SLIBMathematics

Mathematics: Applications and Interpretation

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

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Course Overview

IB Mathematics: Applications and Interpretation SL is organized into 5 units and 39 focused topics with 629 Key Concepts sections, 558 questions, and 780 flashcards.

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Begin with Topic 1.1: Scientific notation and operations with numbers.

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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
Number and Algebra provides the quantitative tools used in finance, modelling and technology-supported problem solving. AI SL adds approximation, financial technology, annuities and numerical equation solving to the shared IB mathematics foundations of scientific notation, sequences and logarithms.
8 topics
Unit 1 progress0%
Topics in this unit
Topic 1.1Not started
Scientific notation and operations with numbers
Scientific notation represents extremely large and small quantities compactly. This AI SL lesson develops accurate operations, significant-figure decisions, calculator fluency and interpretation in authentic scientific, financial and technological contexts.
180 min estimated
Topic 1.1 progress0%
Key Concepts0/19 sections complete Question Bank0/14 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Recognize and write normalized scientific notation $a\times10^k$ , where $1\le |a|<10$ and $k\in\mathbb Z$ .
Convert large and small quantities between ordinary notation and scientific notation.
Explain how positive, zero and negative powers of ten affect place value and scale.
Compare quantities and interpret their relative size or order of magnitude.
Multiply and divide quantities in scientific notation, then normalize the result.
Add and subtract quantities by first expressing them with a common power of ten.
Apply powers to numbers in scientific notation using the laws of indices.
Round values to a stated number of significant figures and preserve guard digits in multi-step work.
Enter, read and check scientific notation correctly on a calculator or GDC.
Replace calculator E-notation with conventional mathematical notation in final written answers.
Use scientific notation with units in scientific, financial and technological applications.
Judge whether an answer is reasonable by checking its sign, units, coefficient and exponent.
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 quantities with constant or approximately constant additive change. This AI SL lesson connects terms, finite sums, sigma notation, technology and real-data interpretation from first principles.
190 min estimated
Topic 1.2 progress0%
Key Concepts0/19 sections complete Question Bank0/20 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Distinguish a sequence of ordered terms from the series formed by adding those terms.
Recognize an arithmetic sequence and identify its first term and common difference.
Generate terms using the recurrence relation $u_{n+1}=u_n+d$ .
Use $u_n=a+(n-1)d$ to find any term directly.
Find an unknown first term, common difference, term value or term number from given information.
Calculate finite arithmetic-series sums using either the first and last terms or the first term and common difference.
Find the number of terms from a term or sum condition and check integer constraints.
Interpret and evaluate sigma notation for finite arithmetic sums.
Build arithmetic models for simple interest, seating, saving, production and other constant-change contexts.
Use tables, spreadsheets, graphing tools and equation solvers while still identifying the first term and common difference.
Approximate the common difference when real data are not perfectly arithmetic and evaluate the model limitations.
Communicate an IB AI SL solution with formulas, units, technology evidence, validation and contextual conclusions.
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 multiplication. This application-focused AI SL lesson develops common ratios, nth terms, finite and relevant infinite sums, technology methods, growth, decay and model evaluation.
200 min estimated
Topic 1.3 progress0%
Key Concepts0/21 sections complete Question Bank0/9 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Distinguish multiplicative geometric change from additive arithmetic change.
Identify the first term and common ratio, including fractional and negative ratios.
Generate terms from the recurrence relation $u_{n+1}=ru_n$ .
Use $u_n=ar^{n-1}$ to find a specified term.
Find an unknown term, first term, ratio or term number and check all valid solutions.
Calculate finite geometric-series sums and handle the special case where the ratio is 1.
Decide whether an infinite geometric series converges and find its sum when relevant.
Interpret sigma notation for finite and infinite geometric sums.
Build and interpret repeated-percentage growth and decay models.
Use GDC tables, graphs, solvers, spreadsheets and exponential regression appropriately.
Evaluate approximate geometric models using ratios, residuals, interpolation and extrapolation.
Present an IB AI SL solution with formulas, units, rounding, integer constraints and contextual conclusions.
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
Model investments, depreciation, inflation, regular savings and loans using compound growth, financial technology and careful interpretation.
190 min estimated
Topic 1.4 progress0%
Key Concepts0/13 sections complete Question Bank0/26 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Distinguish simple interest from compound interest and select the appropriate model.
Convert percentage rates to decimal multipliers for growth, appreciation, decay and depreciation.
Calculate future value, present value, interest rate or time using compound-interest models.
Match nominal annual rates, periodic rates, compounding frequency and elapsed time consistently.
Calculate and interpret effective annual interest rates.
Model regular savings deposits using finite geometric-series formulas.
Model amortizing loans, calculate periodic repayments and interpret outstanding balances.
Use GDC financial or TVM functions with a consistent cash-flow sign convention.
Solve financial word problems with appropriate units, accuracy, integer constraints and contextual interpretation.
Evaluate whether a financial model and its assumptions are reasonable in a real situation.
Calculate the real value of a future amount after allowing for inflation.
Compare financial options and evaluate the assumptions, reliability and limitations of a model.
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
Integer exponents and logarithms
Integer exponents model repeated growth and reciprocal change, while logarithms reverse exponentiation. This application-focused guide develops exponent laws, logarithmic notation, equation solving, growth and decay models, logarithmic scales, GDC methods, and IB AI SL interpretation.
