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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 a 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 $\left(x_{2}, y_{2}\right)$	$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$

Topic 1: Number and Algebra - SL and HL


$\text{SL 1.2}$	The $n$ th term of an arithmetic sequence	$u_{n}=u_{1}+(n-1) d$
	The sum of $n$ terms of an arithmetic sequence	$S_{n}=\frac{n}{2}\left(2 u_{1}+(n-1) d\right) ; S_{n}=\frac{n}{2}\left(u_{1}+u_{n}\right)$
$\text{SL 1.3}$	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	$S_{n}=\frac{u_{1}\left(r^{n}-1\right)}{r-1}=\frac{u_{1}\left(1-r^{n}\right)}{1-r}, r \neq 1$
$\text{SL 1.4}$	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
$\text{SL 1.5}$	Exponents and logarithms	$a^{x}=b \Leftrightarrow x=\log _{a} b$ , where $a>0, b>0, a \neq 1$
$\text{SL 1.7}$	Exponents and logarithms	$\log _{a} x y=\log _{a} x+\log _{a} y$ $\log _{a} \frac{x}{y}=\log _{a} x-\log _{a} y$ $\log _{a} x^{m}=m \log _{a} x$ $\log _{a} x=\frac{\log _{b} x}{\log _{b} a}$
$\text{SL 1.8}$	The sum of an infinite geometric sequence	$S_{\infty}=\frac{u_{1}}{1-r},\\|r\\|<1$
$\text{SL 1.9}$	Binomial theorem, $n \in \mathbb{N}$	$(a+b)^{n}=a^{n}+\binom{n}{1} a^{n-1} b+\ldots+\binom{n}{r} a^{n-r} b^{r}+\ldots+b^{n}$ $\binom{n}{r}=\frac{n!}{r!(n-r)!}$

Topic 1: Number and Algebra - HL only


$\text{AHL 1.10}$	Combinations Permutations	$\binom{n}{r}=\frac{n!}{r!(n-r)!}$ $P(n,r)=\frac{n!}{(n-r)!}$
$\text{AHL 1.10}$	Extension of binomial theorem, $n \in \mathbb{Q}$	$(a+b)^{n}=a^{n}\left(1+n\left(\frac{b}{a}\right)+\frac{n(n-1)}{2!}\left(\frac{b}{a}\right)^{2}+\ldots\right)$
$\text{AHL 1.12}$	Complex numbers	$z=a+b \mathrm{i}$
$\text{AHL 1.13}$	Modulus-argument (polar) and exponential (Euler) form	$z=r(\cos \theta+\mathrm{i} \sin \theta)=r \mathrm{e}^{\mathrm{i} \theta}$ $=r \operatorname{cis} \theta$
$\text{AHL 1.14}$	De Moivre's theorem	$[r(\cos \theta+\mathrm{i} \sin \theta)]^{n}=r^{n}(\cos n \theta+\mathrm{i} \sin n \theta)$ $=r^{n} \mathrm{e}^{\mathrm{i} n \theta}=r^{n} \operatorname{cis} n \theta$

Topic 2: Functions - SL and HL


$\text{SL 2.1}$	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	$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
$\text{SL 2.6}$	Axis of symmetry of the graph of a quadratic function	$f(x)=a x^{2}+b x+c \Rightarrow$ axis of symmetry is $x=-\frac{b}{2 a}$
$\text{SL 2.7}$	Solutions of a quadratic equation	$a x^{2}+b x+c=0 \Rightarrow x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}, a \neq 0$
	Discriminant	$\Delta=b^{2}-4 a c$
$\text{SL 2.9}$	Exponential and logarithmic functions	$a^{x}=\mathrm{e}^{x \ln a} ; \log _{a} a^{x}=x=a^{\log _{a} x}$ where $a, x>0, a \neq 1$

Topic 2: Functions - HL only


$\text{AHL 2.12}$	Sum and product of the roots of polynomial equations of the form $\sum_{r=0}^{n} a_{r} x^{r}=0$	Sum is $\frac{-a_{n-1}}{a_n}$ ; product is $\frac{(-1)^{n} a_0}{a_n}$

