system whose state changes randomly over time.
Identify states, transitions and one-step transition probabilities.
State and apply the Markov property.
Construct a transition diagram from contextual information.
Read probabilities correctly from a directed transition diagram.
Construct a column-stochastic transition matrix.
Check that a transition matrix is valid.
Distinguish column-vector and row-vector conventions.
Construct and interpret an initial state vector.
Calculate a one-step future state using matrix multiplication.
Calculate an n-step future state using a matrix power.
Interpret entries of a powered transition matrix.
Convert probabilities into expected population counts.
Describe transient and long-term behaviour.
Define a steady-state distribution.
Find a steady state by solving a system of equations.
Verify a proposed steady state by multiplication.
Interpret the eigenvalue-one relationship for a steady state.
Recognize a regular Markov chain.
Explain why some chains do not converge.
Recognize absorbing states and simple absorption behaviour.
Model customer movement between categories.
Model migration between regions.
Model movement between market states.
Use GDC matrix operations and powers efficiently.
Use technology to approximate long-term distributions.
Build a Markov model from observed transition counts.
Evaluate the assumptions and limitations of a Markov model.
Avoid orientation, rounding and interpretation errors.
Communicate Markov-chain conclusions in an IB AI HL context.
Key formulas
State process
$X_0,X_1,X_2,\ldots$
State space
$\mathcal S=\{s_1,s_2,\ldots,s_k\}$
One-step transition
$X_n=s_j\longrightarrow X_{n+1}=s_i$
n-step transition
$X_0\longrightarrow X_n$
Markov property
$P(X_{n+1}=i\mid X_n=j,X_{n-1},\ldots,X_0)=P(X_{n+1}=i\mid X_n=j)$
Time-homogeneous transition
$P(X_{n+1}=i\mid X_n=j)=t_{ij}$
Transition probability
$t_{ij}=P(X_{n+1}=s_i\mid X_n=s_j)$
Probability total from current state j
$\sum\limits_{i=1}^{k}t_{ij}=1$
Outgoing total
$\sum\limits_{i=1}^{k}P(j\to i)=1$
Visual matrix model
$\mathbf v_{n+1}=T\mathbf v_n$
Column convention
$t_{ij}=P(\text{next }i\mid\text{current }j)$
Column evolution
$\mathbf v_{n+1}=T\mathbf v_n$
Row evolution
$\mathbf r_{n+1}=\mathbf r_nR$
Convention conversion
$T=R^{\mathsf T}$
Column checks
$0\le t_{ij}\le1,\qquad \sum\limits_{i=1}^{k} t_{ij}=1$
Probability conservation
$\mathbf 1^{\mathsf T}T=\mathbf 1^{\mathsf T}$
Initial probability vector
$\mathbf v_0=\begin{pmatrix}p_1\\p_2\\\vdots\\p_k\end{pmatrix},\qquad \sum\limits_{i=1}^{k} p_i=1$
Counts to proportions
$\mathbf v_0=\frac{1}{N}\mathbf n_0$
One-step evolution
$\mathbf v_{n+1}=T\mathbf v_n$
Component form
$(\mathbf v_{n+1})_i=\sum\limits_{j=1}^{k} t_{ij}(\mathbf v_n)_j$
n-step state
$\mathbf v_n=T^n\mathbf v_0$
Recursive form
$\mathbf v_n=T\mathbf v_{n-1}$
n-step entry
$(T^n)_{ij}=P(X_n=s_i\mid X_0=s_j)$
Pure-state start
$T^n\mathbf e_j=\text{column }j\text{ of }T^n$
Component recurrence
$v_{i,n+1}=\sum\limits_{j=1}^{k} t_{ij}v_{j,n}$
Two-state reduction
$b_n=1-a_n$
Long-term limit
$\lim_{n\to\infty}T^n\mathbf v_0$
Convergent case
$\lim_{n\to\infty}\mathbf v_n=\boldsymbol\pi$
Steady-state equation
$T\boldsymbol\pi=\boldsymbol\pi$
Probability condition
$\mathbf 1^{\mathsf T}\boldsymbol\pi=1,\qquad \pi_i\ge0$
Eigenvector form
$(T-I)\boldsymbol\pi=\mathbf0$
Linear system
$(T-I)\boldsymbol\pi=\mathbf0,\qquad \mathbf1^{\mathsf T}\boldsymbol\pi=1$
Normalize eigenvector w
$\boldsymbol\pi=\frac{\mathbf w}{\mathbf1^{\mathsf T}\mathbf w}$
Two-state balance
$P(A\to B)\pi_A=P(B\to A)\pi_B$
Normalization
$\pi_A+\pi_B=1$
Customer matrix
$T=\begin{pmatrix}0.70&0.15&0.05\\0.20&0.70&0.20\\0.10&0.15&0.75\end{pmatrix}$
Customer steady state
$\boldsymbol\pi=\begin{pmatrix}9/35\\14/35\\12/35\end{pmatrix}$
Regularity condition
$\exists m\in\mathbb N\text{ such that }(T^m)_{ij}>0\text{ for all }i,j$
Regular-chain limit
$\lim_{n\to\infty}T^n\mathbf v_0=\boldsymbol\pi$
Periodic swap
$T^2=I$
Closed class
$P(\text{leave the class})=0$
Absorbing state
$t_{ii}=1$
Simple survival
$P(\text{transient after }n)=q^n$
Simple absorption by n
$P(\text{absorbed by }n)=1-q^n$
Expected next counts
$\mathbf n_{n+1}=T\mathbf n_n$
Long-run expected count
$N\boldsymbol\pi$
Closed total
$\mathbf1^{\mathsf T}\mathbf n_{n+1}=\mathbf1^{\mathsf T}\mathbf n_n=N$
Net change
$\Delta n_i=(\mathbf n_{n+1})_i-(\mathbf n_n)_i$
Pure-state forecast
$\mathbf v_1=T\mathbf e_j$
Most likely state
$\operatorname*{arg\,max}_i (\mathbf v_n)_i$
GDC prediction
$T^n\mathbf v_0$
GDC verification
$T\boldsymbol\pi-\boldsymbol\pi\approx\mathbf0$
Augmented system
$\begin{bmatrix}T-I\\\mathbf1^{\mathsf T}\end{bmatrix}\boldsymbol\pi=\begin{bmatrix}\mathbf0\\1\end{bmatrix}$
Empirical transition
$\widehat t_{ij}=\frac{n_{ij}}{\sum\limits_{i=1}^{k} n_{ij}}$
Column total
$N_j=\sum\limits_{i=1}^{k} n_{ij}$
Time-homogeneous assumption
$t_{ij}^{(n)}=t_{ij}\text{ for all }n$
Changing-transition model
$\mathbf v_n=T_nT_{n-1}\cdots T_1\mathbf v_0$
Customer one-step state
$\mathbf v_1=\begin{pmatrix}0.405\\0.350\\0.245\end{pmatrix}$
Customer limit
$\boldsymbol\pi=\begin{pmatrix}9/35\\14/35\\12/35\end{pmatrix}$
Migration matrix
$T=\begin{pmatrix}0.92&0.12\\0.08&0.88\end{pmatrix}$
Migration balance
$0.08\pi_N=0.12\pi_S$
Steady eigenvector
$T\boldsymbol\pi=1\boldsymbol\pi$
Two-state convergence pattern
$\mathbf v_n-\boldsymbol\pi=\lambda_2^n(\mathbf v_0-\boldsymbol\pi)$
Wrong operation
$nT\ne T^n$
Correct order check
$\text{order}(\mathbf v)=\text{row/column state order of }T$
Prediction template
$\mathbf v_n=T^n\mathbf v_0$
Steady template
$(T-I)\boldsymbol\pi=\mathbf0,\quad\sum\limits_{i=1}^{k}\pi_i=1$
Practice prediction
$\mathbf v_n=T^n\mathbf v_0$
Practice steady state
$T\boldsymbol\pi=\boldsymbol\pi$
Practice normalization
$\sum\limits_{i=1}^{k}\pi_i=1$
Transition probability
$t_{ij}=P(X_{n+1}=i\mid X_n=j)$
Future state
$\mathbf v_n=T^n\mathbf v_0$
Steady state
$T\boldsymbol\pi=\boldsymbol\pi$
Regularity
$\exists m:(T^m)_{ij}>0\ \forall i,j$
Unit 5
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Calculus
AI calculus begins with rates of change, polynomial differentiation, basic integration and optimization, then extends at HL into standard derivative rules, second derivatives, richer integration, volumes, kinematics, differential equations, slope fields, Euler approximation and phase portraits for coupled systems.
17 topics
Unit 5 progress0%
Topics in this unit
Topic 5.1Not started
Limits and the derivative as a rate of change
Calculus begins with the idea that local behaviour can be studied through limiting change. AI emphasizes graphical and contextual meaning.
210 min estimated
Topic 5.1 progress0%
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Learning objectives
Understand a limit conceptually from a graph or table.
Interpret the derivative as the gradient of a tangent.
Interpret derivative units as a rate of change.
Estimate or calculate derivatives with appropriate technology.
Connect positive/negative derivative values to increasing/decreasing behaviour.
Key formulas
Finite limit
$\lim_{x\to a}f(x)=L$
Continuity at a point
$\lim_{x\to a}f(x)=f(a)$
Direct-substitution rule for a continuous function
$\lim_{x\to a}f(x)=f(a)$
Difference-of-squares limit
$\lim_{x\to a}\frac{x^2-a^2}{x-a}=2a$
Condition for a two-sided limit
$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\quad\Longrightarrow\quad\lim_{x\to a}f(x)=L$