170 min estimated
Topic 1.5 progress0%
Key Concepts0/12 sections complete Question Bank0/3 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Interpret positive, zero and negative integer exponents, including all restrictions needed for an expression to be defined.
Apply the product, quotient, power, product-to-a-power and quotient-to-a-power laws accurately.
Simplify numerical and algebraic expressions involving powers and rewrite answers with positive exponents when required.
Explain a logarithm as an exponent and convert fluently between exponential and logarithmic form.
Use basic logarithm laws with valid positive arguments and recognize expressions that cannot be split.
Solve exponential equations exactly when a common base is available and numerically using logarithms or a GDC otherwise.
Solve simple logarithmic equations and reject solutions that violate domain restrictions.
Build, use and interpret exponential growth and decay models in real-world contexts.
Use logarithmic scales such as pH, decibels and earthquake magnitude to compare real quantities.
Use calculator and GDC power, logarithm, graph, table and solver methods efficiently.
Communicate IB AI SL solutions with equations, units, suitable accuracy, integer constraints and model limitations.
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
Reported values are often rounded measurements rather than exact numbers. This application-focused guide develops appropriate accuracy, decimal places, significant figures, error intervals, worst-case bounds, absolute and percentage error, accumulated rounding effects, GDC methods, and reliable interpretation in context.
180 min estimated
Topic 1.6 progress0%
Key Concepts0/13 sections complete Question Bank0/20 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Distinguish exact values from measurements and approximations and select an appropriate degree of accuracy for a context.
Round positive and negative values correctly to a stated number of decimal places or significant figures.
Identify the rounding unit and construct lower and upper bounds using half of that unit.
Write and interpret error intervals using the correct inclusive lower and exclusive upper endpoint.
Use bounds to find possible ranges for sums, differences, products, quotients, areas, volumes and rates.
Calculate and interpret absolute error, relative error and percentage error when a reference value is known.
Determine maximum possible errors caused by rounded measurements when an exact value is unknown.
Explain how premature rounding and repeated calculations can affect a final result and retain suitable guard digits.
Apply approximation and error methods to measurement, finance, science, construction and data contexts.
Use calculator and GDC tools to estimate, verify, graph and report answers with appropriate accuracy.
Communicate IB AI SL solutions with inequalities, units, reasoning, assumptions and a reasonableness check.
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
Annuities model equal payments made at regular intervals, while amortization explains how loan payments divide between interest and principal. This technology-led guide develops cash-flow timing, present and future value, loan repayments, outstanding balances, amortization schedules, savings plans, spreadsheets, TVM/GDC methods, and IB AI SL interpretation.
185 min estimated
Topic 1.7 progress0%
Key Concepts0/13 sections complete Question Bank0/13 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Explain an annuity as a finite sequence of equal cash flows made at regular intervals.
Distinguish ordinary annuities from annuities due and identify the timing of the first and last payment.
Convert annual rates and time spans into a consistent periodic rate and total number of payment periods.
Use financial technology to find the future value of regular deposits and the deposit required for a target.
Use present value to model regular withdrawals, pensions, leases and loan repayments.
Explain amortization and calculate a regular loan payment using a GDC or spreadsheet.
Separate each loan payment into interest and principal components and construct an amortization schedule.
Determine and interpret an outstanding balance after a specified number of payments.
Calculate total payments and total interest and compare realistic financial options.
Use TVM variables, payment frequencies, END/BEGIN settings and cash-flow signs consistently.
Validate financial-technology output using estimation, formula structure, units and contextual constraints.
Communicate IB AI SL financial solutions with assumptions, suitable currency rounding and interpretation.
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
Technology can locate numerical solutions, zeros, roots and intersections efficiently, but students must still build the correct equation, choose a meaningful domain and window, find every required solution, and validate the output. This guide develops the AI SL requirements for systems of up to three linear equations and polynomial equations, then extends the same GDC workflow to graphical, tabular, exponential, logarithmic and contextual problems.
205 min estimated
Topic 1.8 progress0%
Key Concepts0/17 sections complete Question Bank0/7 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Explain a solution as a value that makes an equation true and distinguish a verified solution from a calculator candidate.
Represent the equation $f(x)=g(x)$ either as an intersection problem or as the zero problem $f(x)-g(x)=0$ .
Use graphs, numerical solvers and tables to locate approximate solutions.
Use technology to solve systems of two or three linear equations and interpret the ordered solution.
Enter polynomial coefficients correctly, including zero coefficients for missing powers, and find all real roots.
Solve selected nonlinear, exponential and logarithmic equations using appropriate technology.
Choose domains and graph windows that reveal all solutions relevant to the problem.
Distinguish exact solutions from decimal approximations and report suitable accuracy.
Interpret GDC messages, roots, intersections and numerical solver outputs rather than copying the display blindly.
Check solutions by substitution, residuals, domain restrictions and contextual constraints.
Recognize repeated roots, missed roots, extraneous candidates and solutions outside a real-world domain.
Present an IB AI SL solution with the mathematical model, technological method, results and contextual conclusion.
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
$a_1x+b_1y=c_1,\quad a_2x+b_2y=c_2$
Unit 2
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Functions