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	$l=r \theta$ , 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
$\text{SL 3.5}$	Identity for $\tan \theta$	$\tan \theta=\frac{\sin \theta}{\cos \theta}$
$\text{SL 3.6}$	Pythagorean identity	$\cos ^{2} \theta+\sin ^{2} \theta=1$
	Double angle identities	$\sin 2 \theta=2 \sin \theta \cos \theta$
		$\cos 2 \theta=\cos ^{2} \theta-\sin ^{2} \theta$ $=2 \cos ^{2} \theta-1=1-2 \sin ^{2} \theta$

Topic 3: Geometry and trigonometry - HL only


$\text{AHL 3.9}$	Reciprocal trigonometric identities Pythagorean identities	$\sec \theta=\frac{1}{\cos \theta}$ $\operatorname{cosec} \theta=\frac{1}{\sin \theta}$ $1+\tan ^{2} \theta=\sec ^{2} \theta$ $1+\cot ^{2} \theta=\operatorname{cosec}^{2} \theta$
$\text{AHL 3.10}$	Compound angle identities Double angle identity for tan	$\sin (A \pm B)=\sin A \cos B \pm \cos A \sin B$ $\cos (A \pm B)=\cos A \cos B \mp \sin A \sin B$ $\tan (A \pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ $\tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta}$
$\text{AHL 3.12}$	Magnitude of a vector	$\\|\boldsymbol{v}\\|=\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}$ , where $\boldsymbol{v}=\left(\begin{array}{l} v_{1} \\ v_{2} \\ v_{3} \end{array}\right)$
$\text{AHL 3.13}$	Scalar product Angle between two vectors	$\boldsymbol{v} \cdot \boldsymbol{w}=v_1 w_1+v_2 w_2+v_3 w_3$ , where $\boldsymbol{v}=\left(\begin{array}{l}v_1 \\ v_2 \\ v_3\end{array}\right), \boldsymbol{w}=\left(\begin{array}{l}w_1 \\ w_2 \\ w_3\end{array}\right)$ $\boldsymbol{v} \cdot \boldsymbol{w}=\\|\boldsymbol{v}\\| \\|\boldsymbol{w}\\| \cos \theta$ , where $\theta$ is the angle between $\boldsymbol{v}$ and $\boldsymbol{w}$ $\cos \theta=\frac{v_1 w_1+v_2 w_2+v_3 w_3}{\\|\boldsymbol{v}\\| \\|\boldsymbol{w}\\|}$
$\text{AHL 3.14}$	Vector equation of a line Parametric form of the equation of a line Cartesian equations of a line	$\boldsymbol{r}=\boldsymbol{a}+\lambda \boldsymbol{b}$ $x=x_0+\lambda l, y=y_0+\lambda m, z=z_0+\lambda n$ $\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}$
$\text{AHL 3.16}$	Vector product Area of a parallelogram	$\boldsymbol{v} \times \boldsymbol{w}=\left(\begin{array}{l}v_2 w_3-v_3 w_2 \\ v_3 w_1-v_1 w_3 \\ v_1 w_2-v_2 w_1\end{array}\right)$ , 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 $\boldsymbol{w}$ $A=\\|\boldsymbol{v} \times \boldsymbol{w}\\|$ where $\boldsymbol{v}$ and $\boldsymbol{w}$ form two adjacent sides of a parallelogram
$\text{AHL 3.17}$	Vector equation of a plane Equation of a plane (using the normal vector) Cartesian equation of a plane	$\boldsymbol{r}=\boldsymbol{a}+\lambda \boldsymbol{b}+\mu \boldsymbol{c}$ $\boldsymbol{r} \cdot \boldsymbol{n}=\boldsymbol{a} \cdot \boldsymbol{n}$ $a x+b y+c z=d$

Topic 4: Statistics and probability - SL and HL


$\text{SL 4.2}$	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 Mutually exclusive events Conditional probability Independent events	$\mathrm{P}(A \cup B)=\mathrm{P}(A)+\mathrm{P}(B)-\mathrm{P}(A \cap B)$ $\mathrm{P}(A \cup B)=\mathrm{P}(A)+\mathrm{P}(B)$ $\mathrm{P}(A \mid B)=\frac{\mathrm{P}(A \cap B)}{\mathrm{P}(B)}$ $\mathrm{P}(A \cap B)=\mathrm{P}(A) \mathrm{P}(B)$
$\text{SL 4.7}$	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 Variance	$\mathrm{E}(X)=n p$ $\operatorname{Var}(X)=n p(1-p)$
$\text{SL 4.12}$	Standardized normal variable	$z=\frac{x-\mu}{\sigma}$