Gradient through two points
$m=\frac{y_2-y_1}{x_2-x_1},\qquad x_2\ne x_1$
Average rate of change
$\frac{f(b)-f(a)}{b-a},\qquad a\ne b$
Secant gradient using an increment
$\frac{f(x+h)-f(x)}{h},\qquad h\ne0$
Instantaneous rate at x=a
$f'(a)=\text{gradient of the tangent to }y=f(x)\text{ at }x=a$
Equivalent derivative notation
$f'(x)=y'=\frac{dy}{dx}$
Tangent equation at x=a
$y-f(a)=f'(a)(x-a)$
Rate units
$\text{derivative units}=\frac{\text{output units}}{\text{input units}}$
Velocity from displacement
$v(t)=\frac{ds}{dt}$
Centered-difference estimate
$f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}$
Straight-line gradient
$m=\frac{y_2-y_1}{x_2-x_1}$
Average rate of change
$\frac{f(b)-f(a)}{b-a}$
Centered derivative estimate
$f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}$
Rate units
$\text{rate units}=\frac{\text{output units}}{\text{input units}}$
Topic 5.2Not started
Increasing and decreasing functions
The sign of the derivative describes where a function rises or falls.
215 min estimated
Topic 5.2 progress0%
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Learning objectives
Use $f'(x)>0$ for increasing intervals.
Use $f'(x)<0$ for decreasing intervals.
Recognize $f'(x)=0$ as a candidate stationary point.
Read derivative information from a graph or context.
Link a rate-of-change graph to the original quantity.
Key formulas
Increasing test
$f'(x)>0$
Decreasing test
$f'(x)<0$
Stationary-value condition
$f'(c)=0$
Critical-value search
$f'(x)=0\quad\text{or}\quad f'(x)\text{ is undefined}$
Local maximum sign change
$f':\ +\longrightarrow-$
Local minimum sign change
$f':\ -\longrightarrow+$
No extremum
$f':\ +\longrightarrow+\quad\text{or}\quad f':\ -\longrightarrow-$
Increasing
$f'(x)>0$
Decreasing
$f'(x)<0$
Stationary value
$f'(c)=0$
Local maximum
$f':\ +\longrightarrow-$
Local minimum
$f':\ -\longrightarrow+$
Topic 5.3Not started
Derivatives of polynomial functions
The power rule gives an analytic method for polynomial rates of change and underpins later optimization.
220 min estimated
Topic 5.3 progress0%
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Learning objectives
Differentiate integer powers.
Differentiate sums of power terms.
Evaluate a derivative at a specified value.
Use derivatives in context rather than as an isolated algebra exercise.
Key formulas
Derivative from first principles
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Equivalent derivative notation
$f'(x)=y'=\frac{dy}{dx}$
Gradient at x=a
$m_{\text{tangent}}=f'(a)$
Power rule
$\frac{d}{dx}\left(x^n\right)=nx^{n-1}$
Power rule with a constant coefficient
$\frac{d}{dx}\left(ax^n\right)=anx^{n-1}$
Derivative of a constant
$\frac{d}{dx}(c)=0$
Constant multiple rule
$\frac{d}{dx}\left[af(x)\right]=af'(x)$
Derivative of a linear term
$\frac{d}{dx}(ax)=a$
Sum rule
$\frac{d}{dx}\left[f(x)+g(x)\right]=f'(x)+g'(x)$
Difference rule
$\frac{d}{dx}\left[f(x)-g(x)\right]=f'(x)-g'(x)$
General polynomial derivative
$\frac{d}{dx}\left(\sum\limits_{k=0}^{n}a_kx^k\right)=\sum\limits_{k=1}^{n}ka_kx^{k-1}$
Gradient at a point
$m=f'(a)$
Inputs with gradient m
$f'(x)=m$
Tangent equation
$y-f(a)=f'(a)(x-a)$
Normal gradient
$m_{\text{normal}}=-\frac{1}{f'(a)}$
Stationary condition
$f'(c)=0$
Stationary point coordinates
$\left(c,f(c)\right)$
Velocity from displacement
$v(t)=s'(t)$
Acceleration from displacement
$a(t)=v'(t)=s''(t)$
Power rule
$\frac{d}{dx}\left(ax^n\right)=anx^{n-1}$
Constant rule
$\frac{d}{dx}(c)=0$
Gradient at x=a
$m=f'(a)$
Topic 5.4Not started
Tangents and normals
A derivative gives the tangent gradient; the normal is perpendicular to that tangent.
220 min estimated
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Learning objectives
Find the tangent gradient at a point.
Write the tangent equation.
Find the normal gradient and equation.
Interpret tangent approximations locally.
Handle horizontal/vertical special cases carefully.
Key formulas
Tangent gradient
$m_{\text{tangent}}=f'(a)$
Point-slope form
$y-y_1=m(x-x_1)$
Tangent at x=a
$y-f(a)=f'(a)(x-a)$
Perpendicular-gradient condition
$m_1m_2=-1$
Normal gradient
$m_{\text{normal}}=-\frac{1}{f'(a)}$
Normal at x=a
$y-f(a)=-\frac{1}{f'(a)}(x-a)$
Given tangent gradient
$f'(x)=m$
Given normal gradient
$f'(x)=-\frac{1}{n}$
Horizontal tangent
$f'(a)=0\quad\Longrightarrow\quad y=f(a)$
Corresponding vertical normal
$x=a$
Parallel-gradient condition
$m_1=m_2$
Tangent approximation
$f(x)\approx f(a)+f'(a)(x-a)$
Tangent equation
$y-f(a)=f'(a)(x-a)$
Normal equation
$y-f(a)=-\frac{1}{f'(a)}(x-a)$
Perpendicular gradients
$m_{\text{tangent}}m_{\text{normal}}=-1$
Topic 5.5Not started
Introduction to integration
Integration reverses differentiation and accumulates continuous change. At SL-level AI, technology handles many definite-integral calculations.
230 min estimated
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Learning objectives
Find simple antiderivatives of powers.
Use a boundary condition to determine the integration constant.
Evaluate definite integrals with technology.
Find area between a positive curve and the x-axis.
Interpret integral units as accumulated quantity.
Key formulas
Antiderivative condition
$F'(x)=f(x)$
Reverse relationship
$\frac{d}{dx}\left(\int f(x)\,dx\right)=f(x)$
Indefinite integral notation
$\int f(x)\,dx=F(x)+C\quad\text{when }F'(x)=f(x)$
Definite integral notation
$\int_a^b f(x)\,dx$
Integration power rule
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\neq-1$
Constant multiple rule
$\int kf(x)\,dx=k\int f(x)\,dx$
Sum and difference rule
$\int\bigl(f(x)\pm g(x)\bigr)\,dx=\int f(x)\,dx\pm\int g(x)\,dx$
Integral of a constant
$\int k\,dx=kx+C$
Family of antiderivatives
$F(x)+C,\qquad C\in\mathbb{R}$
Polynomial rule
$\int\left(\sum\limits_{k=0}^{m}a_kx^k\right)dx=\sum\limits_{k=0}^{m}\frac{a_k}{k+1}x^{k+1}+C$
Condition from a point
$F(a)=b\quad\text{for a point }(a,b)$
Verification statement
$\frac{d}{dx}\bigl(F(x)+C\bigr)=f(x)$
Initial-value pattern
$y'=f(x),\ y(a)=b\quad\Longrightarrow\quad y=\int f(x)\,dx\text{ and then use }y(a)=b$
Fundamental Theorem of Calculus
$\int_a^b f(x)\,dx=\bigl[F(x)\bigr]_a^b=F(b)-F(a)$
Area above the x-axis
$A=\int_a^b f(x)\,dx\quad\text{when }f(x)\geq0$
First-quadrant region bounded by both axes
$A=\int_0^b f(x)\,dx$
Total area
$A_{\mathrm{total}}=\int_a^b|f(x)|\,dx$
Accumulated change
$Q(b)-Q(a)=\int_a^b Q'(t)\,dt$
Final amount from a rate
$Q(b)=Q(a)+\int_a^b r(t)\,dt$
Antiderivative definition
$F'(x)=f(x)\quad\Longleftrightarrow\quad\int f(x)\,dx=F(x)+C$
Power rule
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\neq-1$
Constant rule
$\int k\,dx=kx+C$
Linearity
$\int(af(x)+bg(x))\,dx=a\int f(x)\,dx+b\int g(x)\,dx$
Definite integral
$\int_a^b f(x)\,dx=F(b)-F(a)$
Topic 5.6Not started
Stationary points
Stationary points occur where the first derivative is zero and are candidates for local maxima or minima.
230 min estimated
Topic 5.6 progress0%
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Learning objectives
Solve $f'(x)=0$ .
Find coordinates of stationary points.
Use graph or sign information to classify local maxima/minima.
Interpret stationary points in a model.
Distinguish a stationary point from an endpoint optimum.
Key formulas
Stationary-point condition
$f'(a)=0$
Local minimum test
$f'(a)=0,\quad f''(a)>0$
Local maximum test
$f'(a)=0,\quad f''(a)<0$
Shared stationary condition
