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Prior learning - SL and HL


Area of a parallelogram	$A=b h$ , where $b$ is the base, $h$ is the height
Area of a triangle	$A=\frac{1}{2}(b h)$ , 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	$V=l w h$ , 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	$V=A h$ , 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 $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$	$d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}$
Coordinates of the midpoint of a line segment with endpoints $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$	$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$

Prior learning - HL only


Solutions of a quadratic equation	The solutions of $a x^{2}+b x+c=0$ are $x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}, a \neq 0$
$\text{SL 1.2}$	The $n$ th term of an arithmetic sequence The sum of $n$ terms of an arithmetic sequence
$\text{SL 1.3}$	The $n$ th term of a geometric sequence The sum of $n$ terms of a finite geometric sequence
$\text{SL 1.4}$	$\text{Compound interest}$
$\text{SL 1.5}$	$\text{Exponents and logarithms}$
$\text{SL 1.6}$	Percentage error
$\text{AHL 1.9}$	Laws of logarithms
$\text{AHL 1.11}$	$\text{The sum of an infinite geometric sequence}$
$\text{AHL 1.12}$	$\text{Complex numbers}$ Discriminant
$\text{AHL 1.13}$	Modulus-argument (polar) and exponential (Euler) form
$\text{AHL 1.14}$	Determinant of a $2 \times 2$ matrix Inverse of a $2 \times 2$ matrix
$\text{AHL 1.15}$	Power formula for a matrix

Topic 2: Functions - SL and HL


$\text{SL 2.1}$	Equations of a straight line Gradient formula
$\text{SL 2.5}$	Axis of symmetry of the graph of a quadratic function

Topic 2: Functions - HL only


$\text{AHL 2.9}$	Logistic function	$f(x)=\frac{L}{1+C \mathrm{e}^{-k x}}, L, k, C>0$

Topic 3: Geometry and trigonometry - SL and HL


$\text{SL 3.1}$	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
$\text{SL 3.2}$	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	$A=\frac{1}{2} a b \sin C$
$\text{SL 3.4}$	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 3: Geometry and trigonometry - HL only


$\text{AHL 3.7}$	Length of an arc	$l=r \theta$ , where $r$ is the radius, $\theta$ is the angle measured in radians
	Area of a sector	$A=\frac{1}{2} r^{2} \theta$ , where $r$ is the radius, $\theta$ is the angle measured in radians
$\text{AHL 3.8}$	Identities	$\begin{aligned} & \cos ^{2} \theta+\sin ^{2} \theta=1 \\ & \tan \theta=\frac{\sin \theta}{\cos \theta} \end{aligned}$
$\text{AHL 3.9}$	Transformation matrices	$\left(\begin{array}{cc}\cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & -\cos 2 \theta\end{array}\right)$ , reflection in the line $y=(\tan \theta) x$ $\left(\begin{array}{ll}k & 0 \\ 0 & 1\end{array}\right)$ , horizontal stretch / stretch parallel to $x$ -axis with a scale factor of $k$ $\left(\begin{array}{ll}1 & 0 \\ 0 & k\end{array}\right)$ , vertical stretch / stretch parallel to $y$ -axis with a scale factor of $k$ $\left(\begin{array}{ll}k & 0 \\ 0 & k\end{array}\right)$ , enlargement, with a scale factor of $k$ , centre $(0,0)$ $\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$ , anticlockwise/counter-clockwise rotation of angle $\theta$ about the origin $(\theta>0)$ $\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right)$ , clockwise rotation of angle $\theta$ about the origin $(\theta>0)$
$\text{AHL 3.10}$	Magnitude of a vector	$\\|\boldsymbol{v}\\|=\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}$ , where $\boldsymbol{v}=\left(\begin{array}{l}v_{1} \\ v_{2} \\ v_{3}\end{array}\right)$
$\text{AHL 3.11}$	Vector equation of a line	$\boldsymbol{r}=\boldsymbol{a}+\lambda \boldsymbol{b}$
	Parametric form of the equation of a line	$x=x_{0}+\lambda l, y=y_{0}+\lambda m, z=z_{0}+\lambda n$
$\text{AHL 3.13}$	Scalar product	$\boldsymbol{v} \cdot \boldsymbol{w}=v_{1} w_{1}+v_{2} w_{2}+v_{3} w_{3}$ , where $\boldsymbol{v}=\left(\begin{array}{l}v_{1} \\ v_{2} \\ v_{3}\end{array}\right), \boldsymbol{w}=\left(\begin{array}{l}w_{1} \\ w_{2} \\ w_{3}\end{array}\right)$
	Angle between two vectors	$\boldsymbol{v} \cdot \boldsymbol{w}=\\|\boldsymbol{v}\\| \\|\boldsymbol{w}\\| \cos \theta$ , where $\theta$ is the angle between $\boldsymbol{v}$ and $\boldsymbol{w}$
		$\cos \theta=\frac{v_{1} w_{1}+v_{2} w_{2}+v_{3} w_{3}}{\\|\boldsymbol{v}\\| \\|\boldsymbol{w}\\|}$
	Vector product	$\boldsymbol{v} \times \boldsymbol{w}=\left(\begin{array}{l}v_{2} w_{3}-v_{3} w_{2} \\ v_{3} w_{1}-v_{1} w_{3} \\ v_{1} w_{2}-v_{2} w_{1}\end{array}\right)$ , where $\boldsymbol{v}=\left(\begin{array}{l}v_{1} \\ v_{2} \\ v_{3}\end{array}\right), \boldsymbol{w}=\left(\begin{array}{l}w_{1} \\ w_{2} \\ w_{3}\end{array}\right)$
		$\\|\boldsymbol{v} \times \boldsymbol{w}\\|=\\|\boldsymbol{v}\\| \\|\boldsymbol{w}\\| \sin \theta$ , where $\theta$ is the angle between $\boldsymbol{v}$ and $\boldsymbol{w}$
	Area of a parallelogram	$A=\\|\boldsymbol{v} \times \boldsymbol{w}\\|$ where $\boldsymbol{v}$ and $\boldsymbol{w}$ form two adjacent sides of a parallelogram