Functions are the central modelling language of AI SL. Students graph, fit and compare linear, quadratic, exponential, power, cubic and sinusoidal models and then use those models to interpret and predict real behaviour. Technology is integrated throughout.
6 topics
Unit 2 progress0%
Topics in this unit
Topic 2.1Not started
Equations of Straight Lines
Straight-line equations describe constant rates of change. This application-focused guide develops gradients, equations, intercepts, line relationships, intersections, GDC methods and real-world linear models from first principles to IB AI SL exam level.
170 min estimated
Topic 2.1 progress0%
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Learning objectives
Calculate and interpret the gradient of a straight line from a graph, two points or a real-world rate.
Use gradient-intercept, point-slope and general forms and convert correctly between them.
Find a line from a point and gradient, from two points, or from intercept information.
Find and interpret x- and y-intercepts.
Identify and construct parallel and perpendicular lines, including horizontal and vertical special cases.
Find intersections of straight lines algebraically and using a GDC.
Interpret gradient and intercept with appropriate contextual units.
Build, use and evaluate linear models over meaningful domains.
Communicate calculator-supported solutions with sufficient mathematical working, accuracy and context.
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
The Concept of a Function
Functions connect permitted inputs to exactly one output. This application-focused guide develops notation, evaluation, variables, domain and range, representations, function tests, inverse ideas, GDC methods and real-world interpretation to IB AI SL exam level.
205 min estimated
Topic 2.2 progress0%
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Learning objectives
Explain what makes a relation a function using the one-output rule.
Use function notation such as f(x), v(t) and C(n), and evaluate functions accurately.
Identify and interpret independent and dependent variables with units.
Determine algebraic and contextual domains and ranges from formulas, tables and graphs.
Represent functions using equations, tables, graphs, mappings, ordered pairs and verbal descriptions.
Decide whether a relation is a function using mappings, repeated inputs and the vertical line test.
Interpret functions as real-world models and choose realistic discrete or continuous domains.
Recognize one-to-one functions, find simple inverses and connect inverse graphs by reflection in $y = x$ .
Use a GDC to evaluate, tabulate and graph functions while validating and communicating results.
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
Use graphing technology accurately and purposefully: enter functions, choose windows, combine graph and table views, locate key features, interpret models and communicate GDC-supported conclusions at IB AI SL level.
230 min estimated
Topic 2.3 progress0%
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Learning objectives
Enter linear, quadratic, exponential, reciprocal and other relevant functions accurately in a GDC.
Choose graph windows from mathematical domains, contextual restrictions and expected output values.
Use zoom, trace and table features appropriately while recognizing their limitations.
Read coordinates, domain, range, intercepts, extrema, trends and asymptotic behavior from graphs.
Find zeros, x- and y-intercepts, intersections, maximum points and minimum points using suitable GDC commands.
Use tables alongside graphs to locate, bracket and validate numerical features.
Interpret graph features in real-world contexts with variables, units and discrete or continuous restrictions.
Diagnose common entry, window, search and reporting errors.
Present reproducible IB AI SL solutions with mathematical setup, appropriate accuracy and contextual conclusions.
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 and intersections of graphs
Read, calculate and interpret domains, ranges, intercepts, roots, extrema, intervals and graph intersections using algebra, graphs and GDC technology.
220 min estimated
Topic 2.4 progress0%
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Learning objectives
Determine domain and range from equations, graphs and application restrictions.
Distinguish roots from x-intercepts and calculate x- and y-intercepts.
Identify local and global maximum and minimum points, including endpoints on restricted domains.
State increasing and decreasing regions using correct intervals of input values.
Find roots, target values and intersections graphically and with a GDC.
Choose graph windows that reveal every relevant feature and validate calculator results.
Interpret intersection points, thresholds, break-even points and extrema with variables and units.
Communicate numerical graph solutions clearly at IB AI SL exam standard.
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 key function families
Recognize, fit, interpret and evaluate linear, quadratic, exponential, power, cubic and periodic models for real data and situations.
250 min estimated
Topic 2.5 progress0%
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Learning objectives
Recognize linear, quadratic, exponential, power, cubic and sinusoidal patterns from graphs, tables and contexts.
Choose a function family whose behavior and assumptions match the situation.
Find model parameters from given information and interpret them with units.
Use suitable GDC regression commands and store the fitted equation.
Make and communicate interpolated and extrapolated predictions.
Use graphical features and parameters to answer contextual questions.
Evaluate model fit, assumptions, domain, limitations and prediction reliability.
Present technology-supported modelling answers at IB AI SL exam standard.
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: fit, test, interpret and refine
Mathematical modelling is an evidence-based cycle: define a question, examine data, fit and test a model, interpret it in context, and refine it when the evidence or assumptions demand a change.
260 min estimated
Topic 2.6 progress0%
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Learning objectives
Explain the purpose and iterative stages of the mathematical modelling cycle.
Define variables, units, assumptions and a realistic domain before fitting a model.
Collect, organize and examine data for patterns, errors, outliers and limitations.
Choose and fit an appropriate function model using technology and contextual evidence.
Calculate and interpret residuals and use them to test model fit.
Interpret model parameters, predictions and goodness-of-fit information in context.
Distinguish interpolation from extrapolation and evaluate prediction reliability.
Refine or replace a model when residuals, context or new evidence reveal weaknesses.