$f'(a)=0$
Stationary condition
$f'(a)=0$
First-derivative maximum pattern
$f':\quad +\;\to\;-$
First-derivative minimum pattern
$f':\quad -\;\to\;+$
Second-derivative minimum test
$f'(a)=0,\quad f''(a)>0$
Second-derivative maximum test
$f'(a)=0,\quad f''(a)<0$
Inflection requirement
$f''\text{ changes sign at }x=a$
Topic 5.7Not started
Optimization in context
Optimization uses a mathematical model to maximize or minimize a quantity such as cost, area, volume, profit or time.
260 min estimated
Topic 5.7 progress0%
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Learning objectives
Define the objective quantity.
Reduce a multi-variable problem to a function of one variable where necessary.
Find candidates using calculus/technology.
Compare candidates and domain endpoints.
Report the quantity the question actually asks for with units and context.
Key formulas
Interior stationary candidate
$f'(a)=0$
Profit relationship
$P(x)=R(x)-C(x)$
Revenue relationship
$R(x)=x\,p(x)$
Constraint substitution
$y=g(x)\quad\Longrightarrow\quad Q(x)=Q\bigl(x,g(x)\bigr)$
Rectangle area
$A=xy$
Rectangular-prism volume
$V=lwh$
Cylinder volume
$V=\pi r^2h$
Stationary-candidate equation
$Q'(x)=0$
Second-derivative maximum test
$Q'(a)=0,\;Q''(a)<0\;\Longrightarrow\;\text{local maximum}$
Second-derivative minimum test
$Q'(a)=0,\;Q''(a)>0\;\Longrightarrow\;\text{local minimum}$
Closed-interval comparison
$\max\{Q(a),Q(c_1),\ldots,Q(c_n),Q(b)\}$
Three-sided fencing constraint
$2x+y=40$
Enclosure objective
$A(x)=40x-2x^2$
Closed-cylinder surface area
$S=2\pi r^2+2\pi rh$
Open-box volume
$V(x)=x(L-2x)(W-2x)$
Average cost
$\overline C(x)=\frac{C(x)}{x}$
Optimization condition
$Q'(a)=0$
Maximum sign pattern
$Q':\quad +\;\longrightarrow\;-$
Minimum sign pattern
$Q':\quad -\;\longrightarrow\;+$
Second-derivative maximum
$Q'(a)=0,\;Q''(a)<0$
Second-derivative minimum
$Q'(a)=0,\;Q''(a)>0$
Profit
$P=R-C$
Topic 5.8Not started
Trapezoidal approximation of area
The trapezoidal rule estimates an integral from function values or tabulated data when an exact antiderivative is unavailable or unnecessary.
250 min estimated
Topic 5.8 progress0%
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Learning objectives
Apply the trapezoidal rule on equal subintervals.
Use tabulated values accurately.
Interpret the estimate as an approximation.
Compare with a technology integral when asked.
Keep track of the interval width.
Key formulas
One-trapezium area
$A=\frac{h}{2}(y_0+y_1)$
Equal interval width
$h=\frac{b-a}{n}$
Number of ordinates
$\text{ordinates}=n+1$
Composite trapezoidal rule
$T_n=\frac{h}{2}\left[y_0+2(y_1+y_2+\cdots+y_{n-1})+y_n\right]$
Summation form
$T_n=\frac{h}{2}\left[y_0+2\sum\limits_{i=1}^{n-1}y_i+y_n\right]$
Integral approximation
$\int_a^b f(x)\,dx\approx T_n$
Unequal-width trapezoidal sum
$T=\sum\limits_{i=0}^{n-1}\frac{x_{i+1}-x_i}{2}(y_i+y_{i+1})$
Total area from pieces
$A_{\text{total}}\approx |T_1|+|T_2|+\cdots$
Concave-up error direction
$f''(x)>0\;\Longrightarrow\;T_n\text{ usually overestimates}$
Concave-down error direction
$f''(x)<0\;\Longrightarrow\;T_n\text{ usually underestimates}$
Absolute error
$\text{absolute error}=|\text{approximation}-\text{accepted value}|$
Percentage error
$\text{percentage error}=\frac{|\text{approximation}-\text{accepted value}|}{|\text{accepted value}|}\times100\%$
Visual weighting pattern
$1,\;2,\;2,\;\ldots,\;2,\;1$
Equal width
$h=\frac{b-a}{n}$
Composite rule
$T_n=\frac h2\left[y_0+2\sum\limits_{i=1}^{n-1}y_i+y_n\right]$
Irregular data
$T=\sum\limits_{i=0}^{n-1}\frac{x_{i+1}-x_i}{2}(y_i+y_{i+1})$
Topic 5.9Not started
Derivatives of standard functions and differentiation rules
Differentiate polynomial, exponential, logarithmic and trigonometric models using chain, product and quotient rules, then interpret the derivative in applied contexts.
190 min estimated
Topic 5.9 progress0%
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Learning objectives
Review the derivative as gradient and instantaneous rate of change.
Use derivative notation accurately and attach units in context.
Differentiate constants, powers, sums and polynomial functions.
Differentiate natural exponential functions and exponential composites.
Differentiate general exponential functions with a positive base.
Differentiate natural and general logarithmic functions with domain awareness.
Differentiate sine, cosine and tangent functions using radians.
Apply the chain rule to algebraic and standard-function composites.
Apply the product rule to products of differentiable functions.
Apply the quotient rule and preserve the numerator order.
Combine chain, product and quotient rules in multi-layer expressions.
Simplify expressions before differentiating when this reduces error.
Choose an efficient differentiation rule from the structure of a formula.
Evaluate exact derivative values using standard trigonometric and logarithmic values.
Find gradients and equations of tangents in mathematical and applied models.
Interpret derivatives as marginal cost, marginal revenue and changing rates.
Set up and solve elementary related-rate relationships.
Use derivatives to locate candidate optimum points in context.
Use GDC numerical derivatives and graphs as checks rather than substitutes for analytic work.
Communicate assumptions, domains, units and conclusions in IB AI HL style.
Key formulas
Difference quotient
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Tangent gradient
$m_{\text{tangent}}=f'(a)$
Equivalent notation
$f'(x)=y'=\frac{dy}{dx}=\frac{d}{dx}[f(x)]$
Second derivative notation
$f''(x)=\frac{d^2y}{dx^2}$
Derivative units
$[f'] = \frac{[f]}{[x]}$
Constant rule
$\frac{d}{dx}(c)=0$
Power rule
$\frac{d}{dx}(x^n)=nx^{n-1}$
Constant multiple rule
$\frac{d}{dx}[cf(x)]=cf'(x)$
Sum rule
$\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)$
Polynomial derivative
$\frac{d}{dx}\left(\sum\limits_{r=0}^{n}a_rx^r\right)=\sum\limits_{r=1}^{n}ra_rx^{r-1}$
Linear function
$\frac{d}{dx}(mx+c)=m$
Quadratic function
$\frac{d}{dx}(ax^2+bx+c)=2ax+b$
Natural exponential derivative
$\frac{d}{dx}(e^x)=e^x$
Exponential chain rule
$\frac{d}{dx}(e^{u})=u'e^u$
General exponential derivative
$\frac{d}{dx}(a^x)=a^x\ln a$
General exponential chain rule
$\frac{d}{dx}(a^u)=u'a^u\ln a$
Natural logarithm derivative
$\frac{d}{dx}(\ln x)=\frac1x$
Logarithmic chain rule
$\frac{d}{dx}(\ln u)=\frac{u'}{u}$
Change of base
$\log_a x=\frac{\ln x}{\ln a}$
General logarithm derivative
$\frac{d}{dx}(\log_a u)=\frac{u'}{u\ln a}$
Sine derivative
$\frac{d}{dx}(\sin x)=\cos x$
Cosine derivative
$\frac{d}{dx}(\cos x)=-\sin x$
Tangent derivative
$\frac{d}{dx}(\tan x)=\sec^2x$
Sine chain rule
$\frac{d}{dx}(\sin u)=u'\cos u$
Cosine chain rule
$\frac{d}{dx}(\cos u)=-u'\sin u$
Exact value
$\sin\frac{\pi}{6}=\frac12$
Tangent chain rule
$\frac{d}{dx}(\tan u)=u'\sec^2u$
Sinusoidal rate
$\frac{d}{dt}[A\sin(\omega t+\phi)]=A\omega\cos(\omega t+\phi)$
Chain rule
$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$
Power chain rule
$\frac{d}{dx}(u^n)=nu^{n-1}u'$
Exponential composite
$\frac{d}{dx}(e^{u})=u'e^u$
Logarithmic composite
$\frac{d}{dx}(\ln u)=\frac{u'}u$
Repeated chain factors
$\frac{d}{dx}f(g(h(x)))=f'(g(h(x)))g'(h(x))h'(x)$
Product rule
$\frac{d}{dx}(uv)=u'v+uv'$
Quotient rule
$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2}$
Structural rule
$\text{differentiate the outer structure, then each inner component}$
