Appendix 1. Formula Reference {-}
| Exponents and Logarithms | |
|---|---|
| Rules of Exponents | $a^m \cdot a^n = a^{m+n}$ |
| $a^m \div a^n = a^{m-n}$ | |
| $(a^m)^n = a^{mn}$ | |
| $a^0 = 1, \text{ for all } a \text{ in } \mathbb{C}, a \neq 0$ | |
| $a^1 = a$ | |
| $a^{-m} = \frac{1}{a^m}$ | |
| $a^{\frac{1}{m}} = \sqrt[m]{a}$ | |
| $a^{\frac{n}{m}} = (\sqrt[m]{a})^n$ | |
| $(ab)^n = a^n \cdot b^n$ | |
| $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ | |
| Laws of Logarithms (real case: $a, x, y > 0$, and $a \neq 1$) | $a^x = b \Longleftrightarrow \log_a{b}=x$ |
| $\log_a{x} + \log_a{y} = \log_a{(xy)}$ | |
| $\log_a{x} - \log_a{y} = \log_a{(\frac{x}{y})}$ | |
| $n \log_a{x} = \log_a{(x^n)}$ |
| Sequences and Series | |
|---|---|
| Sigma notation properties | |
| Addition property | $\sum_{n=1}^k{(a_n+b_n)}=\sum_{n=1}^k{a_n}+\sum_{n=1}^k{b_n}$ |
| Constant Multiplication Property | $\sum_{n=1}^k{c \cdot u_n} = c \cdot \sum_{n=1}^k{u_n}$ |
| Arithmetic Sequences and Series | |
| The $n^{\text{th}}$ term of an arithmetic sequence | $u_n=u_1+(n-1)d$ |
| The sum of a finite arithmetic series | $S_n = \dfrac{n}{2}(u_1 + u_n)$ |
| $\text{OR} \quad S_n = \dfrac{n}{2}\left(2u_1 + (n - 1)d\right)$ | |
| Geometric Sequences and Series | |
| The $n^{\text{th}}$ term of a geometric sequence | $u_n = u_1 r^{n-1}$ |
| The sum of a finite arithmetic series (where $r \neq 1$) | $S_n = \dfrac{u_1(r^n-1)}{r-1}$ |
| $\text{OR} \quad S_n = \dfrac{u_1(1-r^n)}{1-r}$ | |
| The sum of an infinite geometric series | $S_{\infty}=\dfrac{u_1}{1-r}, \quad \text{where} \quad \lvert r \rvert < 1$ |
| Functions | |
|---|---|
| Equation of a straight line | $y=mx+c \quad \text{OR} \quad y=m(x-x_1)+y_1$ |
| Line of symmetry | If $f(x)=ax^2+bx+c$, the axis of symmetry of is $x=-\frac{b}{2a}$ |
| Discriminant | $\Delta = b^2 -4ac$ |
| Quadratic formula | $\text{The solutions to }ax^2 + bx + c = 0 \text{ are } x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}, a \neq 0$ |
| Finance | |
|---|---|
| Interest paid | $\text{Interest Paid} = \text{Total Repayments} - \text{Amount Borrowed}$ |
| Effective annual interest rate | $r_{\text{eff}} = \left(1 + \frac{r}{n_c}\right)^{n_c} - 1$ |
| Compound interest (discrete) | $V_f=V_0 \left(1 + \frac{r}{n_c}\right)^{n_{c}t}$ |
| Compound interest (continuous) | $V_f = V_0 e^{rt}$ |
| Sets | |
|---|---|
| De Morgan’s Laws | |
| Union | $(A \cup B)^c = A^c \cap B^c$ |
| Intersection | $(A \cap B)^c = A^c \cup B^c$ |
| Combinatorics | |
|---|---|
| Fundamental Counting Principle | |
| $n$ factorial | |
| Binomial theorem |
| Sectors and Radians | |
|---|---|
| Radians and degrees equivalence | $\pi \text{ radians} = 180^\circ$ |
| Arc length | $l=r \theta$ |
| Sector area | $A=\frac{1}{2}r^2\theta$ |
| Geometry | |
|---|---|
| Distance between two points $a$ and $b$ in $n$-dimensional space | $\sqrt{\sum_{i=1}^{n} \left( b_i - a_i \right)^2}$ |
| Midpoint of a line segment with endpoints $a$ and $b$ in $n$-dimensional space | $\left(\frac{1}{2}\left(a_i + b_i \right) \right)_{1 \leq i \leq n}$ |
| Spheres of radius $r$ | |
| Surface area | $A=4 \pi r^2$ |
| Volume | $V=\frac{4}{3} \pi r^3$ |
| Pyramids of height $h$ | |
| Volume | $V=\frac{1}{3}Ah$ |
