IB Mathematics

Applications and Interpretation HLFormula Booklet

Prior learning

Description	Formula
$\displaystyle \text{Area of a parallelogram}$	$\displaystyle A = bh, \text{ where } b \text{ is the base, } h \text{ is the height}$
$\displaystyle \text{Area of a triangle}$	$\displaystyle A = \frac{1}{2}(bh), \text{ where } b \text{ is the base, } h \text{ is the height}$
$\displaystyle \text{Area of a trapezoid}$	$\displaystyle A = \frac{1}{2}(a+b)h, \text{ where } a \text{ and } b \text{ are the parallel sides, } h \text{ is the height}$
$\displaystyle \text{Area of a circle}$	$\displaystyle A = \pi r^2, \text{ where } r \text{ is the radius}$
$\displaystyle \text{Circumference of a circle}$	$\displaystyle C = 2\pi r, \text{ where } r \text{ is the radius}$
$\displaystyle \text{Volume of a cuboid}$	$\displaystyle V = lwh, \text{ where } l \text{ is the length, } w \text{ is the width, } h \text{ is the height}$
$\displaystyle \text{Volume of a cylinder}$	$\displaystyle V = \pi r^2h, \text{ where } r \text{ is the radius, } h \text{ is the height}$
$\displaystyle \text{Volume of a prism}$	$\displaystyle V = Ah, \text{ where } A \text{ is the area of cross-section, } h \text{ is the height}$
$\displaystyle \text{Area of the curved surface of}\\ \text{a cylinder}$	$\displaystyle A = 2\pi rh, \text{ where } r \text{ is the radius, } h \text{ is the height}$
$\displaystyle \text{Distance between two}\\ \text{points $(x_1, y_1)$ and $(x_2, y_2)$}$	$\displaystyle d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$
$\displaystyle \text{Coordinates of the midpoint of}\\ \text{a line segment with endpoints}\\ \text{$(x_1, y_1)$ and $(x_2, y_2)$}$	$\displaystyle \displaystyle\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
$\displaystyle \text{Solutions of a quadratic}\\ \text{equation (HL only)}$	$\displaystyle \text{The solutions of } ax^2 + bx + c = 0 \text{ are } x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \text{ } a \neq 0$

Topic 1: Number and algebra

Topic	Description	Formula
$\displaystyle \text{SL 1.2}$	$\displaystyle \text{The $n$th term of an}\\ \text{arithmetic sequence}$	$\displaystyle u_n = u_1 + (n-1)d$
$\displaystyle \text{SL 1.2}$	$\displaystyle \text{The sum of $n$ terms of an}\\ \text{arithmetic sequence}$	$\displaystyle \displaystyle S_n = \frac{n}{2}(2u_1 + (n-1)d); \; \displaystyle S_n = \frac{n}{2}(u_1 + u_n)$
$\displaystyle \text{SL 1.3}$	$\displaystyle \text{The $n$th term of a}\\ \text{geometric sequence}$	$\displaystyle u_n = u_1r^{n-1}$
$\displaystyle \text{SL 1.3}$	$\displaystyle \text{The sum of $n$ terms of a}\\ \text{finite geometric sequence}$	$\displaystyle \displaystyle S_n = \frac{u_1(r^n - 1)}{r-1} = \frac{u_1(1-r^n)}{1-r}, \; r \neq 1$
$\displaystyle \text{SL 1.4}$	$\displaystyle \text{Compound interest}$	$\displaystyle \displaystyle FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}, \text{ where }FV\text{ is the future value,}\\[1em] \text{where }PV\text{ is the present value, }n\text{ is the number of years,}\\ \text{$k$ is the number of compounding periods per year,}\\ \text{$r\%$ is the nominal annual rate of interest}$
$\displaystyle \text{SL 1.5}$	$\displaystyle \text{Exponents and logarithms}$	$\displaystyle a^x = b \Leftrightarrow x = \log_a b, \text{ where } a > 0, b > 0, a \neq 1$
$\displaystyle \text{SL 1.6}$	$\displaystyle \text{Percentage error}$	$\displaystyle \varepsilon = \left\|\frac{v_A - v_E}{v_E}\right\| \times 100\%, \text{ where } v_E \text{ is the exact value and } v_A \text{ is}\\[1em] \text{the approximate value of } v$
$\displaystyle \text{AHL 1.9}$	$\displaystyle \text{Laws of logarithms}$	$\displaystyle \log_a xy = \log_a x + \log_a y \\[1em] \log_a \frac{x}{y} = \log_a x - \log_a y \\[1em] \log_a x^m = m\log_a x \\[1em] \text{for } a,x,y > 0$
$\displaystyle \text{AHL 1.11}$	$\displaystyle \text{The sum of an infinite}\\ \text{geometric sequence}$	$\displaystyle S_\infty = \frac{u_1}{1-r}, \left\|r\right\| < 1$
$\displaystyle \text{AHL 1.12}$	$\displaystyle \text{Complex numbers}$	$\displaystyle z = a + bi$
$\displaystyle \text{AHL 1.12}$	$\displaystyle \text{Discriminant}$	$\displaystyle \Delta = b^2 - 4ac$
$\displaystyle \text{AHL 1.13}$	$\displaystyle \text{Modulus-argument (polar)}\\ \text{and exponential (Euler)}\\ \text{form}$	$\displaystyle z = r(\cos \theta + i\sin \theta) = re^{i\theta} = r\operatorname{cis} \theta$
$\displaystyle \text{AHL 1.14}$	$\displaystyle \text{Determinant of a } 2\times 2\\ \text{matrix}$	$\displaystyle \mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow \det \mathbf{A} = \|\mathbf{A}\| = ad - bc$
$\displaystyle \text{AHL 1.14}$	$\displaystyle \text{Inverse of a } 2\times 2 \text{ matrix}$	$\displaystyle \mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow \mathbf{A}^{-1} = \frac{1}{\det \mathbf{A}}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}, ad \neq bc$
$\displaystyle \text{AHL 1.15}$	$\displaystyle \text{Power formula for a matrix}$	$\displaystyle \mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1}, \\ \text{where } \mathbf{P} \text{ is the matrix of eigenvectors and } \mathbf{D} \text{ is the diagonal matrix of eigenvalues}$

