## 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\| 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 | $\mathrm{e}^{x}=1+x+\frac{x^{2}}{2!}+\ldots$
$\ln (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\ldots$
$\sin x=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\ldots$
$\cos x=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\ldots$
$\arctan x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\ldots$ |