| Parallelogram | |
| Area (Given $\mathbf{v}$ and $\mathbf{w}$ are adjacent sides of a parallelogram) | $A=\lvert\mathbf{v}\times\mathbf{w}\rvert$ |
| Cone | |
| Surface area of the curved surface | $A=\pi rl$ |
| Volume (right cone) | $V=\frac{1}{3}\pi r^2 h$ |
| Trigonometry | |
|---|---|
| Rules for triangles with sides $a, b, c$, and angles $A, B, C$, where $X$ is opposite to $x$ | |
| Sine Rule | $\dfrac{a}{\sin A}= \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$ |
| Cosine Rule | $c^2 = a^2 + b^2 -2ab \cos C$ |
| Triangle Area | $\dfrac{1}{2}ab \sin C$ |
| Trigonometric Identities | |
| Pythagorean Identity | $\cos^2\theta + \sin^2\theta = 1$ |
| Tangent Identity | $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ |
| Double-Angle Identity | $\cos(2\theta) = \cos^2\theta - \sin^2\theta$ |
| Reciprocal Identities | $\sin\theta = \dfrac{1}{\csc\theta}$ |
| $\cos\theta = \dfrac{1}{\sec\theta}$ | |
| $\tan\theta = \dfrac{1}{\cot\theta}$ |
| Statistics | |
|---|---|
| Percentage error | $\varepsilon = \lvert \dfrac{v_a - v_e}{v_e} \rvert \cdot 100$ |
| Descriptive statistics | |
| Interquartile Range | $\text{IQR} = Q_3 - Q_1$ |
| Arithmetic Mean | $\bar{x} = \dfrac{\sum_{i=1}^{n}{f_ix_i}}{\sum_{i=1}^{n}{f_i}}$ |
| Sampling | |
| Unbiased estimator of the population variance | $\hat{\sigma}^2 = \dfrac{n}{n-1} s_{n}^2$ |
| Probability | |
|---|---|
| Probability of an event $E$ | $\mathbb{P}(E)=\dfrac{\lvert E \rvert}{\lvert U \rvert}$ |
| Combined events | $\mathbb{P}(E \cup F)=\mathbb{P}(E) + \mathbb{P}(F) - \mathbb{P}(E \cap F)$ |
| Mutually exclusive events | $\mathbb{P}(E \cap F) = 0$ |
| Probability of $E$ given $F$ | $\mathbb{P}(E \vert F) = \dfrac{\mathbb{P}(E \cap F)}{\mathbb{P}(F)}$ |
| Random Variables and Probability Distributions | |
|---|---|
| Expected value of a single discrete random variable $X$ | $\mathbb{E}[X] = \sum_{i} x_i \, \mathbb{P}(X = x_i)$ |
| Linear Transformation of a single random variable $X$ | |
| Expected Value | $\mathbb{E}[aX+b] = a\mathbb{E}[X]+b$ |
| Variance | $\mathrm{Var}[aX+b]=a^2 \, \mathrm{Var}[X]$ |
| Linear combinations of random variables $X_1, X_2, \dots, X_n$ | |
| Expected Value | $\mathbb{E}!\left[\sum_{i=1}^n a_i X_i\right] = \sum_{i=1}^n a_i \mathbb{E}[X_i]$ |
| Linear combinations of independent random variables $X_1, X_2, \dots, X_n$ | |
| Variance | $\mathrm{Var}!\left[\sum_{i=1}^n a_i X_i\right]=\sum_{i=1}^{n}{a_i^2 \, \mathrm{Var}[X_i]}$ |
| Uniform Distribution | |
| $X \sim \mathcal{U}(a, b)$ | $a = \text{lower bound}$ |
| $b = \text{upper bound}$ | |
| Mean ($\mu$) | $\mathbb{E}[X]=\dfrac{a+b}{2}$ |
| Variance ($\sigma^2$) | $\mathrm{Var}[X]=\dfrac{(b-a)^2}{12}$ |
| Normal (Gaussian) Distribution | |
| $X \sim \mathcal{N}(\mu, \sigma^2)$ | $\mu = \text{mean}$ |
| $\sigma^2 = \text{variance}$ | |
| Central Limit Theorem (CLT) | For large $n$, $\bar{X} \sim \mathcal{N}!\left(\mu, \dfrac{\sigma^2}{n}\right)$ (approximately) |
| Binomial Distribution | |
| $X \sim \mathcal{B}(n, p)$ | $n = \text{number of trials}$ |
| $p = \text{probability of success}$ | |
| Mean ($\mu$) | $\mathbb{E}[X]=np$ |
| Variance ($\sigma^2$) | $\mathrm{Var}[X]=np(1-p)$ |
| Poisson Distribution | |