Tangent equation
$y-f(a)=f'(a)(x-a)$
Instantaneous rate
$\text{rate at }x=a=f'(a)$
Marginal profit
$P'(q)=R'(q)-C'(q)$
Point elasticity
$\mathcal E(x)=\frac{x}{f(x)}f'(x)$
Marginal approximation
$\Delta f\approx f'(x)\Delta x$
Circle related rate
$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$
Sphere related rate
$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$
Product related rate
$\frac{d}{dt}(xy)=x\frac{dy}{dt}+y\frac{dx}{dt}$
Stationary condition
$f'(x)=0$
Optimization comparison
$\max\{f(\text{endpoints}),f(\text{feasible stationary points})\}$
Numerical derivative check
$f'(a)\approx\operatorname{nDeriv}(f(x),x,a)$
Power
$\frac{d}{dx}(x^n)=nx^{n-1}$
Natural exponential
$\frac{d}{dx}(e^u)=u'e^u$
General exponential
$\frac{d}{dx}(a^u)=u'a^u\ln a$
Natural logarithm
$\frac{d}{dx}(\ln u)=\frac{u'}u$
Sine
$\frac{d}{dx}(\sin u)=u'\cos u$
Cosine
$\frac{d}{dx}(\cos u)=-u'\sin u$
Tangent
$\frac{d}{dx}(\tan u)=u'\sec^2u$
Product
$(uv)'=u'v+uv'$
Quotient
$\left(\frac uv\right)'=\frac{u'v-uv'}{v^2}$
Chain
$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$
Topic 5.10Not started
Second derivative and classification
Use first and second derivatives to describe curvature, classify stationary points, identify inflection points and interpret changing rates in applied models.
200 min estimated
Topic 5.10 progress0%
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Learning objectives
Explain the second derivative as the rate of change of the first derivative.
Use and interpret second-derivative notation accurately.
Find second derivatives of polynomial and standard functions.
Interpret a second derivative as acceleration or a changing marginal rate.
Find stationary points by solving $f'x=0$ .
Distinguish local maxima, local minima and stationary points of inflection.
Apply the second-derivative test to classify stationary points.
Recognize when the second-derivative test is inconclusive.
Use first-derivative sign changes when the second-derivative test fails.
Determine intervals of concavity from the sign of f'' $x$ .
Find candidate points of inflection and confirm a change in concavity.
Distinguish stationary and non-stationary points of inflection.
Connect the graphs and signs of f, f' and f''.
Reconstruct qualitative behaviour from derivative information.
Classify endpoints and interior critical points in constrained models.
Interpret curvature and classification in motion, cost and growth models.
Use GDC graphs, numerical derivatives and solvers appropriately.
State exact values before using decimal approximations where appropriate.
Avoid common errors involving $f''c=0$ and inflection conditions.
Communicate derivative evidence, domains, units and contextual conclusions in IB AI HL style.
Key formulas
Second derivative
$f''(x)=\frac{d}{dx}[f'(x)]$
Leibniz notation
$\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$
Prime notation
$y''=f''(x)$
Evaluate after differentiating
$f''(a)=\left.\frac{d^2y}{dx^2}\right|_{x=a}$
Natural exponential
$\frac{d^2}{dx^2}(e^{kx})=k^2e^{kx}$
Natural logarithm
$\frac{d^2}{dx^2}(\ln x)=-\frac{1}{x^2}$
Sine
$\frac{d^2}{dx^2}(\sin x)=-\sin x$
Cosine
$\frac{d^2}{dx^2}(\cos x)=-\cos x$
Velocity
$v(t)=s'(t)$
Acceleration
$a(t)=v'(t)=s''(t)$
Second-derivative units
$[f'']=\frac{[f]}{[x]^2}$
Stationary condition
$f'(c)=0$
Stationary coordinates
$\bigl(c,f(c)\bigr)$
Interior extremum condition
$f'(c)=0\quad\text{for a differentiable interior local extremum}$
Local minimum test
$f'(c)=0 \text{and} f''(c)>0\Longrightarrow\text{local minimum}$
Local maximum test
$f'(c)=0 \text{and} f''(c)<0\Longrightarrow\text{local maximum}$
Inconclusive case
$f'(c)=0, f''(c)=0\quad\text{requires another test}$
Concave up
$f''(x)>0$
Concave down
$f''(x)<0$
Candidate separators
$f''(x)=0\quad\text{or}\quad f''(x)\text{ undefined}$
Inflection criterion
$f''\text{ changes sign at }x=c$
Common candidate condition
$f''(c)=0\quad\text{is not sufficient}$
Stationary inflection
$f'(c)=0\quad\text{and}\quad f''\text{ changes sign at }c$
Non-stationary inflection
$f'(c)\ne0\quad\text{and}\quad f''\text{ changes sign at }c$
Increasing function
$f'(x)>0$
Decreasing function
$f'(x)<0$
Increasing derivative
$f''(x)>0\Longleftrightarrow f'\text{ is increasing}$
Decreasing derivative
$f''(x)<0\Longleftrightarrow f'\text{ is decreasing}$
Four cases
$(f',f'')\in\{(+,+),(+,-),(-,+),(-,-)\}$
First-derivative maximum
$f':+\to-\Longrightarrow\text{local maximum}$
First-derivative minimum
$f':-\to+\Longrightarrow\text{local minimum}$
No extremum
$f'\text{ has no sign change}\Longrightarrow\text{neither maximum nor minimum}$
Classification sequence
$f'(c)=0 \longrightarrow f''(c) \longrightarrow \text{type and coordinates}$
Even-order conclusion
$f^{(n)}(c)>0, n\text{ even}\Longrightarrow\text{local minimum}$
Odd-order conclusion
$f^{(n)}(c)\ne0, n>1\text{ odd}\Longrightarrow\text{stationary inflection}$
Maximum stationary points
$\deg f=n\Longrightarrow\deg f'=n-1$
Exponential curvature
$\frac{d^2}{dx^2}(Ae^{kx})=Ak^2e^{kx}$
Logarithmic curvature
$\frac{d^2}{dx^2}[A\ln(x+b)]=-\frac{A}{(x+b)^2}$
Sinusoidal curvature
$\frac{d^2}{dx^2}[A\sin(Bx+C)]=-AB^2\sin(Bx+C)$
Turning candidate
$v(t)=s'(t)=0$
Position classification
$s''(t)=a(t)$
Changing marginal cost
$C''(q)=\frac{d}{dq}[C'(q)]$
Profit test
$P'(q)=0, P''(q)<0\Longrightarrow\text{local maximum profit}$
Growth-rate maximum
$P''(t)=0 \text{with sign }+\to-\Longrightarrow P'\text{ has a local maximum}$
Closed-interval candidates
${\text{stationary points in }[a,b],a,b}$
Strict concavity
$f''(x)>0 \text{on an interval}\Longrightarrow f\text{ is strictly convex there}$
Point condition
$f(c)=d$
Inflection candidate
$f''(c)=0$
Numerical first derivative
$f'(a)\approx\operatorname{nDeriv}(f(x),x,a)$
Numerical second derivative
$f''(a)\approx\operatorname{nDeriv}(f'(x),x,a)$
Essential test order
$f'(c)=0 \text{first, then inspect }f''(c)$
Stationary point
$f'(c)=0$
Local minimum
$f'(c)=0, f''(c)>0$
Local maximum
$f'(c)=0, f''(c)<0$
Inconclusive
$f'(c)=0, f''(c)=0$
Inflection
$f''\text{ changes sign}$
Maximum sign test
$f':+\to-$
Minimum sign test
$f':-\to+$
Velocity and acceleration
$v=s',\quad a=s''$
Topic 5.11Not started
Integration of standard functions and substitution
Integrate polynomial, exponential, logarithmic and trigonometric forms, use substitution for composite integrands, apply initial conditions, and evaluate definite integrals accurately.
205 min estimated
Topic 5.11 progress0%
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Learning objectives
Explain integration as reverse differentiation and accumulation.
Distinguish indefinite integrals from definite integrals.
Use correct integral notation, limits and differentials.
Include and interpret the constant of integration.
Integrate polynomial and general power functions.
Recognize the logarithmic exception to the power rule.
Integrate natural and general exponential functions.
Integrate reciprocal and logarithmic-derivative forms.
Integrate standard trigonometric functions in radians.
Check antiderivatives by differentiation.
Apply the Fundamental Theorem of Calculus.
Interpret definite integrals as signed accumulation.
Recognize reverse-chain-rule patterns.
Choose an efficient substitution from the structure of an integrand.
Carry out substitution in indefinite integrals and return to the original variable.
