IB Math AI SL Revision Notes

Formulas, conditions, key connections and common exam traps across the five AI SL topics. Use it alongside the formula booklet and regular question practice.

Written by IB Math tutor Abdulhamid

IB Math Master · Updated November 2026

For the current AI SL syllabus, for exams up to November 2028.

Get the printable PDF

How to use these notes

Each topic is split into short sections. Every section keeps its formulas, conditions and diagrams together.

Connection (blue dot): links between formulas that the formula booklet does not state.
Exam note (gold diamond): short reminders and common traps.
Video: a link to my explanation of the topic.

Topic 1 Number and Algebra

Numbers and Approximation

Standard form and accuracy

Name	Formula	Condition
Standard form	$a \times 10^{k}$	$1 \leq a < 10$ , $k \in ℤ$

Connection: Standard form products: multiply the $a$ parts, add the powers of 10, then rescale to $1 \leq a < 10$ .
Exam note: The GDC shows 4.52E−4. Write $4.52 \times 10^{- 4}$ .
Exam note: Give answers exactly or to 3 significant figures, unless the question says otherwise.
Exam note: Keep full calculator values in your working. Round only the final answer.

Percentage error

Name	Formula	Condition
Percentage error	$ε = \| \frac{v_{A} - v_{E}}{v_{E}} \| \times 100 %$	$v_{E}$ = exact value $v_{A}$ = approximate value

Exam note: Percentage error divides by the exact value $v_{E}$ .

Back to topics

Sequences and Series

Arithmetic and geometric sequences

Item	Arithmetic common difference $d$	Geometric common ratio $r$
$n$ th term	$u_{n} = u_{1} + (n - 1) d$	$u_{n} = u_{1} r^{n - 1}$
Sum of $n$ terms	$\begin{aligned} S_{n} & = \frac{n}{2} (2 u_{1} + (n - 1) d) \\ = \frac{n}{2} (u_{1} + u_{n}) \end{aligned}$	$S_{n} = \frac{u_{1} (r^{n} - 1)}{r - 1} = \frac{u_{1} (1 - r^{n})}{1 - r}, r \neq 1$
Finding $d$ or $r$	$u_{2} - u_{1} = u_{3} - u_{2} = \dots = d$	$\frac{u_{2}}{u_{1}} = \frac{u_{3}}{u_{2}} = \dots = r$

Exam note: $n$ counts terms, so $n \in ℤ^{+}$ .

Sums and sigma notation

Name	Formula	Condition
Sigma notation	$\sum_{i = 1}^{n} u_{i} = u_{1} + u_{2} + \dots + u_{n}$	$i$ runs through whole numbers

Connection: For any sequence:
$S_{1} = u_{1}$
$S_{2} = u_{1} + u_{2}$
$S_{3} = u_{1} + u_{2} + u_{3}$
$u_{n} = S_{n} - S_{n - 1}$ for $n \geq 2$
Exam note: In sigma notation, check the first and last values of $i$ .

Back to topics

Financial Mathematics

Video: Financial mathematics (opens YouTube in a new tab)

Compound interest and depreciation

Name	Formula	Condition
Compound interest	$F V = P V \times {(1 + \frac{r}{100 k})}^{k n}$ $F V$ = future value, $P V$ = present value, $n$ = number of years, $k$ = number of compounding periods per year, $r %$ = nominal annual rate of interest	geometric, ratio $1 + \frac{r}{100 k}$ per period
Depreciation	$V_{n} = V_{0} \times {(1 - \frac{r}{100})}^{n}$ $V_{0}$ = starting value, $V_{n}$ = value after $n$ years, $r %$ = annual rate of depreciation	geometric, ratio $1 - \frac{r}{100}$ per year

Exam note: Number of years to reach a target: the answer is not a whole number. Round up to the next whole compounding period (to whole years when compounding is yearly).

Loans and annuities

TVM solver entry	Meaning
$N$	number of payments
$I %$	annual interest rate
$P V$	present value: the amount borrowed or invested at the start
$P M T$	payment each period
$F V$	future value: the amount at the end
$P / Y$	payments per year
$C / Y$	compounding periods per year

Connection: Loan fully repaid: $F V = 0$
Connection: Total repaid: add up all the payments. That is the number of payments times the amount of each payment.
Connection: Interest paid: the total repaid minus the amount borrowed.
Exam note: Money paid out is negative and money received is positive.
Exam note: Write down the values you enter in the TVM solver. They earn method marks.
Exam note: Monthly payments: $P / Y = 12$ . Set $C / Y$ from the question.