| $X \sim \mathcal{P}(\lambda)$ | $\lambda = \text{mean and variance}$ |
| Hypothesis Testing | |
|---|---|
| $z$ (mean, $\sigma$ known) | $z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$ |
| $t$ (mean, $\sigma$ unknown) | $t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}, \quad df = n-1$ |
| $z$ (proportion) | $z = \dfrac{\hat{p} - p_0}{\sqrt{\tfrac{p_0(1-p_0)}{n}}}$ |
| $\chi^2$ (GoF/independence) | $\chi^2 = \sum \dfrac{(O - E)^2}{E}$ |
| Two sample | |
| $t$ (two-sample, pooled variance) | $t = \dfrac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\tfrac{1}{n_1} + \tfrac{1}{n_2}}}, \quad s_p^2 = \dfrac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}$ |
| $z$ (two-sample proportions) | $z = \dfrac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\tfrac{1}{n_1}+\tfrac{1}{n_2})}}, \quad \hat{p} = \dfrac{x_1+x_2}{n_1+n_2}$ |
| Differentiation | |
|---|---|
| Derivative of $f(x)$ using first principles | $f’(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$ |
| Derivative of $x^n$ | $f(x)=x^n \;\Rightarrow\; f’(x)=nx^{n-1}$ |
| Standard Derivatives | |
| $f(x)=\sin x \Rightarrow f’(x)=\cos x$ | |
| $f(x)=\cos x \Rightarrow f’(x)= -\sin x$ | |
| $f(x)=\tan x \Rightarrow f’(x)= \dfrac{1}{\cos^2 x}= \sec^2 x$ | |
| $f(x)=\sin x \Rightarrow f’(x)=\cos x$ | |
| $f(x)=\sin x \Rightarrow f’(x)=\cos x$ | |
| Standard Differentiation Rules | |
| Chain rule | $\dfrac{d}{dx} f(g(x)) = f’(g(x)) \cdot g’(x)$ |
| $\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}$ | |
| Product rule | $\dfrac{d}{dx}[u(x)v(x)] = u(x)v’(x) + u’(x)v(x)$ |
| Quotient rule | $\dfrac{d}{dx}\left[\dfrac{u(x)}{v(x)}\right] = \dfrac{u’(x)v(x) - u(x)v’(x)}{v(x)^2}$ |
| L’Hôpital’s Rule | $\lim_{x \to a} \dfrac{f(x)}{g(x)} = \lim_{x \to a} \dfrac{f’(x)}{g’(x)}$, for $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$ forms, with $g’(x)\neq 0$ |
| Integration | |
|---|---|
| Integral of $x^n$ | $\int x^n \, dx = \dfrac{x^{n+1}}{n+1}+C, \quad n \neq -1$ |
| Area of the region between the $x$-axis and $f(x)$, for the range $(a, b)$ | $A = \displaystyle \int_a^b y \, dx$ |
| Trapezoidal Rule | $\displaystyle \int_a^b y \, dx \approx \frac{h}{2} (y_0 + y_n + 2(y_1 + \dots + y_{n-1}))$ |
| Standard Integrals | |
| $\int \dfrac{1}{x} \, dx = \ln \lvert x \rvert+C$ | |
| $\int \sin x \, dx = -\cos x + C$ | |
| $\int \cos x \, dx = \sin x + C$ | |
| $\int e^x \, dx = e^x + C$ | |
| $\int \tan x \, dx = -\ln \lvert \cos x \rvert + C$ | |
| $\int \cot x \, dx = \ln \lvert \sin x \rvert + C$ | |
| $\displaystyle \int \sec^2 x \, dx = \tan x + C$ | |
| $\int \sec x \, dx = \ln \lvert \sec x + \tan x \rvert + C$ | |
| $\displaystyle \int \dfrac{1}{x^2 - a^2} \, dx = \dfrac{1}{2a} \ln \lvert \dfrac{x - a}{x + a} \rvert + C$ | |
| $\displaystyle \int \dfrac{1}{x^2 + 1} \, dx = \arctan (x) + C$ |
| Differential Equations | |
|---|---|
| Euler’s method (step size $h$) | $y_{n+1} = y_n + h f(x_n, y_n)$ |
| $x_{n+1} = x_n + h$ | |
| Coupled linear differential equations (diagonalisable case) | $\mathbf{x}(t) = \begin{pmatrix} x(t) \ y(t) \end{pmatrix} = C_1 e^{\lambda_1 t} \vec{p_1} + C_2 e^{\lambda_2 t} \vec{p_2}$ |
| Complex Numbers | |
|---|---|
| Rectangular form | $z = a + bi$ |