Change limits correctly in definite integrals.
Use initial or boundary conditions to determine unknown constants.
Apply integration to accumulated change in contextual models.
Use GDC numerical integration appropriately and preserve accuracy.
Communicate methods, domains, units and conclusions in IB AI HL style.
Key formulas
Antiderivative relationship
$F'(x)=f(x)\Longleftrightarrow\int f(x)\,dx=F(x)+C$
Derivative check
$\frac{d}{dx}\left[\int f(x)\,dx\right]=f(x)$
Indefinite integral notation
$\int f(x)\,dx$
Definite integral notation
$\int_a^b f(x)\,dx$
Family of antiderivatives
$\int f(x)\,dx=F(x)+C$
Sum rule
$\int[f(x)\pm g(x)]\,dx=\int f(x)\,dx\pm\int g(x)\,dx$
Constant multiple
$\int kf(x)\,dx=k\int f(x)\,dx$
Fundamental evaluation
$\int_a^b f(x)\,dx=F(b)-F(a)=[F(x)]_a^b$
Reverse limits
$\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$
Same limits
$\int_a^a f(x)\,dx=0$
Accumulation derivative
$\frac{d}{dx}\int_a^x f(t)\,dt=f(x)$
Composite upper limit
$\frac{d}{dx}\int_a^{g(x)} f(t)\,dt=f(g(x))g'(x)$
Net change
$\Delta Q=\int_a^b Q'(t)\,dt$
Final amount
$Q(b)=Q(a)+\int_a^b Q'(t)\,dt$
Total area
$A=\int_a^b|f(x)|\,dx$
Power rule
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\ne-1$
Constant
$\int k\,dx=kx+C$
Polynomial
$\int\left(\sum\limits_{r=0}^{n}a_rx^r\right)\,dx=\sum\limits_{r=0}^{n}\frac{a_r}{r+1}x^{r+1}+C$
Reciprocal rule
$\int\frac1x\,dx=\ln|x|+C$
Logarithmic derivative pattern
$\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+C$
Natural exponential
$\int e^x\,dx=e^x+C$
Linear exponent
$\int e^{kx+c}\,dx=\frac1k e^{kx+c}+C,\quad k\ne0$
Reverse-chain exponential
$\int f'(x)e^{f(x)}\,dx=e^{f(x)}+C$
General exponential
$\int a^x\,dx=\frac{a^x}{\ln a}+C$
Composite general exponential
$\int a^{kx+c}\,dx=\frac{a^{kx+c}}{k\ln a}+C$
Sine
$\int\sin x\,dx=-\cos x+C$
Cosine
$\int\cos x\,dx=\sin x+C$
Secant squared
$\int\sec^2x\,dx=\tan x+C$
Cosecant squared
$\int\csc^2x\,dx=-\cot x+C$
Tangent
$\int\tan x\,dx=-\ln|\cos x|+C$
Inverse tangent
$\int\frac1{1+x^2}\,dx=\arctan x+C$
Scaled inverse tangent
$\int\frac1{a^2+x^2}\,dx=\frac1a\arctan\left(\frac{x}{a}\right)+C,\quad a>0$
Power composite
$\int f'(x)[f(x)]^n\,dx=\frac{[f(x)]^{n+1}}{n+1}+C,\quad n\ne-1$
Log composite
$\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+C$
Cosine composite
$\int f'(x)\cos(f(x))\,dx=\sin(f(x))+C$
Substitution rule
$u=g(x),\quad du=g'(x)\,dx$
Transformed integral
$\int f(g(x))g'(x)\,dx=\int f(u)\,du$
Selection test
$\frac{du}{dx}=g'(x)\quad\text{should match a factor in the integrand}$
Linear power
$\int(ax+b)^n\,dx=\frac{(ax+b)^{n+1}}{a(n+1)}+C,\quad n\ne-1$
Linear reciprocal
$\int\frac1{ax+b}\,dx=\frac1a\ln|ax+b|+C$
Exponential substitution
$\int f'(x)e^{f(x)}\,dx=e^{f(x)}+C$
Nested logarithm
$\int\frac{\ln x}{x}\,dx=\frac12(\ln x)^2+C$
Sine composite
$\int f'(x)\sin(f(x))\,dx=-\cos(f(x))+C$
Secant-squared composite
$\int f'(x)\sec^2(f(x))\,dx=\tan(f(x))+C$
Changed limits
$x=a\Rightarrow u=g(a),\qquad x=b\Rightarrow u=g(b)$
Definite substitution
$\int_a^b f(g(x))g'(x)\,dx=\int_{g(a)}^{g(b)}f(u)\,du$
Consistent alternative
$\left[F(u)\right]_{u=g(a)}^{u=g(b)}=\left[F(g(x))\right]_{x=a}^{x=b}$
Particular antiderivative
$F(x)=\int f(x)\,dx+C,\quad F(a)=b$
Accumulation form
$F(x)=F(a)+\int_a^x f(t)\,dt$
Accumulation from rate
$Q(t)=Q(t_0)+\int_{t_0}^{t}R(s)\,ds$
Units
$[\text{integral}]=[\text{rate}]\times[\text{input}]$
Average value
$f_{\text{avg}}=\frac1{b-a}\int_a^b f(x)\,dx$
Method decision
$\text{simplify}\to\text{standard form}\to\text{substitution if an inner derivative matches}$
Generic numerical command
$\int_a^b f(x)\,dx\approx\operatorname{fnInt}(f(x),x,a,b)$
Verification
$\frac{d}{dx}[\text{answer}]=\text{integrand}$
Power
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\ne-1$
Reciprocal
$\int\frac1x\,dx=\ln|x|+C$
Log pattern
$\int\frac{f'}f\,dx=\ln|f|+C$
Substitution
$u=g(x),\quad du=g'(x)\,dx$
Definite evaluation
$\int_a^b f(x)\,dx=[F(x)]_a^b$
Net change
$Q(b)=Q(a)+\int_a^bQ'(x)\,dx$
Average value
$f_{\text{avg}}=\frac1{b-a}\int_a^bf(x)\,dx$
Topic 5.12Not started
Areas and volumes of revolution
Use definite integration to model signed and total area, areas between boundaries, and volumes formed by rotating regions about coordinate axes. Build accurate disc and washer models, use technology effectively, and interpret results in practical contexts.
260 min estimated
Topic 5.12 progress0%
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Learning objectives
Interpret a definite integral as signed accumulation and distinguish it from total geometric area.
Find the area under a curve and between a curve and a coordinate axis.
Find exact or numerical intersection points and use them as justified limits of integration.
Find areas between two curves using upper minus lower or right minus left.
Split an area integral when a graph crosses an axis or the boundary order changes.
Choose vertical or horizontal slices to create an efficient integral.
Explain how rotating a thin strip creates a disc or washer cross-section.
Find volumes of revolution about the x-axis and y-axis using discs and washers.
Identify outer and inner radii as perpendicular distances from the axis of rotation.
Set correct bounds and integrate with respect to the variable matching the cross-sections.
Model real-world cross-sectional areas, capacities, hollow components and manufactured solids.
Find unknown parameters from specified areas or volumes.
Use a GDC to graph boundaries, locate intersections and evaluate definite integrals.
Retain full calculator precision and report appropriate units and accuracy.
Check answers using sketches, dimensions, positivity and geometric reasonableness.
Present complete IB AI HL solutions with a model, limits, calculation and contextual conclusion.
Key formulas
Fundamental Theorem of Calculus
$\int_a^b f(x)\,dx=F(b)-F(a),\qquad F'(x)=f(x)$
Reversing limits
$\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$
Total area
$A=\int_a^b|f(x)|\,dx$
Area under a non-negative curve
$A=\int_a^b f(x)\,dx\qquad\text{when }f(x)\ge0$
Area between curves with vertical slices
$A=\int_a^b\bigl(f(x)-g(x)\bigr)\,dx\qquad\text{when }f(x)\ge g(x)$
Intersection condition
$f(x)=g(x)$
Area with horizontal slices
$A=\int_c^d\bigl(R(y)-L(y)\bigr)\,dy$
Symmetric area
$A=2\int_0^a h(x)\,dx\qquad\text{for a symmetric region}$
Cross-sectional volume principle
$V=\int A_{\text{cross-section}}\,d\ell$
Disc method
$V=\pi\int_a^b [R(x)]^2\,dx$
Volume about the x-axis
$V=\pi\int_a^b [f(x)]^2\,dx$
Washer area
$A_{\text{washer}}=\pi(R^2-r^2)$
Washer volume about the x-axis
$V=\pi\int_a^b\left([R(x)]^2-[r(x)]^2\right)\,dx$
Disc volume about the y-axis
$V=\pi\int_c^d[R(y)]^2\,dy$
Washer volume about the y-axis
$V=\pi\int_c^d\left([R(y)]^2-[r(y)]^2\right)\,dy$
Distance from a horizontal axis
$\text{radius}=|f(x)-k|\qquad\text{about }y=k$
Distance from a vertical axis
$\text{radius}=|g(y)-h|\qquad\text{about }x=h$
Area between measured profiles
$A=\int_a^b\bigl(y_{\text{upper}}-y_{\text{lower}}\bigr)\,dx$
Accumulation from a rate
$\Delta Q=\int_{t_1}^{t_2}q(t)\,dt$
Capacity from a radius profile
$V=\pi\int_c^d[R(y)]^2\,dy$
Material volume of a hollow profile
$V=\pi\int_a^b\left(R(x)^2-r(x)^2\right)\,dx$
Radius scaling
$\widetilde R=kR\quad\Longrightarrow\quad\widetilde V=k^2V$
Bounding-cylinder check
$0\le V\le\pi R_{\max}^2L$
Area between curves
$A=\int_a^b(\text{upper}-\text{lower})\,dx$