Topic 4: Statistics and probability - HL only


$\text{AHL 4.13}$	Bayes' theorem	$\mathrm{P}(B \mid A)=\frac{\mathrm{P}(B) \mathrm{P}(A \mid B)}{\mathrm{P}(B) \mathrm{P}(A \mid B)+\mathrm{P}\left(B^{\prime}\right) \mathrm{P}\left(A \mid B^{\prime}\right)}$ $\mathrm{P}\left(B_i \mid A\right)=\frac{\mathrm{P}\left(B_i\right) \mathrm{P}\left(A \mid B_i\right)}{\mathrm{P}\left(B_1\right) \mathrm{P}\left(A \mid B_1\right)+\mathrm{P}\left(B_2\right) \mathrm{P}\left(A \mid B_2\right)+\mathrm{P}\left(B_3\right) \mathrm{P}\left(A \mid B_3\right)}$
$\text{AHL 4.14}$	Variance $\sigma^2$	$\sigma^2=\frac{\sum_{i=1}^k f_i\left(x_i-\mu\right)^2}{n}=\frac{\sum_{i=1}^k f_i x_i^2}{n}-\mu^2$
	Standard deviation $\sigma$	$\sigma=\sqrt{\frac{\sum_{i=1}^k f_i\left(x_i-\mu\right)^2}{n}}$
	Linear transformation of a single random variable	$\mathrm{E}(a X+b)=a \mathrm{E}(X)+b$ $\operatorname{Var}(a X+b)=a^2 \operatorname{Var}(X)$
	Expected value of a continuous random variable $X$	$\mathrm{E}(X)=\mu=\int_{-\infty}^{\infty} x f(x) \mathrm{d} x$
	Variance	$\operatorname{Var}(X)=\mathrm{E}\left[(X-\mu)^2\right]=\mathrm{E}\left(X^2\right)-[\mathrm{E}(X)]^2$
	Variance of a discrete random variable $X$	$\operatorname{Var}(X)=\sum(x-\mu)^2 \mathrm{P}(X=x)=\sum x^2 \mathrm{P}(X=x)-\mu^2$
	Variance of a continuous random variable $X$	$\operatorname{Var}(X)=\int_{-\infty}^{\infty}(x-\mu)^2 f(x) \mathrm{d} x=\int_{-\infty}^{\infty} x^2 f(x) \mathrm{d} x-\mu^2$

Topic 5: Calculus - SL


$\text{SL 5.3}$	Derivative of $x^{n}$	$f(x)=x^{n} \Rightarrow f^{\prime}(x)=n x^{n-1}$
$\text{SL 5.5}$	Integral of $x^{n}$ Area between a curve $y=f(x)$ and the $x$ -axis, where $f(x)>0$	$\int x^{n} \mathrm{~d} x=\frac{x^{n+1}}{n+1}+C, n \neq-1$ $A=\int_{a}^{b} y \mathrm{~d} x$
$\text{SL 5.6}$	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 $\mathrm{e}^{x}$	$f(x)=\mathrm{e}^{x} \Rightarrow f^{\prime}(x)=\mathrm{e}^{x}$
	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	$y=\frac{u}{v} \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{v \frac{\mathrm{d} u}{\mathrm{~d} x}-u \frac{\mathrm{d} v}{\mathrm{~d} x}}{v^{2}}$
$\text{SL 5.9}$	Acceleration Distance travelled from $t_{1}$ to $t_{2}$ Displacement from $t_{1}$ to $t_{2}$	$a=\frac{\mathrm{d} v}{\mathrm{~d} t}=\frac{\mathrm{d}^{2} s}{\mathrm{~d} t^{2}}$ $\text{distance}=\int_{t_{1}}^{t_{2}}\\|v(t)\\| \mathrm{d} t$ $\text{displacement}=\int_{t_{1}}^{t_{2}} v(t) \mathrm{d} t$
$\text{SL 5.10}$	Standard integrals	$\int \frac{1}{x} \mathrm{~d} x=\ln \\|x\\|+C$
		$\int \sin x \mathrm{~d} x=-\cos x+C$
		$\int \cos x \mathrm{~d} x=\sin x+C$
$\text{SL 5.11}$	Area of region enclosed by a curve and $x$ -axis	$A=\int_a^b\\|y\\| \mathrm{d} x$