Use a GDC to perform regression, store a model, graph residuals and make predictions.
Communicate a complete IB AI SL modelling conclusion with appropriate accuracy and limitations.
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
$\hat y=ab^x$
Unit 3
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Geometry and Trigonometry
Geometry and Trigonometry in AI SL is strongly practical. Students solve two- and three-dimensional measurement problems, use triangle trigonometry, work with arcs and sectors, construct perpendicular bisectors and interpret Voronoi diagrams for nearest-location problems.
6 topics
Unit 3 progress0%
Topics in this unit
Topic 3.1Not started
Distance, midpoint and three-dimensional solids
Students extend coordinate geometry into three dimensions and combine it with surface area, volume and right-triangle trigonometry.
260 min estimated
Topic 3.1 progress0%
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Learning objectives
Calculate the distance between two points in three-dimensional space.
Calculate the midpoint of a segment in three dimensions.
Calculate volumes and surface areas of right pyramids, right cones, spheres, hemispheres and composite solids.
Identify useful right triangles inside a three-dimensional object.
Find the angle between intersecting lines or between a line and a plane in practical right-triangle settings.
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 right and non-right triangles
AI SL uses right-triangle ratios plus the sine and cosine rules to solve practical triangle problems.
230 min estimated
Topic 3.2 progress0%
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Learning objectives
Use sine, cosine and tangent in right-angled triangles.
Use the sine rule to find unknown sides or angles when appropriate.
Use the cosine rule to find a side or angle when appropriate.
Calculate the area of a non-right triangle from two sides and the included angle.
Select the correct rule from the information provided.
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 and diagram construction
Many AI SL questions hide the mathematics inside a written description. Constructing the right diagram is often the most important first step.
230 min estimated
Topic 3.3 progress0%
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Learning objectives
Use Pythagoras’ theorem and trigonometry in contextual problems.
Solve angles of elevation and depression.
Work with bearings where prior learning is assumed.
Create a clear labelled diagram from a verbal description.
Use consistent units and state practical answers appropriately.
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
Radian measure, arcs and sectors
Radian measure links angle directly to arc length and sector area and is especially useful when modelling circular motion or geometry.
225 min estimated
Topic 3.4 progress0%
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Learning objectives
Convert between degrees and radians when needed.
Use a radian angle to calculate arc length.
Calculate the area of a sector.
Rearrange circular-measure formulas to find radius or angle.
Recognize when the angle in a formula must be measured in radians.
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 and nearest-point geometry
The perpendicular bisector of a segment contains all points equidistant from its endpoints. This idea is the 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 gradient of the original segment and the perpendicular gradient.
Write the equation of the perpendicular bisector using point-gradient form.
Interpret the bisector as a boundary separating points closer to one site from points closer to another.
Use intersections of bisectors to locate points equidistant from multiple 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
Voronoi diagrams divide a region into cells according to which site is nearest. They are useful for service areas, facility placement, navigation and optimization.
240 min estimated
Topic 3.6 progress0%
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Learning objectives
Identify sites, cells, edges and vertices.
Interpret an existing Voronoi diagram in context.
Use perpendicular-bisector geometry to construct or extend parts of a diagram.
Add a new site and determine how nearby cells change.
Use nearest-neighbour interpolation.
Solve location and optimization problems such as choosing a point farthest from existing sites or assigning a new point to the nearest site.
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
$\text{maximize }\min_i d(P,S_i)$
Unit 4
Not startedSubscription
Statistics and Probability
Statistics and Probability is the largest AI SL topic by recommended teaching time. The course emphasizes choosing suitable techniques, using technology correctly and interpreting what statistical output means. Calculation is important, but statistical judgment is equally important.
11 topics
Unit 4 progress0%
Topics in this unit
Topic 4.1Not started
Populations, samples, data quality and sampling
Define populations and samples, classify data, select and evaluate sampling methods, and judge whether real-world survey evidence is reliable, valid and fit for purpose.
240 min estimated
Topic 4.1 progress0%
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Learning objectives
Distinguish a population, sample, census, sampling frame, parameter and statistic.
Classify categorical and numerical data, including nominal, ordinal, discrete and continuous variables.
Compare primary and secondary data and evaluate their quality and relevance.
Design simple random, systematic and proportional stratified samples and evaluate convenience samples.
Identify coverage, selection, non-response, response, measurement and processing bias or error.
Evaluate reliability, validity, questionnaire design, ethics and the justified scope of a statistical conclusion.
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
Presenting and reading data
Organize, construct and interpret statistical displays, moving confidently among tables, bar charts, histograms, frequency polygons, cumulative-frequency graphs, box plots and stem-and-leaf diagrams.
250 min estimated
Topic 4.2 progress0%
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Learning objectives
Construct and interpret frequency tables for grouped and ungrouped categorical or numerical data.
Distinguish bar charts from histograms and calculate frequency density for unequal class widths.
Construct and compare frequency polygons using class midpoints and suitable vertical measures.
Use cumulative-frequency graphs to estimate medians, quartiles, percentiles and interval frequencies.
Construct and interpret box-and-whisker plots and stem-and-leaf diagrams.
Choose suitable displays, compare distributions using center and spread, and identify misleading graphical choices.