Back to topics

Exponents and Logarithms

Exponent laws

Name	Formula	Condition
Same base	$a^{m} \times a^{n} = a^{m + n}$ $\frac{a^{m}}{a^{n}} = a^{m - n}$ ${(a^{m})}^{n} = a^{m n}$
Zero and negative	$a^{0} = 1$ $a^{- n} = \frac{1}{a^{n}}$	$a \neq 0$

Logarithms

Name	Formula	Condition
Definition of a log	$a^{x} = b ⟺ x = \log_{a} b$	$a > 0$ , $b > 0$ , $a \neq 1$
Base 10 and base $e$	$\log x = \log_{10} x$ $\ln x = \log_{e} x$	$e \approx 2.718$

Connection: $\log_{a} 1 = 0$ and $\log_{a} a = 1$
Connection: $a^{\log_{a} x} = x$ and $\log_{a} a^{x} = x$
Exam note: On the GDC, log is base 10 and ln is base $e$ .
Exam note: To find an unknown power, use the GDC solver.

Back to topics

Systems of Linear Equations

Systems of linear equations

Exam note: Systems of up to 3 equations and polynomial equations: solve on the GDC, but write the equations first.

Back to topics

Topic 2 Functions

Straight Lines

Equations and gradients

Name	Formula	Condition
Straight line	$y = m x + c$ $a x + b y + d = 0$ $y - y_{1} = m (x - x_{1})$
Gradient	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$	$x_{1} \neq x_{2}$
Parallel lines	$m_{1} = m_{2}$
Perpendicular lines	$m_{1} \times m_{2} = - 1$ $m_{2} = - \frac{1}{m_{1}}$

Connection: In a linear model, $m$ is the change in $y$ for each unit increase in $x$ and $c$ is the value at $x = 0$ .
Exam note: $a x + b y + d = 0$ usually needs whole numbers $a$ , $b$ and $d$ .
Exam note: A vertical line is $x = k$ . Its gradient is undefined.

Back to topics

Functions and Inverses

Functions

Name	Formula	Condition
Function	Each $x$ in the domain gives exactly one $y$
Domain, range	Domain: the allowed inputs. Range: the outputs.

Exam note: In a model, the domain comes from the context, for example $t \geq 0$ .

Inverse functions

Connection: The graph of $f^{- 1}$ is the reflection of $f$ in $y = x$ .
Connection: Domain of $f^{- 1}$ = range of $f$
Range of $f^{- 1}$ = domain of $f$
Exam note: $f^{- 1} (x)$ is not $\frac{1}{f (x)}$ .

(1, 2)

f

and

(2, 1)

f^{- 1}

Back to topics

Polynomial and Variation Models

Video: Graphing quadratics (opens YouTube in a new tab)

Quadratic models

Form	Equation	What it shows
Standard	$f (x) = a x^{2} + b x + c,$ $a \neq 0$	$y$ -intercept $(0, c)$ axis of symmetry $x = - \frac{b}{2 a}$ vertex $(- \frac{b}{2 a}, f (- \frac{b}{2 a}))$
Factor	$f (x) = a (x - p) (x - q)$	zeros $p$ and $q$ axis of symmetry $x = \frac{p + q}{2}$
Vertex	$f (x) = a (x - h)^{2} + k$	vertex $(h, k)$ axis of symmetry $x = h$

Connection: $a > 0$ : minimum at the vertex. $a < 0$ : maximum at the vertex.
Exam note: Check the maximum or minimum lies inside the domain of the model.

f (x)

= x^{2} - 6 x + 5

= (x - 1) (x - 5)

= (x - 3)^{2} - 4

Cubic models

Name	Formula	Condition
Cubic	$f (x) = a x^{3} + b x^{2} + c x + d, a \neq 0$ At most 2 turning points

Exam note: Zeros, turning points and intersections: use the GDC.