Topic 5: Calculus - HL only


$\text{AHL 5.12}$	Derivative of $f(x)$ from first principles	$y=f(x) \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=f^{\prime}(x)=\lim _{h \rightarrow 0}\left(\frac{f(x+h)-f(x)}{h}\right)$
$\text{AHL 5.15}$	Standard derivatives
	$f(x)=\tan x$	$f^{\prime}(x)=\sec ^{2} x$
	$f(x)=\sec x$	$f^{\prime}(x)=\sec x \tan x$
	$f(x)=\operatorname{cosec} x$	$f^{\prime}(x)=-\operatorname{cosec} x \cot x$
	$f(x)=\cot x$	$f^{\prime}(x)=-\operatorname{cosec}^{2} x$
	$f(x)=a^{x}$	$f^{\prime}(x)=a^{x}(\ln a)$
	$f(x)=\log _{a} x$	$f^{\prime}(x)=\frac{1}{x \ln a}$
	$f(x)=\arcsin x$	$f^{\prime}(x)=\frac{1}{\sqrt{1-x^{2}}}$
	$f(x)=\arccos x$	$f^{\prime}(x)=-\frac{1}{\sqrt{1-x^{2}}}$
	$f(x)=\arctan x$	$f^{\prime}(x)=\frac{1}{1+x^{2}}$
$\text{AHL 5.15}$	Standard integrals
	$\int a^{x} \mathrm{~d} x$	$\frac{1}{\ln a} a^{x}+C$
	$\int \frac{1}{a^{2}+x^{2}} \mathrm{~d} x$	$\frac{1}{a} \arctan \left(\frac{x}{a}\right)+C$
	$\int \frac{1}{\sqrt{a^{2}-x^{2}}} \mathrm{~d} x$	$\arcsin \left(\frac{x}{a}\right)+C,\\|x\\|<a$
$\text{AHL 5.16}$	Integration by parts	$\int u \frac{\mathrm{~d} v}{\mathrm{~d} x} \mathrm{~d} x=u v-\int v \frac{\mathrm{~d} u}{\mathrm{~d} x} \mathrm{~d} x \text { or } \int u \mathrm{~d} v=u v-\int v \mathrm{~d} u$
$\text{AHL 5.17}$	Area of region enclosed by a curve and $y$ -axis Volume of revolution about the $x$ or $y$ -axes	$A=\int_{a}^{b}\\|x\\| \mathrm{d} y$ $V=\int_{a}^{b} \pi y^{2} \mathrm{~d} x$ or $V=\int_{a}^{b} \pi x^{2} \mathrm{~d} y$
$\text{AHL 5.18}$	Euler's method Integrating factor for $y^{\prime}+P(x) y=Q(x)$	$y_{n+1}=y_{n}+h \times f\left(x_{n}, y_{n}\right) ; x_{n+1}=x_{n}+h$ , where $h$ is a constant (step length) $\mathrm{e}^{\int P(x) \mathrm{d} x}$
$\text{AHL 5.19}$	Maclaurin series	$f(x)=f(0)+x f^{\prime}(0)+\frac{x^{2}}{2!} f^{\prime \prime}(0)+\ldots$
	Maclaurin series for special functions	$\mathrm{e}^{x}=1+x+\frac{x^{2}}{2!}+\ldots$ $\ln (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\ldots$ $\sin x=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\ldots$ $\cos x=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\ldots$ $\arctan x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\ldots$

IB Mathematics Analysis and Approaches Data Booklet

Prior learning - SL and HL

Topic 1: Number and Algebra - SL and HL

Topic 1: Number and Algebra - 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

Topic 5: Calculus - HL only