Use GDC and spreadsheet tools with correct classes, labels, scales and contextual interpretation.
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
Calculate, interpret and choose measures of center and spread for raw, frequency and grouped data, including the effects of transformations and outliers in practical contexts.
250 min estimated
Topic 4.3 progress0%
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Learning objectives
Calculate and interpret mean, median and mode for ungrouped data and frequency tables.
Determine quartiles, percentiles, range and interquartile range using a stated convention.
Calculate or interpret population and sample variance and standard deviation, including units.
Estimate the mean and standard deviation of grouped data using class midpoints.
Identify potential outliers with the 1.5×IQR rule and evaluate their effects without automatically deleting them.
Compare distributions using appropriate center-and-spread pairs, shape, outliers and context.
Use GDC one-variable statistics correctly with raw values, frequencies and grouped midpoints.
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
Scatter plots, Pearson correlation and linear regression
Use paired data, scatter plots, Pearson’s correlation coefficient and linear regression technology to describe associations, make predictions and evaluate real-world models responsibly.
250 min estimated
Topic 4.4 progress0%
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Learning objectives
Recognize bivariate data as paired observations of two quantitative variables measured on the same statistical units.
Construct and interpret scatter plots using appropriate axes, scales, units and contextual labels.
Describe the form, direction and strength of an association and identify unusual or influential observations.
Use technology to calculate Pearson’s product-moment correlation coefficient and interpret its sign and magnitude.
Explain why Pearson’s correlation coefficient measures linear association and why correlation does not establish causation.
Use a GDC to obtain and interpret the least-squares regression line of y on x.
Interpret regression gradients and intercepts in context, including their units and practical limitations.
Use a regression equation for interpolation and make cautious judgments about extrapolation.
Calculate and interpret predicted values and residuals.
Assess how outliers and high-leverage observations can change correlation, regression and predictions.
Communicate AI SL statistical conclusions with appropriate precision, technology evidence and contextual language.
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
Basic probability concepts
Model uncertainty using experiments, sample spaces, events, theoretical and experimental probability, complements and basic combined-event rules in practical contexts.
250 min estimated
Topic 4.5 progress0%
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Learning objectives
Distinguish experiments, trials and outcomes and construct complete sample spaces.
Represent sample spaces using lists, tables, grids or organized diagrams.
Define events as subsets of a sample space and use union, intersection and complement notation accurately.
Interpret probabilities on the scale from impossible to certain.
Calculate theoretical probability when finite outcomes are equally likely.
Calculate experimental probability and relative frequency and compare estimates with theoretical models.
Use complementary events to simplify ‘not’, ‘none’ and ‘at least one’ calculations.
Recognize mutually exclusive events and apply their addition rule.
Use the general addition rule for overlapping events.
Use basic multiplication rules for successive events, distinguishing independent and dependent situations.
Solve combined-event problems and communicate conclusions with appropriate assumptions, units and precision.
Use simulation or GDC technology to estimate probability and evaluate long-run behavior.
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
Venn diagrams, trees, conditional probability and independence
Represent combined events with Venn diagrams, tables and probability trees, then use conditional probability and independence to solve applied multi-stage problems.
250 min estimated
Topic 4.6 progress0%
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Learning objectives
Represent combined events using Venn diagrams, tree diagrams, sample-space diagrams and two-way tables.
Translate verbal conditions into union, intersection and complement notation.
Use the general addition rule without double-counting an intersection.
Recognize mutually exclusive events and distinguish them from independent events.
Construct tree diagrams whose outgoing branches total 1 at every node.
Multiply probabilities along a path and add disjoint paths for a combined event.
Calculate and interpret conditional probability using $P(A\mid B)$ .
Apply the multiplication rule $P(A\cap B)=P(A)P(B\mid A)$ .
Test independence using product or conditional-probability conditions.
Update probabilities correctly in experiments with and without replacement.
Choose an efficient representation and solve multi-stage application problems.
Communicate AI SL probability solutions with defined events, exact working and contextual conclusions.
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 and expected value
Model countable outcomes with discrete probability distributions, then use expected value and spread to evaluate games, risk and financial decisions.
250 min estimated
Topic 4.7 progress0%
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Learning objectives
Define a random variable and distinguish a discrete random variable from a continuous one.
Map experiment outcomes to numerical values of a discrete random variable.
Construct, read and graph probability distributions using tables, formulas and contexts.
Verify that every probability is valid and that a complete distribution totals 1.
Find unknown probabilities using the normalization condition.
Calculate expected value and interpret it as a long-run mean.
Use expected net gain to assess games, warranties and financial choices.
Calculate and interpret $E(X^2)$ , variance and standard deviation where relevant.
Compare decisions using both expected return and variability rather than mean alone.
Apply linear transformations to the mean, variance and standard deviation of a random variable.
Use probability weights correctly in calculator or GDC one-variable statistics.
Communicate AI SL solutions with notation, units, sufficient precision and contextual conclusions.
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
Recognize valid binomial settings, use GDC exact and cumulative functions, and interpret expected counts and variability in practical situations.
250 min estimated
Topic 4.8 progress0%
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Learning objectives