Direct and inverse variation

Name	Formula	Condition
Direct variation	$y$ is directly proportional to $x^{n}$ : $y = a x^{n}, n \in ℤ^{+}, a \neq 0$ The graph passes through the origin. With $n = 1$ it is a straight line.	$y = 2 x$ $(n = 1)$
Inverse variation	$y$ is inversely proportional to $x^{n}$ : $y = \frac{a}{x^{n}}, n \in ℤ^{+}, a \neq 0$ Asymptotes $x = 0$ and $y = 0$ .	$y = \frac{4}{x}$ $(n = 1)$

Name

Formula

Condition

Direct variation

$y$ is directly proportional to $x^{n}$ :

y = a x^{n}, n \in ℤ^{+}, a \neq 0

The graph passes through the origin. With $n = 1$ it is a straight line.

y = 2 x

(n = 1)

Inverse variation

$y$ is inversely proportional to $x^{n}$ :

y = \frac{a}{x^{n}}, n \in ℤ^{+}, a \neq 0

Asymptotes $x = 0$ and $y = 0$ .

y = \frac{4}{x}

(n = 1)

Exam note: “ $y$ is proportional to $x^{2}$ ” means $y = a x^{2}$ . “ $y$ is inversely proportional to $x$ ” means $y = \frac{a}{x}$ . Find $a$ from one data point.

Back to topics

Exponential Models

Exponential models

Name	Formula	Condition
Exponential	$f (x) = k a^{x} + c, a > 0, a \neq 1$ $f (x) = k e^{r x} + c$ asymptote $y = c$

Connection: The value at $x = 0$ is $k + c$ .
Connection: $k e^{r x}$ : $r > 0$ growth and $r < 0$ decay
Exam note: In $f (t) = k e^{r t} + c$ , the initial value is $k + c$ , not $k$ .
Exam note: Long-term value: let $x \to \infty$ . Exponential decay tends to $c$ .

y = 4 e^{- 0.5 x} + 1

: starts at

k + c = 5

and tends to

c = 1

Back to topics

Sinusoidal Models

Sinusoidal models

Name	Formula	Condition
Sinusoidal	$f (x) = a \sin (b x) + d$ $f (x) = a \cos (b x) + d$	$b > 0$

Feature	From the function	From the graph
Amplitude	$\| a \|$	$\frac{max - min}{2}$
Period	$\frac{360^{\circ}}{b}$	from a maximum to the next maximum, or from a minimum to the next minimum then $b$ $= \frac{360^{\circ}}{period}$
Principal axis	$y = d$	$y = \frac{max + min}{2}$
Maximum, minimum	$d + \| a \|, d - \| a \|$	the highest and lowest $y$ -values

Exam note: Set the GDC to degrees.

Back to topics

Modelling

Choosing and using a model

Exam note: Choose a model from the shape of the data and the context. Give a reasonable domain.
Exam note: Predictions far outside the data are not reliable.
Exam note: “Sketch”: show the shape and key features. “Draw”: plot accurately to scale.
Exam note: Label intercepts, turning points, asymptotes and end points of the domain.

Back to topics

Topic 3 Geometry and Trigonometry

3D Geometry and Triangles

Points in 3D

Name	Formula	Condition
Distance	$d = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} + (z_{1} - z_{2})^{2}}$
Midpoint	$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}, \frac{z_{1} + z_{2}}{2})$

Solids

Name	Formula	Condition
Right pyramid	$V = \frac{1}{3} A h,$ $A$ = area of base, $h$ = height
Right cone	$V = \frac{1}{3} π r^{2} h$ curved surface $A = π r l$ total surface $= π r l + π r^{2}$ $l^{2} = r^{2} + h^{2}$ , $l$ = slant height
Sphere	$V = \frac{4}{3} π r^{3}$ $A = 4 π r^{2}$
Hemisphere	$V = \frac{2}{3} π r^{3}$ curved surface $2 π r^{2}$

Right-angled triangles

Name	Formula	Condition
Ratios	$\sin θ = \frac{opp}{hyp}$ $\cos θ = \frac{adj}{hyp}$ $\tan θ = \frac{opp}{adj}$
Pythagoras	$a^{2} + b^{2} = c^{2},$ $c$ = hypotenuse

Opposite and adjacent are named from

θ

Angle between a line and a plane

Connection: Drop a perpendicular from a point on the line to the plane. This gives the projection of the line on the plane.
Connection: $θ$ is the angle between the line and its projection. Find it in that right-angled triangle.