Area with horizontal slices
$A=\int_c^d(\text{right}-\text{left})\,dy$
Disc volume
$V=\pi\int R^2\,d\ell$
Washer volume
$V=\pi\int(R^2-r^2)\,d\ell$
Topic 5.13Not started
Kinematics and calculus
Use calculus to model one-dimensional motion, connect position, velocity and acceleration, interpret motion graphs, distinguish displacement from total distance, and solve realistic IB AI HL kinematics problems.
300 min estimated
Topic 5.13 progress0%
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Learning objectives
Define position, displacement, velocity, speed and acceleration with correct signs and units.
Interpret position as a function of time relative to a chosen origin and positive direction.
Differentiate position to obtain velocity and differentiate velocity to obtain acceleration.
Integrate acceleration to obtain velocity and integrate velocity to obtain position.
Use initial conditions to determine integration constants.
Find all times when a particle is at rest within a stated domain.
Use the sign of velocity to determine the direction of motion.
Use the signs of velocity and acceleration to decide whether speed is increasing or decreasing.
Find and classify local and absolute maximum or minimum positions.
Calculate net displacement from endpoint positions or the signed area under a velocity graph.
Calculate total distance by splitting at every change of direction or integrating speed.
Calculate and distinguish average velocity and average speed.
Interpret gradients and signed areas on position–time, velocity–time and acceleration–time graphs.
Analyse polynomial, exponential, trigonometric and piecewise motion models.
Build and evaluate real-world models for braking, vertical motion and approach to terminal velocity.
Use a GDC to solve motion equations, evaluate derivatives and integrals, and inspect graphs.
Check domains, units, assumptions and physical reasonableness.
Present complete IB AI HL solutions with analytical setup, technological evidence and contextual conclusions.
Key formulas
Velocity from displacement
$v(t)=\frac{ds}{dt}=s'(t)$
Acceleration from velocity
$a(t)=\frac{dv}{dt}=v'(t)=s''(t)$
Average velocity
$\bar v=\frac{s(t_2)-s(t_1)}{t_2-t_1}$
Speed
$\text{speed}=|v(t)|$
Velocity change
$v(t_2)-v(t_1)=\int_{t_1}^{t_2}a(t)\,dt$
Displacement change
$s(t_2)-s(t_1)=\int_{t_1}^{t_2}v(t)\,dt$
Speed sign test
$v(t)a(t)>0\Rightarrow\text{speeding up},\qquad v(t)a(t)<0\Rightarrow\text{slowing down}$
Displacement maximum test
$v(r)=0,\quad a(r)<0$
Displacement minimum test
$v(r)=0,\quad a(r)>0$
Net displacement
$\Delta s=s(t_2)-s(t_1)$
Total distance
$D=\int_{t_1}^{t_2}|v(t)|\,dt$
Average velocity
$\bar v=\frac{s(t_2)-s(t_1)}{t_2-t_1}=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}v(t)\,dt$
Average speed
$\text{average speed}=\frac{\int_{t_1}^{t_2}|v(t)|\,dt}{t_2-t_1}$
Derivative chain
$s(t)\xrightarrow{\frac{d}{dt}}v(t)\xrightarrow{\frac{d}{dt}}a(t)$
Total distance on an interval
$D=\int_{t_1}^{t_2}|v(t)|\,dt$
Distance check
$D\ge|s(t_2)-s(t_1)|$
Oscillatory derivative pattern
$s=A\sin(\omega t+\phi)\Rightarrow v=A\omega\cos(\omega t+\phi)\Rightarrow a=-\omega^2s$
Exponential chain rule
$\frac{d}{dt}e^{kt}=ke^{kt}$
Terminal velocity limit
$\lim_{t\to\infty}V\left(1-e^{-kt}\right)=V\quad(k>0)$
Stopping distance
$D=\int_0^{t_\text{stop}}|v(t)|\,dt,\qquad v(t_\text{stop})=0$
Vertical acceleration near Earth
$a=-g\approx-9.8\text{ m s}^{-2}$
Velocity
$v(t)=s'(t)=\frac{ds}{dt}$
Acceleration
$a(t)=v'(t)=s''(t)=\frac{d^2s}{dt^2}$
Topic 5.14Not started
Modelling with differential equations
Construct, solve and evaluate first-order differential-equation models for growth, decay, finance, populations, cooling and input-output systems, with careful use of initial conditions, technology, units and assumptions.
320 min estimated
Topic 5.14 progress0%
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Learning objectives
Explain what a differential equation represents in a rate-based model.
Identify dependent variables, independent variables, parameters and their units.
Determine the order of a differential equation.
Distinguish general, particular and equilibrium solutions.
Use initial conditions to determine constants accurately.
Translate verbal proportional-rate statements into differential equations.
Construct net-rate equations using input minus output.
Recognize and solve separable first-order differential equations.
Preserve equilibrium solutions and valid solution branches.
Model and interpret exponential growth and decay.
Use doubling time and half-life in context.
Model Newton cooling and interpret ambient temperature.
Model logistic growth and interpret carrying capacity and equilibria.
Construct continuous financial growth and contribution models.
Analyze population models with constant harvesting or intervention.
Estimate model parameters from observations.
Use a GDC to solve parameter and threshold equations and graph solutions.
Check solutions by differentiation and substitution.
Interpret predictions with units and domain restrictions.
Evaluate assumptions, sensitivity, extrapolation and model limitations.
Key formulas
General first-order form
$\frac{dy}{dx}=f(x,y)$
General second-order form
$F\left(x,y,\frac{dy}{dx},\frac{d^2y}{dx^2}\right)=0$
Rate-balance principle
$\frac{dQ}{dt}=\text{rate in}-\text{rate out}$
Dimensional consistency
$[\frac{dQ}{dt}]=[\text{right-hand side}]$
Initial condition
$y(x_0)=y_0$
Separation rule
$\frac{dy}{dx}=g(x)h(y)\quad\Longrightarrow\quad\int\frac{1}{h(y)}\,dy=\int g(x)\,dx$
Equilibrium condition
$F(c)=0\quad\Longrightarrow\quad y=c$
Exponential model
$N(t)=N_0e^{kt}$
Doubling time
$T_2=\frac{\ln2}{k}\quad(k>0)$
Half-life
$T_{\frac{1}{2}}=\frac{\ln2}{|k|}\quad(k<0)$
Continuous compounding
$A(t)=A_0e^{rt}$
Continuous contributions
$A(t)=\left(A_0+\frac{c}{r}\right)e^{rt}-\frac{c}{r}\quad(r\ne0)$
Newton cooling solution
$T(t)=T_a+(T_0-T_a)e^{-kt}$
Logistic differential equation
$\frac{dN}{dt}=kN(M-N)$
Logistic solution
$N(t)=\frac{M}{1+Ae^{-kMt}}$
Logistic partial fractions
$\frac{1}{N(M-N)}=\frac1M\left(\frac1N+\frac1{M-N}\right)$
Constant-harvest model
$\frac{dP}{dt}=kP-H$
Constant-harvest solution
$P(t)=\frac{H}{k}+\left(P_0-\frac{H}{k}\right)e^{kt}$
Input-output model
$\frac{dQ}{dt}=a-bQ$
Input-output solution
$Q(t)=\frac{a}{b}+\left(Q_0-\frac{a}{b}\right)e^{-bt}\quad(b>0)$
Two-point exponential calibration
$k=\frac{\ln(\frac{Q_2}{Q_1})}{t_2-t_1}$
Proportional growth or decay
$Q'=kQ$
Logistic growth
$P'=rP\left(1-\frac{P}{K}\right)$
Newton cooling
$T'=-k(T-T_a)$
Numerical parameter equation
$F(k)=0\quad\text{solved for the context-appropriate root}$
Relative sensitivity measure
$\text{relative change}=\frac{|\text{new prediction}-\text{old prediction}|}{|\text{old prediction}|}$
Separable equation
$\frac{dy}{dx}=g(x)h(y)\quad\Longrightarrow\quad\int\frac{1}{h(y)}\,dy=\int g(x)\,dx$
Euler update
$x_{n+1}=x_n+h,\qquad y_{n+1}=y_n+h f(x_n,y_n)$
Homogeneous substitution
$y=vx,\qquad\frac{dy}{dx}=v+x\frac{dv}{dx}$
Integrating factor
$\mu=e^{\int P(x)\,dx}\quad\text{for}\quad y'+P(x)y=Q(x)$
Linear solution
$y=\frac{1}{\mu}\left(\int\mu Q\,dx+C\right)$
Topic 5.15Not started
Slope fields
Construct and interpret slope fields for first-order differential equations, sketch initial-value solution curves, analyze nullclines and equilibria, match fields to equations, and communicate long-term behaviour in applied models.