Recognize and justify the four conditions required for a binomial model in context.
Define a binomial random variable and identify the parameters n and p.
Write and interpret binomial notation and the integer support from 0 to n.
Calculate exact binomial probabilities using a GDC and understand the underlying formula.
Translate exactly, at most, fewer than, at least and more than into correct probability notation.
Use cumulative probabilities, complements and CDF differences with correct integer boundaries.
Calculate and interpret the expected value, variance and standard deviation of a binomial variable.
Use binomial PDF and CDF technology accurately and report calculator inputs clearly.
Apply binomial models to quality control, surveys, reliability and other real-world settings.
Assess independence and constant-probability assumptions, especially for sampling without replacement.
Check answers for suitable range, accuracy and reasonableness relative to the mean.
Present complete IB AI SL solutions using model notation, probability statements, technology and contextual conclusions.
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
Model continuous measurements with a normal distribution, use GDC normal CDF and inverse-normal functions, and interpret probabilities and cutoffs in practical contexts.
250 min estimated
Topic 4.9 progress0%
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Learning objectives
Describe the bell shape, symmetry, total area and other defining properties of a normal distribution.
Interpret the mean as the center and the standard deviation as the spread of a normal model.
Use the notation X distributed normally with mean mu and variance sigma squared.
Connect shaded areas under a normal density curve with lower-tail, upper-tail and interval probabilities.
Use symmetry and the 68–95–99.7 empirical rule to estimate and check probabilities.
Use a GDC normal CDF function to calculate probabilities from given boundaries.
Use an inverse-normal function to find values from cumulative probabilities.
Convert highest, lowest and central percentages into the correct left-tail area.
Find and interpret percentiles, cutoffs and equal-tail intervals in context.
Apply normal models to measurements, quality control and other real-world situations.
Evaluate whether a normal model is plausible from the context and distribution shape.
Present complete IB AI SL solutions with notation, shading, technology inputs, rounding, units and interpretation.
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
Rank paired data, handle ties correctly, calculate and interpret Spearman’s rank correlation, and choose appropriately between Spearman and Pearson correlation in real applications.
260 min estimated
Topic 4.10 progress0%
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Learning objectives
Explain rank correlation and monotonic association in accessible contextual language.
Recognize when Spearman rank correlation is more appropriate than Pearson correlation.
Rank paired data consistently and preserve the pairing between observations.
Assign average ranks correctly when observations are tied.
Calculate rank differences and their squared values in an organized table.
Use the Spearman shortcut formula correctly when there are no tied ranks.
Calculate Spearman correlation as Pearson correlation of average ranks when ties occur.
Use a GDC or spreadsheet workflow to obtain or verify Spearman’s rank correlation coefficient.
Interpret the sign and magnitude of Spearman’s coefficient in context.
Compare monotonic and linear association and distinguish Spearman from Pearson correlation.
Explain why correlation does not establish causation.
Present complete IB AI SL solutions with ranks, technology evidence, interpretation and limitations.
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 t-tests
Hypothesis testing uses sample evidence to evaluate population claims. AI SL students formulate hypotheses, use technology for chi-square and pooled two-sample t-tests, interpret p-values, check assumptions, and communicate conclusions in context.
300 min estimated
Topic 4.11 progress0%
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Learning objectives
Explain the purpose and limitations of hypothesis testing.
State null and alternative hypotheses in words and appropriate notation.
Interpret significance levels, p-values, and statistical significance.
Apply the p-value and critical-value decision rules correctly.
Calculate and interpret observed and expected frequencies.
Carry out chi-square goodness-of-fit tests using technology.
Carry out chi-square tests of independence using contingency tables.
Determine the correct degrees of freedom for each chi-square test.
Carry out pooled, unpaired two-sample t-tests using raw data or summary statistics.
Distinguish one-tailed and two-tailed alternatives.
Check the assumptions and conditions of each test.
Explain Type I and Type II errors conceptually in context.
Interpret GDC output and write complete contextual conclusions.
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
$df=(r-1)(c-1)$
Unit 5
Not startedSubscription
Calculus
Calculus in AI SL is introductory and application-focused. Students interpret rates of change, identify increasing and decreasing behaviour, find tangents and normals, optimize quantities and use integration for area. Technology supports the process, but basic derivative and antiderivative skills are still required.
8 topics
Unit 5 progress0%
Topics in this unit
Topic 5.1Not started
Limits, gradients and rates of change
Limits describe nearby behavior, while gradients measure how one quantity changes relative to another. Graphs, tables, secants, tangents and technology connect average rates to instantaneous rates in practical models.
210 min estimated
Topic 5.1 progress0%
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Learning objectives
Interpret finite limits from notation, graphs and numerical tables.
Distinguish a limiting value from the actual function value at a point.
Calculate and interpret the gradient of a straight line.
Calculate average rates of change and connect them to secant gradients.
Interpret instantaneous rates of change as tangent gradients and derivatives.
Estimate gradients from graphs and centered numerical data.
Use a GDC to investigate limits and numerical derivatives.
Interpret rates with correct signs, input values, units and real-world meaning.
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 shows where a modeled quantity rises or falls. Graphs, derivative graphs, sign diagrams and GDC methods identify increasing and decreasing intervals and classify stationary points in context.
215 min estimated
Topic 5.2 progress0%
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Learning objectives