Topic 2: Functions

Topic	Description	Formula
$\displaystyle \text{SL 2.1}$	$\displaystyle \text{Equations of a straight line}$	$\displaystyle y = mx + c; \; ax + by + d = 0; \\ y - y_1 = m(x - x_1)$
$\displaystyle \text{SL 2.1}$	$\displaystyle \text{Gradient formula}$	$\displaystyle \displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$
$\displaystyle \text{SL 2.6}$	$\displaystyle \text{Axis of symmetry of the}\\ \text{graph of a quadratic}\\ \text{function}$	$\displaystyle \displaystyle f(x) = ax^2 + bx + c \Rightarrow \text{ axis of symmetry is } x = -\frac{b}{2a}$
$\displaystyle \text{AHL 2.9}$	$\displaystyle \text{Logistic function}$	$\displaystyle f(x) = \frac{L}{1 + Ce^{-kx}}, \text{ } L, k, C > 0$

Topic 3: Geometry and trigonometry

Topic	Description	Formula
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Distance between two}\\\text{points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$}$	$\displaystyle d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Coordinates of the}\\ \text{midpoint of a line segment}\\ \text{with endpoints $(x_1, y_1, z_1)$} \\ \text{and $(x_2, y_2, z_2)$}$	$\displaystyle \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Volume of a right-pyramid}$	$\displaystyle V = \frac{1}{3}Ah \text{, where } A \text{ is the area of the base, } h \text{ is the height}$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Volume of a right cone}$	$\displaystyle V = \frac{1}{3}\pi r^2h \text{, where } r \text{ is the radius, } h \text{ is the height}$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Area of the curved surface}\\\text{of a cone}$	$\displaystyle A = \pi rl \text{, where } r \text{ is the radius, } l \text{ is the slant height}$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Volume of a sphere}$	$\displaystyle V = \frac{4}{3}\pi r^3 \text{, where } r \text{ is the radius}$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Surface area of a sphere}$	$\displaystyle A = 4\pi r^2 \text{, where } r \text{ is the radius}$
$\displaystyle \text{SL 3.2}$	$\displaystyle \text{Sine rule}$	$\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
$\displaystyle \text{SL 3.2}$	$\displaystyle \text{Cosine rule}$	$\displaystyle c^2 = a^2 + b^2 - 2ab\cos C ; \; \cos C = \frac{a^2 + b^2 - c^2}{2ab}$
$\displaystyle \text{SL 3.2}$	$\displaystyle \text{Area of a triangle}$	$\displaystyle A = \frac{1}{2}ab\sin C$
$\displaystyle \text{SL 3.4}$	$\displaystyle \text{Length of an arc}$	$\displaystyle l = \frac{\theta}{360} \times 2\pi r, \\[1em] \text{where } \theta \text{ is the angle measured in degrees, } r \text{ is the radius}$
$\displaystyle \text{SL 3.4}$	$\displaystyle \text{Area of a sector}$	$\displaystyle A = \frac{\theta}{360} \times \pi r^2, \\[1em] \text{where } \theta \text{ is the angle measured in degrees, } r \text{ is the radius}$
$\displaystyle \text{AHL 3.7}$	$\displaystyle \text{Length of an arc}$	$\displaystyle l = r\theta, \text{ where } r \text{ is the radius, } \theta \text{ is the angle measured in radians}$
$\displaystyle \text{AHL 3.7}$	$\displaystyle \text{Area of a sector}$	$\displaystyle A = \frac{1}{2}r^2\theta, \text{ where } r \text{ is the radius, } \theta \text{ is the angle measured in}\\ \text{radians}$
