Appendix 1. Formula Reference {-}
| Exponents and Logarithms | |
|---|---|
| Rules of Exponents | $a^m \cdot a^n = a^{m+n}$ |
| $a^m \div a^n = a^{m-n}$ | |
| $(a^m)^n = a^{mn}$ | |
| $a^0 = 1$ | |
| $a^1 = a$ | |
| $a^{-m} = \frac{1}{a^m}$ | |
| $a^{\frac{1}{m}} = \sqrt[m]{a}$ | |
| $a^{\frac{n}{m}} = (\sqrt[m]{a})^n$ | |
| $(ab)^n = a^n \cdot b^n$ | |
| $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ | |
| 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 | $S_n = \frac{n}{2}(u_1 + u_n) \quad \text{OR} \quad S_n = \frac{n}{2}\left(2u_1 + (n - 1)d\right)$ |
| 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 | $S_n = \frac{u_1(r^n-1)}{r-1} \quad \text{OR} \quad S_n = \frac{u_1(1-r^n)}{1-r}, \quad \text{where} \quad r \neq 1$ |
| The sum of an infinite geometric series | $S_{\infty}=\frac{u_1}{1-r}, \quad \text{where} \quad \lvert r \rvert < 1$ |
| 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 | |
|---|---|
| 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 | $V=\frac{1}{3}Ah$, where $A$ is the area of the base, and $h$ is perpendicular to the base plane |
| Parallelogram | |
| Area | $A=\lvert\mathbf{v}\times\mathbf{w}\rvert$, where $\mathbf{v}$ and $\mathbf{w}$ are two adjacent sides of a parallelogram |
| Trigonometry | |
|---|---|
| Trigonometric Identities | |
| $\cos^2\theta + \sin^2\theta = 1$ | |
| $\tan\theta = \frac{\sin\theta}{\cos\theta}$ |
| Statistics | |
|---|---|
| Percentage error | $\varepsilon = \lvert \frac{v_a - v_e}{v_e} \rvert \cdot 100$ |
| Descriptive statistics |
| Probability | |
|---|---|
| Random Variables and Probability Distributions | |
|---|---|
| Hypothesis Testing | |
|---|---|
| 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 | $\Delta = b^2 -4ac$ |
| Differentiation | |
|---|---|
| Integration | |
|---|---|
| Differential Equations | |
|---|---|
| Complex Numbers | |
|---|---|
| Rectangular form | $z = a + bi$ |
| Polar form | $z = r(\cos \theta + i\sin(\theta))$ |
| Exponential form | $z = re^{i \theta}$ |
| Vectors | |
|---|---|
| Magnitude | $\lvert \mathbf{v} \rvert = \sqrt{v_1^2 + v_2^2 + v_3^2}$ |
| $\text{OR} \quad \lvert \mathbf{v} \rvert = \sqrt{\sum_{i=1}^{n} v_i^2}$, where $v_i$ is the $i^\text{th}$ component of $\mathbf{v}$ | |
| Scalar product | $\mathbf{v} \cdot \mathbf{w} = \sum_{i=1}^n v_i w_i$ |
| $= \lvert\mathbf{v}\rvert\lvert\mathbf{w}\rvert\cos\theta$, where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w}$ | |
| Vector product | given $\mathbf{v}=\begin{pmatrix}v_1\v_2\v_3\end{pmatrix},$ and $\mathbf{w}=\begin{pmatrix}w_1\w_2\w_3\end{pmatrix}, \mathbf{v}\times \mathbf{w}=\begin{pmatrix}v_2w_3-v_3w_2\v_3w_1-v_1w_3\v_1w_2-v_2w_1\end{pmatrix}$ |
| $\lvert\mathbf{v}\times\mathbf{w}\rvert=\lvert\mathbf{v}\rvert\lvert\mathbf{w}\rvert\sin\theta$ |
| 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 | $A^n = PD^nP^{-1}$, where $A$ is a diagonalisible square matrix, $n \in \mathbb{N}$, $P$ is the matrix of eigenvectors of $A$, and $D$ is the diagonal matrix of the corresponding eigenvalues |
| Transformation Matrices |
| Graph Theory | |
|---|---|
| Adjacency matrices | $a_{ij}=1$ if there exists an edge from vertex $i$ to vertex $j$, otherwise $a_{ij}=0$, for an unweighted graph. |
| If the graph is weighted, $a_{ij}=w_{ij}$, where $w_{ij}$ represents the weight of the edge from $i$ to $j$, or $0$ if there exists no edge. | |
| Transition matrices | $a_{ij}$ is the probability of moving from vertex $i$ to vertex $j$ in a given step. (Note: Some curricula define $a_{ij}$ as the probability of moving from $j$ to $i$) |
| Modelling | |
|---|---|
| Logistic function | $\frac{L}{1+Ce^{-kx}}$, where $L, C, k \in \mathbb{R}^+$ |