300 min estimated
Topic 5.15 progress0%
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Learning objectives
Explain the meaning of a slope or direction field.
Relate each direction segment to $\frac{dy}{dx}=f(x,y)$ .
Calculate local slopes at selected coordinates.
Draw short direction segments with correct sign and steepness.
Construct a slope field systematically from a table and repeated patterns.
Distinguish autonomous and non-autonomous differential equations.
Recognize row, column and diagonal isocline patterns.
Find and interpret nullclines.
Read increasing, decreasing and near-horizontal behaviour from a field.
Sketch smooth solution curves tangent to a direction field.
Use an initial condition to select one solution curve.
Apply the uniqueness and non-crossing principle appropriately.
Find equilibrium solutions in autonomous models.
Classify stable, unstable and side-dependent equilibrium behaviour.
Predict long-term behaviour without solving explicitly.
Match differential equations to slope fields using structural evidence.
Check exact solutions against a field and by differentiation.
Interpret slope fields in population, cooling and rate-balance contexts.
Use technology or a GDC to generate or support field analysis.
Avoid common construction and interpretation errors in IB AI HL problems.
Key formulas
Direction-field rule
$m(x,y)=\frac{dy}{dx}=f(x,y)$
Local gradient
$f(a,b)=\left.\frac{dy}{dx}\right|_{(a,b)}$
Slope table entry
$m_{ij}=f(x_i,y_j)$
Logistic row slope
$m(y)=y\left(1-\frac{y}{4}\right)$
Segment angle
$\theta=\tan^{-1}(m)$
Column-pattern example
$y'=x\quad\Longrightarrow\quad m\text{ depends only on }x$
Autonomous equation
$\frac{dy}{dx}=F(y)$
Non-autonomous equation
$\frac{dy}{dx}=f(x,y)$
Isocline equation
$f(x,y)=c$
Increasing/decreasing test
$f(x,y)\gtrless0\quad\Longleftrightarrow\quad y\text{ increases/decreases locally}$
Nullcline
$f(x,y)=0$
Tangency condition
$y'(x)=f(x,y(x))$
Initial-value problem
$\frac{dy}{dx}=f(x,y),\qquad y(x_0)=y_0$
Uniqueness consequence
$\text{one initial point}\Longrightarrow\text{one local solution}$
Equilibrium condition
$F(c)=0\quad\Longrightarrow\quad y=c$
Stability sign test
$F(y)>0:\ \uparrow,\qquad F(y)<0:\ \downarrow$
Row pattern
$y'=F(y)$
Column pattern
$y'=G(x)$
Diagonal-nullcline example
$y'=x-y$
Exact verification
$y'(x)\stackrel{?}{=}f(x,y(x))$
Logistic field
$\frac{dy}{dt}=y\left(1-\frac{y}{4}\right)$
Newton cooling field
$\frac{dT}{dt}=-k(T-T_a),\qquad k>0$
Logistic population model
$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)$
Maximum logistic growth row
$P=\frac{K}{2}$
Technology input
$f(x,y)=\text{right-hand side of the differential equation}$
Field-analysis chain
$\text{equation}\to\text{slopes}\to\text{field}\to\text{solution behaviour}$
Field rule
$m=f(x,y)$
Autonomous equilibrium
$F(c)=0\Rightarrow y=c$
Initial condition
$y(x_0)=y_0$
Segment angle
$\theta=\tan^{-1}(f(x,y))$
Topic 5.16Not started
Euler method and numerical differential equations
Use Euler’s method to approximate first-order initial-value problems, organize iterations in tables, control step size and error, compare numerical and exact solutions, and apply the method to real-world and coupled models.
340 min estimated
Topic 5.16 progress0%
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Learning objectives
Explain why numerical solutions are needed when exact solutions are unavailable or inconvenient.
Interpret a first-order initial-value problem $\frac{dy}{dx}=f(x,y)$ , $y(x_0)=y_0$ .
Describe Euler’s method as repeated local tangent-line prediction.
Use $x_{n+1}=x_n+h$ and $y_{n+1}=y_n+h f(x_n,y_n)$ .
Identify the initial point, step size, target value and number of steps.
Calculate one Euler step accurately.
Construct a complete Euler table for repeated iterations.
Use forward and backward Euler steps.
Approximate a requested solution value and state it with suitable accuracy.
Explain geometrically why Euler’s method follows tangent segments.
Compare Euler approximations obtained with different step sizes.
Distinguish local truncation error from accumulated global error.
Calculate absolute, relative and percentage error when an exact value is known.
Predict overestimation or underestimation from concavity where appropriate.
Recognize round-off, discretization, data and model error.
Apply Euler’s method to growth, cooling and rate-balance models.
Update two coupled differential equations simultaneously using old-row values.
Implement Euler iteration using a spreadsheet, recurrence table or GDC.
Check numerical results using units, signs, scale and qualitative behaviour.
Communicate a complete IB AI HL numerical-method solution with recurrence, table and interpretation.
Key formulas
General first-order model
$\frac{dy}{dx}=f(x,y)$
Initial-value problem
$\frac{dy}{dx}=f(x,y),\qquad y(x_0)=y_0$
Tangent prediction
$\Delta y\approx \frac{dy}{dx}\,\Delta x$
Local tangent line
$y\approx y_n+f(x_n,y_n)(x-x_n)$
Euler recurrence
$x_{n+1}=x_n+h,\qquad y_{n+1}=y_n+h f(x_n,y_n)$
Euler grid
$x_n=x_0+nh$
Step count
$N=\frac{x_{\mathrm{target}}-x_0}{h}$
Slope column
$m_n=f(x_n,y_n)$
Increment form
$\Delta y_n=h m_n$
Repeated update
$(x_n,y_n)\mapsto\left(x_n+h,\;y_n+h f(x_n,y_n)\right)$
Table update
$y_{n+1}=y_n+h m_n,\qquad m_n=f(x_n,y_n)$
Backward step
$h<0\quad\Longrightarrow\quad x_{n+1}<x_n$
Exponential Euler recurrence
$y_{n+1}=(1+h)y_n\quad\text{for }y'=y$
First-order global error
$E_{\mathrm{global}}=O(h)$
Work count
$N=\frac{b-a}{h}$
Local truncation order
$E_{\mathrm{local}}=O(h^2)$
Global error order
$E_{\mathrm{global}}=O(h)$
Absolute error
$E_a=|y_{\mathrm{exact}}-y_{\mathrm{Euler}}|$
Relative error
$E_r=\frac{E_a}{|y_{\mathrm{exact}}|}$
Percentage error
$100E_r\%$
Concavity guide
$y''>0\Rightarrow\text{tangent below};\qquad y''<0\Rightarrow\text{tangent above}$
Signed error
$e_n=y_n-y(x_n)$
Exact-solution check
$y'(x)=f(x,y(x)),\qquad y(x_0)=y_0$
Total discrepancy concept
$\text{observed discrepancy}=\text{model/data effects}+\text{numerical effects}$
Euler decay multiplier
$y_{n+1}=(1-kh)y_n$
Simple decay stability condition
$|1-kh|<1$
Proportional-growth model
$\frac{dP}{dt}=kP$
Euler population update
$P_{n+1}=(1+kh)P_n$
Cooling model
$\frac{dT}{dt}=-k(T-T_a)$
Cooling Euler update
$T_{n+1}=T_n-hk(T_n-T_a)$
Rate-balance model
$\frac{dQ}{dt}=a-bQ$
Rate-balance equilibrium
$Q^*=\frac{a}{b}$
Nonlinear Euler rule
$y_{n+1}=y_n+h f(x_n,y_n)$
Coupled Euler update
$\begin{aligned}t_{n+1}&=t_n+h\\u_{n+1}&=u_n+hF(t_n,u_n,v_n)\\v_{n+1}&=v_n+hG(t_n,u_n,v_n)\end{aligned}$
Spreadsheet recurrence
$\text{next y cell}=\text{current y cell}+h\times\text{current slope cell}$
Essential update
$\boxed{y_{n+1}=y_n+h f(x_n,y_n)}$
Complete response pattern
$\text{model}\to\text{recurrence}\to\text{table}\to\text{value}\to\text{interpretation}$
Practice anchor
$(x_{n+1},y_{n+1})=(x_n+h,\;y_n+h f(x_n,y_n))$
Euler method
$\boxed{x_{n+1}=x_n+h,\quad y_{n+1}=y_n+h f(x_n,y_n)}$
Euler error orders
$E_{\mathrm{local}}=O(h^2),\qquad E_{\mathrm{global}}=O(h)$
Topic 5.17Not started
Phase portraits for coupled differential equations
Analyze two-variable coupled differential equations in the phase plane, construct and interpret trajectories, classify equilibria with eigenvalues, predict stability and long-term behaviour, and connect portraits to oscillation, damping, populations and numerical technology.