$\displaystyle \text{AHL 3.8}$	$\displaystyle \text{Identities}$	$\displaystyle \cos^2 \theta + \sin^2 \theta = 1 \\[1em] \tan \theta = \frac{\sin \theta}{\cos \theta}$
$\displaystyle \text{AHL 3.9}$	$\displaystyle \text{Transformation matrices}$	$\displaystyle \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}, \text{ reflection in the line } y = (\tan \theta)x \\[1em] \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}, \text{ horizontal stretch / stretch parallel to $x$-axis with a scale}\\ \text{factor of } k \\[1em] \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix}, \text{ vertical stretch / stretch parallel to $y$-axis with a scale}\\ \text{factor of } k \\[1em] \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}, \text{ enlargement, with a scale factor of } k, \text{ centre } (0, 0) \\[1em] \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}, \text{ anticlockwise/counter-clockwise rotation of}\\ \text{angle } \theta \text{ about the origin } (\theta > 0) \\[1em] \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}, \text{ clockwise rotation of angle } \theta \text{ about the origin}\\ (\theta > 0)$
$\displaystyle \text{AHL 3.10}$	$\displaystyle \text{Magnitude of a vector}$	$\displaystyle \\|\mathbf{v}\\| = \sqrt{v_1^2 + v_2^2 + v_3^2}, \text{ where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$
$\displaystyle \text{AHL 3.11}$	$\displaystyle \text{Vector equation of a line}$	$\displaystyle \mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$
$\displaystyle \text{AHL 3.11}$	$\displaystyle \text{Parametric form of the}\\ \text{equation of a line}$	$\displaystyle x = x_0 + \lambda l, \; y = y_0 + \lambda m, \; z = z_0 + \lambda n$
$\displaystyle \text{AHL 3.13}$	$\displaystyle \text{Scalar product}$	$\displaystyle \mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 + v_3w_3, \text{ where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}, \mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}\\[1em] \mathbf{v} \cdot \mathbf{w} = \|\mathbf{v}\|\|\mathbf{w}\|\cos \theta, \text{ where } \theta \text{ is the angle between } \mathbf{v} \text{ and } \mathbf{w}$
$\displaystyle \text{AHL 3.13}$	$\displaystyle \text{Angle between two}\\ \text{vectors}$	$\displaystyle \cos \theta = \frac{v_1w_1 + v_2w_2 + v_3w_3}{\|\mathbf{v}\|\|\mathbf{w}\|}$
$\displaystyle \text{AHL 3.13}$	$\displaystyle \text{Vector product}$	$\displaystyle \mathbf{v} \times \mathbf{w} = \begin{pmatrix} v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \end{pmatrix}, \text{ where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}, \mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}\\[2em] \|\mathbf{v} \times \mathbf{w}\| = \|\mathbf{v}\|\|\mathbf{w}\|\sin \theta, \text{ where } \theta \text{ is the angle between } \mathbf{v} \text{ and } \mathbf{w}$
$\displaystyle \text{AHL 3.13}$	$\displaystyle \text{Area of a parallelogram}$	$\displaystyle A = \|\mathbf{v} \times \mathbf{w}\| \text{ where } \mathbf{v} \text{ and } \mathbf{w} \text{ form two adjacent sides of a}\\ \text{parallelogram}$