θ

is measured to the projection on the plane.

Any triangle

Name	Formula	Condition
Sine rule	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Cosine rule	$c^{2} = a^{2} + b^{2} - 2 a b \cos C$ $\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$
Area	$A = \frac{1}{2} a b \sin C$

Side

a

is opposite angle

A

, and so on.

Use the sine rule when you know	Use the cosine rule when you know
two angles and one side	two sides and the angle between them
two sides and an angle not between them	all three sides

Exam note: Draw and label a diagram from the words first.

Elevation, depression and bearings

Name	Formula	Condition
Elevation, depression	Measured from the horizontal
Bearings	Measured clockwise from north, with three figures, for example $065^{\circ}$

Back to topics

Circles

Arcs and sectors

Name	Formula	Condition
Arc length	$l = \frac{θ}{360} \times 2 π r$	degrees
Sector area	$A = \frac{θ}{360} \times π r^{2}$	degrees

Back to topics

Voronoi Diagrams

Perpendicular bisectors

Name	Formula	Condition
Perpendicular bisector of $A B$	Through the midpoint $M$ of $A B$ , gradient $- \frac{1}{m_{A B}}$ Every point on it is equidistant from $A$ and $B$ .

Name

Formula

Condition

Perpendicular bisector of

A B

Through the midpoint $M$ of $A B$ , gradient $- \frac{1}{m_{A B}}$

Every point on it is equidistant from $A$ and $B$ .

Connection: Equation: $y - y_{M}$ $= - \frac{1}{m_{A B}} (x - x_{M})$
Connection: $A B$ horizontal: the bisector is $x = x_{M}$ . $A B$ vertical: the bisector is $y = y_{M}$ .

Any point

P

on the bisector:

P A = P B

Voronoi diagrams

Name	Formula	Condition
Cell	All points closer to its site than to any other site
Edge	Part of the perpendicular bisector of two sites
Vertex	Where at least three edges meet. Equidistant from 3 or more sites.
Nearest neighbour interpolation	A point takes the value of the site whose cell contains it

Connection: Adding a site: draw the bisectors between the new site and its neighbours, then remove the old edges inside the new cell.
Connection: The bisectors of $A B$ and $B C$ meet at a point equidistant from $A$ , $B$ and $C$ , so the bisector of $A C$ passes through it too.

Sites in blue, vertices in gold.

Back to topics

Topic 4 Statistics and Probability

Sampling and Data

Data and sampling methods

Name	Formula	Condition
Types of data	Discrete data is counted. Continuous data is measured.
Simple random	Every possible sample of that size is equally likely
Systematic	Every $k$ th member from a list	$k$ $= \frac{population}{sample size}$
Stratified	Each group in proportion: $\frac{group size}{population} \times sample size$
Quota	A fixed number from each group, not chosen at random
Convenience	The members easiest to reach

Connection: Random sampling reduces bias. Quota and convenience samples are not random.

Back to topics

Descriptive Statistics

Mean and spread

Name	Formula	Condition
Mean	$\bar{x} = \frac{\sum f_{i} x_{i}}{n}$	$n = \sum f_{i}$
Grouped data	Mid-interval values as $x_{i}$ . Modal class: highest frequency.	gives an estimate
Standard deviation	$σ$ from the GDC. Variance $= σ^{2}$

Connection: Add $k$ to every value: $mean + k$ . $σ$ and IQR unchanged.
Connection: Multiply every value by $k$ : $mean \times k$ , $σ \times | k |$ , $σ^{2} \times k^{2}$
Exam note: Use $σ_{x}$ from the GDC for the standard deviation of the data.

Quartiles, box plots and histograms

Name	Formula	Condition
Interquartile range	$I Q R = Q_{3} - Q_{1}$
Outlier	below $Q_{1} - 1.5 \times I Q R$ or above $Q_{3} + 1.5 \times I Q R$
Cumulative frequency	read across at $\frac{n}{4}$ for $Q_{1}$ , $\frac{n}{2}$ for the median, $\frac{3 n}{4}$ for $Q_{3}$

Exam note: Remove an outlier only if it is an error.
Exam note: Histograms of continuous data have no gaps between bars.