Topic 4: Statistics and probability - SL and HL


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

Topic 4: Statistics and probability - HL only


$\text{AHL 4.14}$	Linear transformation of a single random variable	$\mathrm{E}(a X+b)=a \mathrm{E}(X)+b$
		$\operatorname{Var}(a X+b)=a^{2} \operatorname{Var}(X)$
	Linear combinations of $n$ independent random variables, $X_{1}, X_{2}, \ldots, X_{n}$	$\mathrm{E}\left(a_{1} X_{1} \pm a_{2} X_{2} \pm \ldots \pm a_{n} X_{n}\right)=a_{1} \mathrm{E}\left(X_{1}\right) \pm a_{2} \mathrm{E}\left(X_{2}\right) \pm \ldots \pm a_{n} \mathrm{E}\left(X_{n}\right)$
		$\operatorname{Var}\left(a_{1} X_{1} \pm a_{2} X_{2} \pm \ldots \pm a_{n} X_{n}\right) =$
		$a_{1}^{2} \operatorname{Var}\left(X_{1}\right)+a_{2}^{2} \operatorname{Var}\left(X_{2}\right)+\ldots+a_{n}^{2} \operatorname{Var}\left(X_{n}\right)$
	Sample statistics
	Unbiased estimate of population variance $s_{n-1}^{2}$	$s_{n-1}^{2}=\frac{n}{n-1} s_{n}^{2}$
$\text{AHL 4.17}$	Poisson distribution $X \sim \operatorname{Po}(m)$
	Mean	$\mathrm{E}(X)=m$
	Variance	$\operatorname{Var}(X)=m$
$\text{AHL 4.19}$	Transition matrices	$\boldsymbol{T}^{n} \boldsymbol{s}_{0}=\boldsymbol{s}_{n}$ , where $\boldsymbol{s}_{0}$ is the initial state

Topic 5: Calculus - SL and HL


$\text{SL 5.3}$	Derivative of $x^{n}$	$f(x)=x^{n} \Rightarrow f^{\prime}(x)=n x^{n-1}$
$\text{SL 5.5}$	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 $f(x)>0$	$A=\int_{a}^{b} y \mathrm{~d} x$
$\text{SL 5.8}$	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 $h=\frac{b-a}{n}$