Topic 4: Statistics and probability

Topic	Description	Formula
$\displaystyle \text{SL 4.2}$	$\displaystyle \text{Interquartile range}$	$\displaystyle \operatorname{IQR} = Q_3 - Q_1$
$\displaystyle \text{SL 4.3}$	$\displaystyle \text{Mean, } \bar{x} \text{, of a set of data}$	$\displaystyle \bar{x} = \frac{\sum\limits_{i=1}^k f_ix_i}{n}, \text{ where } n = \sum\limits_{i=1}^k f_i$
$\displaystyle \text{SL 4.5}$	$\displaystyle \text{Probability of an event } A$	$\displaystyle \operatorname{P}(A) = \frac{n(A)}{n(U)}$
$\displaystyle \text{SL 4.5}$	$\displaystyle \text{Complementary events}$	$\displaystyle \operatorname{P}(A) + \operatorname{P}(A') = 1$
$\displaystyle \text{SL 4.6}$	$\displaystyle \text{Combined events}$	$\displaystyle \operatorname{P}(A \cup B) = \operatorname{P}(A) + \operatorname{P}(B) - \operatorname{P}(A \cap B)$
$\displaystyle \text{SL 4.6}$	$\displaystyle \text{Mutually exclusive events}$	$\displaystyle \operatorname{P}(A \cup B) = \operatorname{P}(A) + \operatorname{P}(B)$
$\displaystyle \text{SL 4.6}$	$\displaystyle \text{Conditional probability}$	$\displaystyle \operatorname{P}(A\|B) = \frac{\operatorname{P}(A \cap B)}{\operatorname{P}(B)}$
$\displaystyle \text{SL 4.6}$	$\displaystyle \text{Independent events}$	$\displaystyle \operatorname{P}(A \cap B) = \operatorname{P}(A)\operatorname{P}(B)$
$\displaystyle \text{SL 4.7}$	$\displaystyle \text{Expected value of a}\\ \text{discrete random variable } X$	$\displaystyle \operatorname{E}(X) = \sum x\operatorname{P}(X = x)$
$\displaystyle \text{SL 4.8}$	$\displaystyle \text{Binomial distribution}\\ X \sim \text{B}(n, p)$	$\displaystyle \operatorname{E}(X) = np$
$\displaystyle \text{AHL 4.14}$	$\displaystyle \text{Linear transformation of a}\\ \text{single random variable}$	$\displaystyle \operatorname{E}(aX + b) = a\operatorname{E}(X) + b \\[1em] \operatorname{Var}(aX + b) = a^2\operatorname{Var}(X)$
$\displaystyle \text{AHL 4.14}$	$\displaystyle \text{Linear combinations of $n$}\\ \text{independent random}\\ \text{variables, } X_1, X_2, ..., X_n$	$\displaystyle \operatorname{E}(a_1X_1 \pm a_2X_2 \pm ... \pm a_nX_n) = a_1\operatorname{E}(X_1) \pm a_2\operatorname{E}(X_2) \pm ... \pm a_n\operatorname{E}(X_n) \\[1em] \operatorname{Var}(a_1X_1 \pm a_2X_2 \pm ... \pm a_nX_n)\\ \quad = a_1^2\operatorname{Var}(X_1) + a_2^2\operatorname{Var}(X_2) + ... + a_n^2\operatorname{Var}(X_n)$
$\displaystyle \text{AHL 4.14}$	$\displaystyle \text{Sample statistics}\\ \text{Unbiased estimate of}\\ \text{population variance } s_{n-1}^2$	$\displaystyle s_{n-1}^2 = \frac{n}{n-1}s_n^2$
$\displaystyle \text{AHL 4.17}$	$\displaystyle \text{Poisson distribution}\\ X \sim \operatorname{Po}(m)\\[1em] \text{Mean}\\[1em] \text{Variance}$	$\displaystyle \\[4em] \operatorname{E}(X) = m \\[1em] \operatorname{Var}(X) = m$
$\displaystyle \text{AHL 4.19}$	$\displaystyle \text{Transition matrices}$	$\displaystyle \mathbf{T}^n\mathbf{s}_0 = \mathbf{s}_n, \text{ where } \mathbf{s}_0 \text{ is the initial state}$