Box plot data: 1, 3, 4, 6, 7, 8, 10, 11, 13, 14, 16. Read each value from the scale.

Back to topics

Correlation and Regression

Video: Regression line (opens YouTube in a new tab)

Correlation

Name	Formula	Condition
Pearson’s $r$	$- 1 \leq r \leq 1$	linear correlation
Spearman’s $r_{s}$	First rank the data for each variable. Then find Pearson’s $r$ of the ranks.	tied ranks: use the average

Connection: $r$ close to $\pm 1$ : strong linear correlation. $r$ close to 0: weak or no linear correlation.
Connection: $r_{s} = 1$ : always increasing, not always linear. $r_{s}$ is less affected by outliers than $r$ .
Connection: Scatter diagram: plot each pair $(x, y)$ as a point. A line of best fit drawn by eye passes through the mean point $(\bar{x}, \bar{y})$ .
Exam note: Use $r$ for linear data and $r_{s}$ for non-linear but monotonic data.
Exam note: Correlation does not mean causation.

r

close to 1: strong positive

r

close to 0: no linear correlation

r_{s} = 1

: always increasing,

r < 1

Regression line

Name	Formula	Condition
Regression line of $y$ on $x$	$y = a x + b$ : predicts $y$ from $x$

Connection: For each increase of 1 in $x$ , $y$ changes by $a$ .
When $x = 0$ , $y = b$ .
Connection: The line passes through $(\bar{x}, \bar{y})$ .
Exam note: Interpolation is reliable when the correlation is strong ( $r$ is close to $\pm 1$ ). Extrapolation is not reliable.

The line passes through

(\bar{x}, \bar{y})

Back to topics

Probability

Video: Basic theoretical probability (opens YouTube in a new tab)

Basic rules

Name	Formula	Condition
Probability of $A$	$P (A)$ $= \frac{n (A)}{n (U)}$ , equally likely outcomes
Complementary events	$P (A) + P (A') = 1$
Combined events	$P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Mutually exclusive	$P (A \cap B) = 0$
Expected number	$n p$ for $n$ trials with probability $p$

Connection: $P (A)$ $= P (A \cap B) + P (A \cap B')$

P (A) + P (B)

counts

A \cap B

twice, so subtract

P (A \cap B)

once.

Conditional probability and independence

Name	Formula	Condition
Conditional	$P (A ∣ B) = \frac{P (A \cap B)}{P (B)}$
Independent	$P (A \cap B) = P (A) P (B)$

Connection: Independent $⟺ P (A ∣ B)$ $= P (A)$
Connection: Mutually exclusive events with non-zero probabilities are never independent.
Exam note: “Given that” means conditional: divide by the probability of the given event.

Tree diagrams

Connection: Multiply along branches and add the end results.
Connection: A conditional probability from a tree: the branch you want over all branches that give the condition.
Exam note: Without replacement, the second-stage probabilities change.

P (B ∣ A)

= \frac{0.01 \times 0.95}{0.01 \times 0.95 + 0.99 \times 0.08}

Back to topics

Binomial Distribution

Video: Binomial distribution (opens YouTube in a new tab)

Discrete random variables

Name	Formula	Condition
Discrete distribution	$\sum P (X = x) = 1$ $E (X) = \sum x P (X = x)$

Connection: Fair game: $E (gain) = 0$

Binomial distribution

Name	Formula	Condition
Binomial	$X \sim B (n, p)$ $E (X) = n p$ $Var (X) = n p (1 - p)$

Connection: Conditions: fixed $n$ , two outcomes, constant $p$ , independent trials
Connection: $P (X \geq k)$ $= 1 - P (X \leq k - 1)$
$P (X > k)$ $= 1 - P (X \leq k)$
Exam note: GDC: pdf for $P (X = x)$ and cdf for $P (X \leq x)$ .
Exam note: “At least”, “more than” and “fewer than”: rewrite with $\leq$ before using the cdf.