380 min estimated
Topic 5.17 progress0%
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Learning objectives
Explain why coupled differential equations model interacting quantities.
Write a two-variable autonomous system as $x'=f(x,y)$ , $y'=g(x,y)$ .
Represent a linear system in matrix form $\mathbf X'=A\mathbf X$ .
Distinguish the time-domain graphs of variables from a phase-plane plot.
Interpret a phase trajectory as the path of the system state.
Calculate direction vectors and trajectory gradients at selected points.
Use nullclines to divide the phase plane into directional regions.
Find equilibrium or fixed points by solving both rate equations simultaneously.
Construct a qualitative phase portrait from equilibria, nullclines, eigenvectors and direction tests.
Define stable, asymptotically stable, unstable and neutrally stable behaviour.
Classify nodes, saddles, centers and spirals from eigenvalues.
Use eigenvectors to identify invariant directions and separatrices.
Apply trace, determinant and discriminant tests to two-dimensional linear systems.
Determine clockwise or counterclockwise motion using a test point.
Describe long-term behaviour for different initial conditions.
Recognize oscillatory, cyclic and damped behaviour in a phase portrait.
Interpret predator–prey and competing-population phase portraits.
Connect a damped mechanical model to a stable spiral.
Use Euler iteration, a spreadsheet or GDC to generate trajectories.
Choose appropriate windows, initial states and time intervals in technology.
Avoid common errors involving axes, arrows, stability and eigenvalue classification.
Communicate complete IB AI HL conclusions with mathematical and contextual evidence.
Key formulas
Two-variable autonomous system
$\frac{dx}{dt}=f(x,y),\qquad \frac{dy}{dt}=g(x,y)$
Matrix form
$\mathbf X'=A\mathbf X,\qquad \mathbf X=\begin{pmatrix}x\\y\end{pmatrix}$
Affine equilibrium
$A\mathbf X^*+\mathbf b=\mathbf0$
State vector
$\mathbf X(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}$
Phase velocity
$\frac{d\mathbf X}{dt}=\begin{pmatrix}f(x,y)\\g(x,y)\end{pmatrix}$
Phase-trajectory slope
$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{g(x,y)}{f(x,y)},\qquad f\ne0$
Initial state
$\mathbf X(0)=\mathbf X_0$
Portrait workflow
$\text{equilibria}\to\text{nullclines}\to\text{eigendirections}\to\text{test arrows}\to\text{trajectories}$
Equilibrium conditions
$f(x^*,y^*)=0,\qquad g(x^*,y^*)=0$
Linear equilibrium equation
$A\mathbf X^*=\mathbf0$
x-nullcline
$f(x,y)=0$
y-nullcline
$g(x,y)=0$
Asymptotic stability idea
$\mathbf X(t)\to\mathbf X^*\quad\text{as }t\to\infty$
Eigenmode
$\mathbf X(t)=c\mathbf v e^{\lambda t}$
Characteristic equation
$\det(A-\lambda I)=0$
Stable-node solution
$\mathbf X(t)=c_1\mathbf v_1e^{\lambda_1t}+c_2\mathbf v_2e^{\lambda_2t},\qquad \lambda_1,\lambda_2<0$
Source growth
$e^{\lambda t}\to\infty\quad\text{when }\lambda>0$
Canonical saddle invariant
$xy=C$
Saddle sign test
$\lambda_1\lambda_2<0$
Center eigenvalues
$\lambda=\pm i\omega$
Circular invariant
$x^2+y^2=C$
Complex modes
$\lambda=\alpha\pm\beta i$
Spiral amplitude
$r(t)\propto e^{\alpha t}$
Repeated discriminant
$(\operatorname{tr}A)^2-4\det A=0$
Degenerate determinant
$\det A=0$
Characteristic equation
$\lambda^2-\tau\lambda+\Delta=0$
Discriminant
$D=\tau^2-4\Delta$
Complex real part
$\operatorname{Re}(\lambda)=\frac{\tau}{2}$
Axis test
$(x',y')\big|_{(1,0)}=\big(f(1,0),g(1,0)\big)$
Basin definition
$\mathcal B(\mathbf X^*)=\{\mathbf X_0:\mathbf X(t;\mathbf X_0)\to\mathbf X^*\}$
Four-type eigenvalue guide
$\begin{array}{c|c}\text{type}&\text{eigenvalues}\\\hline\text{stable node}&\lambda_1,\lambda_2<0\\\text{saddle}&\lambda_1\lambda_2<0\\\text{center}&\pm i\omega\\\text{stable spiral}&\alpha\pm\beta i,\ \alpha<0\end{array}$
Conserved quantity test
$\frac{dH}{dt}=H_xf(x,y)+H_yg(x,y)=0$
Trajectory equation
$\frac{dy}{dx}=\frac{g(x,y)}{f(x,y)}$
Lotka–Volterra model
$\frac{dx}{dt}=\alpha x-\beta xy,\qquad \frac{dy}{dt}=\delta xy-\gamma y$
Coexistence equilibrium
$\left(\frac{\gamma}{\delta},\frac{\alpha}{\beta}\right)$
Damped oscillator system
$\begin{pmatrix}x\\v\end{pmatrix}'=\begin{pmatrix}0&1\\-k&-c\end{pmatrix}\begin{pmatrix}x\\v\end{pmatrix}$
Damped characteristic equation
$\lambda^2+c\lambda+k=0$
Symmetric competition model
$x'=x(1-x-\tfrac12y),\qquad y'=y(1-\tfrac12x-y)$
Coupled Euler update
$\begin{aligned}x_{n+1}&=x_n+h f(x_n,y_n)\\y_{n+1}&=y_n+h g(x_n,y_n)\end{aligned}$
Parametric phase plot
$t\mapsto\big(x(t),y(t)\big)$
Contextual conclusion
$\text{state behaviour}+\text{time direction}+\text{units}+\text{model limitation}$
Classification anchor
$\text{sign of }\operatorname{Re}(\lambda)\Rightarrow\text{decay or growth}$
Exam response chain
$\text{system}\to\text{equilibrium}\to\text{eigen-analysis}\to\text{portrait}\to\text{interpretation}$
Practice classification core
$\tau=\operatorname{tr}A,\qquad \Delta=\det A,\qquad D=\tau^2-4\Delta$
Linear classification
$\lambda_{1,2}=\frac{\tau\pm\sqrt{\tau^2-4\Delta}}{2}$
Direction rule
$(x',y')=(f(x,y),g(x,y))$

Mathematics: Applications and Interpretation

Course Overview

Continue where you left off

Course units

Number and Algebra

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Operations with numbers and scientific notation

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Arithmetic sequences and series

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Geometric sequences and series

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Financial applications

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Exponents and introductory logarithms

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Approximation, bounds and percentage error

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Amortization and annuities

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Solving equations with technology

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Laws of logarithms

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Expressions with rational exponents

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Infinite geometric sequences and series

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Complex numbers in Cartesian form

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Polar and exponential forms of complex numbers

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Matrices

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Eigenvalues, eigenvectors and diagonalization

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Functions

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Equations of a straight line

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Concept of a function

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Graphing functions with technology

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Key features of graphs

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Modelling with common functions

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The modelling cycle

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Composite and inverse functions

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Transformations of functions

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Logarithmic, sinusoidal, logistic and piecewise models

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