| Polar (modulus-argument) form | $z = r(\cos \theta + i\sin(\theta))$ |
| Exponential (Euler) form | $z = re^{i \theta}$ |
| Compliment conjugate | $\bar{z} = a - bi$ |
| De Moivre’s theorem | $(\cos \theta + i\sin \theta)^n = \cos(n\theta) + i\sin(n\theta)$ |
| Vectors | |
|---|---|
| Magnitude $\text{\quad}$ | $\lvert \mathbf{v} \rvert = \sqrt{v_1^2 + v_2^2 + v_3^2}$ |
| $\lvert \mathbf{v} \rvert = \sqrt{\sum_{i=1}^{n} v_i^2}, \quad v_i =$ $i^\text{th}$ component | |
| Scalar product | $\mathbf{v} \cdot \mathbf{w} = \sum_{i=1}^n v_i w_i$ |
| $\mathbf{v} \cdot \mathbf{w} = \lvert\mathbf{v}\rvert \lvert\mathbf{w}\rvert \cos\theta$, $\theta =$ angle between $\mathbf{v},\mathbf{w}$ | |
| Vector product | Given $\mathbf{v}=\begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix},\; \mathbf{w}=\begin{pmatrix} w_1 \ w_2 \ w_3 \end{pmatrix}$ |
| $\mathbf{v}\times\mathbf{w}=\begin{pmatrix} v_2w_3-v_3w_2 \ v_3w_1-v_1w_3 \ v_1w_2-v_2w_1 \end{pmatrix}$ | |
| $\lvert\mathbf{v}\times\mathbf{w}\rvert=\lvert\mathbf{v}\rvert \lvert\mathbf{w}\rvert \sin\theta$ | |
| Vector equation of a line | $\mathbf{r} = \mathbf{a} + \lambda \mathbf{d}, \;\; \lambda \in \mathbb{R}$ |
| Parametric form of a line | $x = x_0 + \lambda l, \;\; y = y_0 + \lambda m, \;\; z = z_0 + \lambda n$ |
| Matrices | |
|---|---|
| Matrix addition | $(A + B){ij} = a{ij} + b_{ij}$ |
| Matrix multiplication | $(AB){ij} = \sum{k=1}^n a_{ik} b_{kj}$ |
| $\text{OR} \quad AB = \left[ \sum_{k=1}^n a_{ik} b_{kj} \right]_{1 \le i \le m,\ 1 \le j \le p}$ | |
| Determinant of a 2×2 matrix | $\det \begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc$ |
| Determinant of a 3×3 matrix | $\det \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} = a(ei-fh)-b(di-fg)+c(dh-eg)$ |
| Invertibility condition | $A$ is invertible if $\det A \neq 0$ |
| Power formula of a diagonalisible matrix $A$ | $A^n = P D^n P^{-1}$, $n \in \mathbb{N}$, $P =$ eigenvectors, $D =$ eigenvalues |
| 2D Transformation Matrices | (Assuming $x$-axis is horizontal, and $y$-axis is vertical) |
| Stretch with scale factor $h$ horizontally, and $v$ vertically | $\begin{pmatrix} h & 0 \ 0 & v \end{pmatrix}$, centred at the origin |
| Anticlockwise rotation of angle $\theta$ | $\begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix}$, about the origin |
| Reflection in the line through the origin with angle $\theta$ from the $x$-axis | $\begin{pmatrix} \cos 2\theta & \sin 2\theta \ \sin 2\theta & -\cos 2\theta \end{pmatrix}$ |
| Shear with factor $h$ horizontally, and $v$ vertically | $\begin{pmatrix} 1 & h \ v & 1 \end{pmatrix}$, about the origin |
| Graph Theory | |
|---|---|
| Adjacency matrix | $a_{ij}=1$ if edge $i\to j$, otherwise $0$. If weighted, $a_{ij}=w_{ij}$ |
| Transition matrix | $a_{ij} =$ probability of moving $i \to j$ (convention can vary, however) |
| State after $n$ transitions (row-stochastic convention) | $s_n = s_0 T^n$ |
| Modelling | |
|---|---|
| Logistic function | $\dfrac{L}{1+Ce^{-kx}}$, where $L, C, k \in \mathbb{R}^+$ |
| Volume of revolution in the range $(a, b)$ | |
| about the $x$-axis | $V = \pi \displaystyle \int_a^b [f(x)]^2 \, dx$ |
| about the $y$-axis | $V = \pi \displaystyle \int_a^b [f(y)]^2 \, dy$ |