X \sim B (6, 0.4)

P (X \geq 3)

= 1 - P (X \leq 2)

Back to topics

Normal Distribution

The normal distribution

Name	Formula	Condition
Normal	$X \sim N (μ, σ^{2})$ , $σ^{2} =$ variance

Exam note: $N (μ, σ^{2})$ gives the variance. Take the square root for $σ$ .
Exam note: Inverse normal: check which tail the GDC uses.

Within

1 σ

: about 68%. Within

2 σ

: about 95%. Within

3 σ

: about 99.7%.

Back to topics

Hypothesis Tests

Setting up a test

Name	Formula	Condition
Hypotheses	$H_{0}$ : no effect or no change. $H_{1}$ : what you are testing for.
One-tailed, two-tailed	One-tailed: $H_{1}$ uses $<$ or $>$ . Two-tailed: $H_{1}$ uses $\neq$ .
Decision	$p$ -value $<$ significance level: reject $H_{0}$

Exam note: Conclude in context: “There is sufficient evidence at the 5% level that…”
Exam note: Never say $H_{0}$ is proved. Say “do not reject $H_{0}$ ”.

Choosing the test

You are testing	Test	$H_{0}$	$H_{1}$	Use when
two means	two-sample $t$ -test	$μ_{1} = μ_{2}$	$μ_{1} \neq μ_{2}$	independent samples from normal populations. Pooled: equal variances.
independence	$χ^{2}$ test for independence	independent	not independent	a contingency table
a distribution	$χ^{2}$ goodness of fit	the data fit	the data do not fit	frequencies in classes

Back to topics

$χ^{2}$ Tests

$χ^{2}$ tests

Test	$H_{0}$	Expected frequency $f_{e}$	Degrees of freedom $ν$
Independence	the two variables are independent	$f_{e} = \frac{row total \times column total}{grand total}$	$ν = (r - 1) (c - 1)$ $r$ rows, $c$ columns
Goodness of fit	the data fit the given distribution	$f_{e} =$ total frequency $\times$ probability of the class	$ν =$ number of classes $- 1$

Connection: $f_{o} =$ observed frequency: the counts in the data. $f_{e} =$ expected frequency: the counts you would expect if $H_{0}$ were true.
Connection: The GDC gives $χ_{c a l c}^{2}$ and the $p$ -value. Reject $H_{0}$ when the $p$ -value $<$ significance level.
Exam note: Every expected frequency must be at least 5. If not, combine classes and reduce $ν$ .

Back to topics

Topic 5 Calculus

Derivatives, Tangents and Normals

The derivative

Name	Formula	Condition
Derivative	$\frac{d y}{d x} = f' (x)$ : the gradient function and the rate of change of $y$ with respect to $x$
Power rule	$f (x) = a x^{n} \Rightarrow f' (x) = a n x^{n - 1}$

Connection: The derivative is the limit of the gradient of a chord as the chord shrinks.
Connection: Rewrite first: $\frac{1}{x^{2}} = x^{- 2}$

h \to 0

the chord turns into the tangent.

Tangents and normals

Name	Formula	Condition
Tangent at $(x_{1}, y_{1})$	$y - y_{1} = m (x - x_{1}), m = f' (x_{1})$
Normal at $(x_{1}, y_{1})$	$y - y_{1} = - \frac{1}{m} (x - x_{1}), m \neq 0$

Connection: The normal is perpendicular to the tangent.
Exam note: Find $y_{1}$ from $f (x_{1})$ , not from $f' (x_{1})$ .
Exam note: GDC: the derivative at a point and the tangent line can be checked.

y = x^{3} - 2 x

(1, - 1)

tangent

y = x - 2

, normal

y = - x

Back to topics

Stationary Points and Optimisation

Shape of a graph

If	Then
$f' (x) > 0$	$f$ is increasing
$f' (x) < 0$	$f$ is decreasing
$f' (a) = 0$	stationary point at $x = a$

Connection: $f$ turns where $f' = 0$ and $f'$ changes sign.

f (x) = x^{3} - 3 x

f' (x) = 3 x^{2} - 3 = 0

gives the turning points

a = - 1

and

a = 1

Optimisation

Connection: Write the quantity in one variable, set the derivative to 0, then confirm max or min.
Exam note: Check the end points of the domain too.
Exam note: Give the value of the quantity as well as the $x$ where it occurs, if asked.