Describe increasing and decreasing behavior from a graph, table or real-world context.
Use $f'(x)>0$ to identify increasing intervals and $f'(x)<0$ to identify decreasing intervals.
Find stationary values by solving $f'(x)=0$ and respect points where the derivative is undefined.
Construct and interpret a first-derivative sign diagram.
Classify stationary points using changes in the sign of f′.
Use the graph of f′ to describe the behavior of f.
Use a GDC to locate stationary points and verify intervals of increase or decrease.
Communicate interval notation, domain restrictions and contextual interpretations accurately.
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
Polynomial differentiation produces a gradient or rate function. The power, constant, sum and difference rules support tangent gradients, stationary points, GDC checks and practical rate interpretations.
220 min estimated
Topic 5.3 progress0%
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Learning objectives
Explain differentiation as finding a gradient function or instantaneous rate of change.
Use f′(x), y′ and dy/dx notation accurately.
Apply the power rule to polynomial terms with nonnegative integer powers.
Differentiate constants, constant multiples, sums and differences.
Differentiate and simplify complete polynomial functions term by term.
Find gradients at specified inputs and equations of tangents where required.
Find and classify stationary points of polynomial functions.
Use GDC methods and interpret polynomial derivatives with correct signs, inputs and units.
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
Tangents describe the instantaneous direction and rate of change of a curve, while normals pass through the same point at right angles. Derivatives, point-slope form and GDC methods connect these lines to practical models.
220 min estimated
Topic 5.4 progress0%
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Learning objectives
Explain a tangent as the line matching a curve's instantaneous gradient at a contact point.
Find tangent gradients and equations using derivatives and point-slope form.
Explain a normal and use the negative-reciprocal gradient relationship correctly.
Find normal equations, including horizontal-tangent and vertical-normal special cases.
Determine contact points from gradient, coordinate, parallel-line or perpendicular-line conditions.
Use a GDC to evaluate derivatives, locate contact points and verify tangent or normal equations.
Interpret tangent gradients and linear approximations with appropriate contextual units.
Solve multi-step IB AI SL tangent and normal problems and validate every solution.
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 and area
Integration reverses differentiation and measures accumulated change. Antiderivatives, definite integrals and carefully interpreted shaded regions connect algebraic rules with practical area and rate models.
230 min estimated
Topic 5.5 progress0%
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Learning objectives
Explain integration as reverse differentiation and identify antiderivatives.
Apply basic constant, power, sum and difference integration rules.
Include and determine the constant of integration in indefinite integrals.
Evaluate definite integrals using antiderivatives and the Fundamental Theorem of Calculus.
Distinguish signed area from total geometric area and split intervals at zeros when necessary.
Find areas bounded by a curve and the coordinate axes in appropriate cases.
Interpret definite integrals as accumulated change with correct units and contextual meaning.
Use a GDC to evaluate integrals, locate zeros and verify area calculations in IB AI SL problems.
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 a smooth curve has zero gradient. Derivative tests, graphs and GDC methods locate and classify local maxima, local minima and stationary inflections in mathematical and practical models.
230 min estimated
Topic 5.6 progress0%
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Learning objectives
Explain a stationary point and use the condition $f'(x)=0$ .
Find complete stationary-point coordinates by solving for x and evaluating the original function.
Distinguish local maxima, local minima and stationary points of inflection.
Classify stationary points using first-derivative sign changes.
Use the second-derivative test when it is conclusive and recognize when it is inconclusive.
Relate features of a function graph to the signs and zeros of its derivative graph.
Interpret stationary points with domains, endpoints, units and contextual meaning.
Use a GDC to locate, check and communicate stationary-point results in IB AI SL problems.
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
Build one-variable models from real constraints, use derivatives and GDC methods to locate feasible maxima or minima, and interpret optimum dimensions, costs, revenues, profits and other practical quantities.
260 min estimated
Topic 5.7 progress0%
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Learning objectives
Explain optimization as finding a feasible maximum or minimum.
Translate real-world information into variables, assumptions, constraints and a feasible domain.
Form a one-variable objective function for geometry, cost, revenue, profit and other applications.
Use derivatives to locate interior optimum candidates.
Classify candidates using derivative signs, second derivatives or complete value comparisons.
Include endpoints, boundaries and discrete restrictions when required.
Use GDC graph, zero, maximum, minimum and table tools responsibly.
Interpret the input and optimum value with suitable units and accuracy.
Evaluate whether a model and its assumptions are reasonable in 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 rule for approximating area
Approximate definite integrals and real-world accumulated quantities from formulas, tables and graphs using the trapezoidal rule, with careful treatment of spacing, error direction, units and technology.
250 min estimated
Topic 5.8 progress0%
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Learning objectives
Explain why numerical integration is needed and interpret the trapezoidal rule geometrically.
Calculate equal interval widths and distinguish trapezia from ordinates.
Apply the composite trapezoidal rule accurately to tables, graphs and functions.
Handle irregular spacing by summing individual trapezia.
Distinguish signed accumulation from total geometric area.
Use concavity to identify likely overestimates and underestimates.
Investigate how increasing the number of intervals affects accuracy.
Use a GDC to generate values and check numerical approximations.
Interpret trapezoidal estimates in real-world contexts with correct units and assumptions.
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
$T=\sum\limits_{i=0}^{n-1}\frac{x_{i+1}-x_i}{2}(y_i+y_{i+1})$