Topic 5: Calculus - HL only


$\text{AHL 5.9}$	Standard derivatives
	Derivative of $\sin x$	$f(x)=\sin x \Rightarrow f^{\prime}(x)=\cos x$
	Derivative of $\cos x$	$f(x)=\cos x \Rightarrow f^{\prime}(x)=-\sin x$
	Derivative of $\tan x$	$f(x)=\tan x \Rightarrow f^{\prime}(x)=\frac{1}{\cos ^{2} x}$
	Derivative of $\mathrm{e}^{x}$	$f(x)=\mathrm{e}^{x} \Rightarrow f^{\prime}(x)=\mathrm{e}^{x}$
	Derivative of $\ln x$	$f(x)=\ln x \Rightarrow f^{\prime}(x)=\frac{1}{x}$
	Chain rule	$y=g(u)$ , where $u=f(x) \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} u} \times \frac{\mathrm{d} u}{\mathrm{~d} x}$
	Product rule	$y=u v \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=u \frac{\mathrm{d} v}{\mathrm{~d} x}+v \frac{\mathrm{d} u}{\mathrm{~d} x}$
	Quotient rule	$y=\frac{u}{v} \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{v \frac{\mathrm{d} u}{\mathrm{~d} x}-u \frac{\mathrm{d} v}{\mathrm{~d} x}}{v^{2}}$
$\text{AHL 5.11}$	Standard integrals
	$\int \frac{1}{x} \mathrm{~d} x$	$\ln \\|x\\|+C$
	$\int \sin x \mathrm{~d} x$	$-\cos x+C$
	$\int \cos x \mathrm{~d} x$	$\sin x+C$
	$\int \frac{1}{\cos ^{2} x}$	$\tan x+C$
	$\int \mathrm{e}^{x} \mathrm{~d} x$	$\mathrm{e}^{x}+C$
$\text{AHL 5.12}$	Area of region enclosed by a curve and $x$ or $y$ -axes	$A=\int_{a}^{b}\\|y\\| \mathrm{d} x$ or $A=\int_{a}^{b}\\|x\\| \mathrm{d} y$
	Volume of revolution about $x$ or $y$ -axes	$V=\int_{a}^{b} \pi y^{2} \mathrm{d} x$ or $V=\int_{a}^{b} \pi x^{2} \mathrm{~d} y$
$\text{AHL 5.13}$	Acceleration	$a=\frac{\mathrm{d} v}{\mathrm{~d} t}=\frac{\mathrm{d}^{2} s}{\mathrm{~d} t^{2}}=v \frac{\mathrm{d} v}{\mathrm{~d} s}$
	Distance travelled from $t_{1}$ to $t_{2}$	$\text{distance} = \int_{t_{1}}^{t_{2}}\\|v(t)\\| \mathrm{d} t$
	Displacement from $t_{1}$ to $t_{2}$	$\text{displacement} = \int_{t_{1}}^{t_{2}} v(t) \mathrm{d} t$
$\text{AHL 5.16}$	Euler's method	$y_{n+1}=y_{n}+h \times f\left(x_{n}, y_{n}\right)$ , $x_{n+1}=x_{n}+h$ , where $h$ is a constant (step length)
	Euler's method for coupled systems	$x_{n+1}=x_{n}+h \times f_{1}\left(x_{n}, y_{n}, t_{n}\right)$ , $y_{n+1}=y_{n}+h \times f_{2}\left(x_{n}, y_{n}, t_{n}\right)$ , $t_{n+1}=t_{n}+h$ , where $h$ is a constant (step length)
$\text{AHL 5.17}$	Exact solution for coupled linear differential equations	$\boldsymbol{x}=A \mathrm{e}^{\lambda_{1} t} \boldsymbol{p}_{1}+B \mathrm{e}^{\lambda_{2} t} \boldsymbol{p}_{2}$

IB Mathematics Applications & Interpretation Data Booklet

Data Booklet

Prior learning - SL and HL

Prior learning - HL only

Topic 2: Functions - SL and HL

Topic 2: Functions - HL only

Topic 3: Geometry and trigonometry - SL and HL

Topic 3: Geometry and trigonometry - HL only

Topic 4: Statistics and probability - SL and HL

Topic 4: Statistics and probability - HL only

Topic 5: Calculus - SL and HL

Topic 5: Calculus - HL only