Back to topics

Integration

Standard integrals

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C, n \neq - 1

Connection: A boundary condition finds $C$ .
Connection: Integration reverses differentiation: check by differentiating.
Exam note: Add $+ C$ to every indefinite integral.

Definite integrals

Name	Formula	Condition
Definite integral	$\int_{a}^{b} f (x) d x$	the GDC gives the value

Back to topics

Areas and the Trapezoidal Rule

Areas

Name	Formula	Condition
With the $x$ -axis	$A = \int_{a}^{b} y d x$ when $y > 0$

Trapezoidal rule

Name	Formula	Condition
Trapezoidal rule	$\int_{a}^{b} y d x \approx \frac{1}{2} h ((y_{0} + y_{n}) + 2 (y_{1} + y_{2} + \dots + y_{n - 1}))$ $h = \frac{b - a}{n}$

Connection: More strips give a better estimate.
Exam note: $n$ strips need $n + 1$ values of $y$ .

h = 0.5

: estimate 4.75, exact

\frac{14}{3} \approx 4.67

Back to topics

Cover of the AI SL Revision Notes PDF, updated November 2026

Prefer a printable copy?

Receive the AI SL Revision Notes as a PDF, ready to print or keep offline. Free to share, unchanged.

19 pages · 5 topics · 26 sections

Need help applying these ideas?

These notes show the formulas and their conditions. One-to-one tutoring looks at your own work, finds what is limiting your grade, chooses the right practice, and gives feedback on your reasoning.

Explore one-to-one tutoring

Prefer a fixed programme? See the revision groups starting January 2027

Related pages and updates

Looking for what the course covers? View the AI SL syllabus
Found an error? Email contact@ibmathmaster.com. Corrections appear in the next update.
Last updated: November 2026.

How to use these notes

Topic 1 Number and Algebra

1Numbers and Approximation

Standard form and accuracy

Percentage error

2Sequences and Series

Arithmetic and geometric sequences

Sums and sigma notation

3Financial Mathematics

Compound interest and depreciation

Loans and annuities

4Exponents and Logarithms

Exponent laws

Logarithms

5Systems of Linear Equations

Systems of linear equations

Topic 2 Functions

1Straight Lines

Equations and gradients

2Functions and Inverses

Functions

Inverse functions

3Polynomial and Variation Models

Quadratic models

Cubic models

Direct and inverse variation

4Exponential Models

Exponential models

5Sinusoidal Models

Sinusoidal models

6Modelling

Choosing and using a model

Topic 3 Geometry and Trigonometry

13D Geometry and Triangles

Points in 3D

Solids

Right-angled triangles

Angle between a line and a plane

Any triangle

Elevation, depression and bearings

2Circles

Arcs and sectors

3Voronoi Diagrams

Perpendicular bisectors

Voronoi diagrams

Topic 4 Statistics and Probability

1Sampling and Data

Data and sampling methods

2Descriptive Statistics

Mean and spread

Quartiles, box plots and histograms

3Correlation and Regression

Correlation

Regression line

4Probability

Basic rules

Conditional probability and independence

Tree diagrams

5Binomial Distribution

Discrete random variables

Binomial distribution

6Normal Distribution

The normal distribution

7Hypothesis Tests

Setting up a test

Choosing the test

8χ2 Tests

χ2 tests

Topic 5 Calculus

1Derivatives, Tangents and Normals

The derivative

Tangents and normals

2Stationary Points and Optimisation

Shape of a graph

Optimisation

3Integration

Standard integrals

Definite integrals

4Areas and the Trapezoidal Rule

Areas

Numbers and Approximation

Sequences and Series

Financial Mathematics

Exponents and Logarithms

Systems of Linear Equations

Straight Lines

Functions and Inverses

Polynomial and Variation Models

Exponential Models

Sinusoidal Models

Modelling

3D Geometry and Triangles

Circles

Voronoi Diagrams

Sampling and Data

Descriptive Statistics

Correlation and Regression

Probability

Binomial Distribution

Normal Distribution

Hypothesis Tests

$χ^{2}$ Tests

$χ^{2}$ tests

Derivatives, Tangents and Normals

Stationary Points and Optimisation

Integration

Areas and the Trapezoidal Rule