Mathematics: Applications and Interpretation

Course Overview

Continue where you left off

Course units

Number and Algebra

Topics in this unit

Scientific notation and operations with numbers

Learning objectives

Key formulas

Arithmetic sequences and series

Learning objectives

Key formulas

Geometric sequences and series

Learning objectives

Key formulas

Financial applications

Learning objectives

Key formulas

Integer exponents and logarithms

Learning objectives

Key formulas

Approximation, bounds and percentage error

Learning objectives

Key formulas

Amortization and annuities

Learning objectives

Key formulas

Solving equations with technology

Learning objectives

Key formulas

Functions

Topics in this unit

Equations of Straight Lines

Learning objectives

Key formulas

The Concept of a Function

Learning objectives

Key formulas

Graphing Functions with Technology

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Key formulas

Key features and intersections of graphs

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Key formulas

Modelling with key function families

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Key formulas

The modelling cycle: fit, test, interpret and refine

Learning objectives

Key formulas

Geometry and Trigonometry

Topics in this unit

Distance, midpoint and three-dimensional solids

Learning objectives

Key formulas

Trigonometry in right and non-right triangles

Learning objectives

Key formulas

Applications of trigonometry and diagram construction

Learning objectives

Key formulas

Radian measure, arcs and sectors

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Key formulas

Perpendicular bisectors and nearest-point geometry

Learning objectives

Key formulas

Voronoi diagrams

Learning objectives

Key formulas

Statistics and Probability

Topics in this unit

Populations, samples, data quality and sampling

Learning objectives

Key formulas

Presenting and reading data

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Key formulas

Measures of central tendency and spread

Learning objectives