IB Mathematics Applications & Interpretation Data Booklet - Formula Reference | RevisionDojo

IB Mathematics Applications & Interpretation Data Booklet

RevisionDojo's Mathematics: Applications and Interpretation (AI) SL and HL Formula Booklet is designed to give IB students a clear, exam-ready reference to every essential formula you need. Unlike generic math resources, our booklet is directly aligned with the IB AI syllabus, ensuring that you revise with the exact formulas required for your exams.

Whether you're working on statistics, probability, finance, or functions, our clean layout and topic-by-topic organization make it simple to locate and apply the right formula under exam conditions. For students aiming to study smarter, this resource pairs perfectly with our notes, question bank, and papers, providing you with a complete toolkit for IB Mathematics AI success.

Topic 1: Number and algebra – SL


$\text{1.2}$	The $n$ th term of an arithmetic sequence	$u_{n}=u_{1}+(n-1) d$
	The sum of $n$ terms of an arithmetic sequence	$S_{n}=\frac{n}{2}\left(2 u_{1}+(n-1) d\right)$
		$S_{n}=\frac{n}{2}\left(u_{1}+u_{n}\right)$
$\text{1.3}$	The $n$ th term of a geometric sequence	$u_{n}=u_{1} r^{n-1}$
	The sum of $n$ terms of a finite geometric sequence	$S_{n}=\frac{u_{1}\left(r^{n}-1\right)}{r-1}=\frac{u_{1}\left(1-r^{n}\right)}{1-r}, r \neq 1$
$\text{1.4}$	Compound interest	$F V=P V \times\left(1+\frac{r}{100 k}\right)^{k n}$
		where $F V$ is the future value, $P V$ is the present value, $n$ is the number of years, $k$ is the number of compounding periods per year, $r \%$ is the nominal annual rate of interest
$\text{1.5}$	Exponents and logarithms	$a^{x}=b \Leftrightarrow x=\log _{a} b$ , where $a>0, b>0, a \neq 1$
$\text{1.6}$	Percentage error	$\varepsilon=\left\\|\frac{v_{\mathrm{A}}-v_{\mathrm{E}}}{v_{\mathrm{E}}}\right\\| \times 100 \%$
		where $v_{\mathrm{E}}$ is the exact value and $v_{\mathrm{A}}$ is the approximate value of $v$

Topic 2: Functions – SL


$\text{2.1}$	Equations of a straight line	$y=m x+c ; a x+b y+d=0 ; y-y_{1}=m\left(x-x_{1}\right)$
	Gradient formula	$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
$\text{2.5}$	Axis of symmetry of the graph of a quadratic function	$f(x)=a x^{2}+b x+c \Rightarrow$ axis of symmetry is $x=-\frac{b}{2 a}$

Topic 3: Geometry and trigonometry – SL


Prior learning – SL
Area of a parallelogram	$A=bh$ , where $b$ is the base, $h$ is the height
Area of a triangle	$A=\frac{1}{2}(bh)$ , where $b$ is the base, $h$ is the height
Area of a trapezoid	$A=\frac{1}{2}(a+b)h$ , where $a$ and $b$ are the parallel sides, $h$ is the height
Area of a circle	$A=\pi r^{2}$ , where $r$ is the radius
Circumference of a circle	$C=2 \pi r$ , where $r$ is the radius
Volume of a cuboid	$V=lwh$ , where $l$ is the length, $w$ is the width, $h$ is the height
Volume of a cylinder	$V=\pi r^{2} h$ , where $r$ is the radius, $h$ is the height
Volume of prism	$V=Ah$ , where $A$ is the area of cross-section, $h$ is the height
Area of the curved surface of a cylinder	$A=2 \pi r h$ , where $r$ is the radius, $h$ is the height
Distance between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$	$d=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}$
Coordinates of the midpoint of a line segment with endpoints $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$	$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$


$\text{3.1}$	Distance between two points $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$	$d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}}$
	Coordinates of the midpoint of a line segment with endpoints $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$	$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)$
	Volume of a right-pyramid	$V=\frac{1}{3} A h$ , where $A$ is the area of the base, $h$ is the height
	Volume of a right cone	$V=\frac{1}{3} \pi r^{2} h$ , where $r$ is the radius, $h$ is the height
	Area of the curved surface of a cone	$A=\pi r l$ , where $r$ is the radius, $l$ is the slant height
	Volume of a sphere	$V=\frac{4}{3} \pi r^{3}$ , where $r$ is the radius
	Surface area of a sphere	$A=4 \pi r^{2}$ , where $r$ is the radius
$\text{3.2}$	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 of a triangle	$A=\frac{1}{2} a b \sin C$
$\text{3.4}$	Length of an arc	$l=\frac{\theta}{360} \times 2 \pi r$ , where $\theta$ is the angle measured in degrees, $r$ is the radius
	Area of a sector	$A=\frac{\theta}{360} \times \pi r^{2}$ , where $\theta$ is the angle measured in degrees, $r$ is the radius

Topic 4: Statistics and probability – SL


$\text{4.2}$	Interquartile range	$\mathrm{IQR}=Q_{3}-Q_{1}$
$\text{4.3}$	Mean, $\bar{x}$ , of a set of data	$\bar{x}=\frac{\sum_{i=1}^{k} f_{i} x_{i}}{n}$ , where $n=\sum_{i=1}^{k} f_{i}$
$\text{4.5}$	Probability of an event $A$	$\mathrm{P}(A)=\frac{n(A)}{n(U)}$
	Complementary events	$\mathrm{P}(A)+\mathrm{P}\left(A^{\prime}\right)=1$
$\text{4.6}$	Combined events	$\mathrm{P}(A \cup B)=\mathrm{P}(A)+\mathrm{P}(B)-\mathrm{P}(A \cap B)$
	Mutually exclusive events	$\mathrm{P}(A \cup B)=\mathrm{P}(A)+\mathrm{P}(B)$
	Conditional probability	$\mathrm{P}(A \mid B)=\frac{\mathrm{P}(A \cap B)}{\mathrm{P}(B)}$
	Independent events	$\mathrm{P}(A \cap B)=\mathrm{P}(A) \mathrm{P}(B)$
$\text{4.7}$	Expected value of a discrete random variable $X$	$\mathrm{E}(X)=\sum x \mathrm{P}(X=x)$
$\text{4.8}$	Binomial distribution $X \sim \mathrm{~B}(n, p)$
	Mean	$\mathrm{E}(X)=n p$
	Variance	$\operatorname{Var}(X)=n p(1-p)$

Topic 5: Calculus – SL


$\text{5.3}$	Derivative of $x^{n}$	$f(x)=x^{n} \Rightarrow f^{\prime}(x)=n x^{n-1}$
$\text{5.5}$	Integral of $x^{n}$	$\int x^{n} \mathrm{~d} x=\frac{x^{n+1}}{n+1}+C, \quad n \neq-1$
	Area of region enclosed by a curve $y=f(x)$ and the $x$ -axis, where $f(x)>0$	$A=\int_{a}^{b} y \mathrm{~d} x$
$\text{5.8}$	The trapezoidal rule	$\int_{a}^{b} y \mathrm{~d} x \approx \frac{1}{2} h\left(\left(y_{0}+y_{n}\right)+2\left(y_{1}+y_{2}+\ldots+y_{n-1}\right)\right)$ , where $h=\frac{b-a}{n}$