Topic 5: Calculus

Topic	Description	Formula
$\displaystyle \text{SL 5.3}$	$\displaystyle \text{Derivative of } x^n$	$\displaystyle f(x) = x^n \Rightarrow f'(x) = nx^{n-1}$
$\displaystyle \text{SL 5.5}$	$\displaystyle \text{Integral of } x^n$	$\displaystyle \int x^n \,dx = \frac{x^{n+1}}{n+1} + C, \text{ } n \neq -1$
$\displaystyle \text{SL 5.5}$	$\displaystyle \text{Area between a curve}\\ \text{$y = f(x)$ and the $x$-axis,}\\ \text{where } f(x) > 0$	$\displaystyle A = \int_a^b y\,dx$
$\displaystyle \text{SL 5.8}$	$\displaystyle \text{The trapezoidal rule}$	$\displaystyle \int_a^b y\,dx \approx \frac{1}{2}h((y_0 + y_n) + 2(y_1 + y_2 + ... + y_{n-1})), \\[1em] \text{where } h = \frac{b-a}{n}$
$\displaystyle \text{AHL 5.9}$	$\displaystyle \text{Derivative of } \sin x$	$\displaystyle f(x) = \sin x \Rightarrow f'(x) = \cos x$
$\displaystyle \text{AHL 5.9}$	$\displaystyle \text{Derivative of } \cos x$	$\displaystyle f(x) = \cos x \Rightarrow f'(x) = -\sin x$
$\displaystyle \text{AHL 5.9}$	$\displaystyle \text{Derivative of } \tan x$	$\displaystyle f(x) = \tan x \Rightarrow f'(x) = \frac{1}{\cos^2 x}$
$\displaystyle \text{AHL 5.9}$	$\displaystyle \text{Derivative of } e^x$	$\displaystyle f(x) = e^x \Rightarrow f'(x) = e^x$
$\displaystyle \text{AHL 5.9}$	$\displaystyle \text{Derivative of } \ln x$	$\displaystyle f(x) = \ln x \Rightarrow f'(x) = \frac{1}{x}$
$\displaystyle \text{AHL 5.9}$	$\displaystyle \text{Chain rule}$	$\displaystyle y = g(u), \text{ where } u = f(x) \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}u} \times \frac{\mathrm{d}u}{\mathrm{d}x}$
$\displaystyle \text{AHL 5.9}$	$\displaystyle \text{Product rule}$	$\displaystyle y = uv \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = u\frac{\mathrm{d}v}{\mathrm{d}x} + v\frac{\mathrm{d}u}{\mathrm{d}x}$
$\displaystyle \text{AHL 5.9}$	$\displaystyle \text{Quotient rule}$	$\displaystyle y = \frac{u}{v} \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{v\frac{\mathrm{d}u}{\mathrm{d}x} - u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}$
$\displaystyle \text{AHL 5.11}$	$\displaystyle \text{Standard integrals}$	$\displaystyle \int \frac{1}{x}\,\mathrm{d}x = \ln\|x\| + C \\[1em] \int \sin x\,\mathrm{d}x = -\cos x + C \\[1em] \int \cos x\,\mathrm{d}x = \sin x + C \\[1em] \int \frac{1}{\cos^2 x}\,\mathrm{d}x = \tan x + C \\[1em] \int e^x\,\mathrm{d}x = e^x + C$
$\displaystyle \text{AHL 5.12}$	$\displaystyle \text{Area of region enclosed}\\ \text{by a curve and $x$ or $y$-axes}$	$\displaystyle A = \int_a^b \|y\|\,\mathrm{d}x \text{ or } A = \int_a^b \|x\|\,\mathrm{d}y$
$\displaystyle \text{AHL 5.12}$	$\displaystyle \text{Volume of revolution}\\ \text{about $x$ or $y$-axes}$	$\displaystyle V = \int_a^b \pi y^2\,\mathrm{d}x \text{ or } V = \int_a^b \pi x^2\,\mathrm{d}y$
$\displaystyle \text{AHL 5.13}$	$\displaystyle \text{Acceleration}$	$\displaystyle a = \frac{\mathrm{d}v}{\mathrm{d}t} = \frac{\mathrm{d}^2s}{\mathrm{d}t^2} = v\frac{\mathrm{d}v}{\mathrm{d}s}$
$\displaystyle \text{AHL 5.13}$	$\displaystyle \text{Distance travelled from}\\ t_1 \text{ to } t_2$	$\displaystyle \text{distance} = \int_{t_1}^{t_2} \|v(t)\|\,\mathrm{d}t$
$\displaystyle \text{AHL 5.13}$	$\displaystyle \text{Displacement from}\\ t_1 \text{ to } t_2$	$\displaystyle \text{displacement} = \int_{t_1}^{t_2} v(t)\,\mathrm{d}t$
$\displaystyle \text{AHL 5.16}$	$\displaystyle \text{Euler's method}$	$\displaystyle y_{n+1} = y_n + h \times f(x_n, y_n); \; x_{n+1} = x_n + h, \text{ where $h$ is a constant}\\ \text{(step length)}$
$\displaystyle \text{AHL 5.16}$	$\displaystyle \text{Euler's method for}\\ \text{coupled systems}$	$\displaystyle x_{n+1} = x_n + h \times f_1(x_n, y_n, t_n) \\[1em] y_{n+1} = y_n + h \times f_2(x_n, y_n, t_n) \\[1em] t_{n+1} = t_n + h \\[1em] \text{where } h \text{ is a constant (step length)}$
$\displaystyle \text{AHL 5.17}$	$\displaystyle \text{Exact solution for coupled}\\ \text{linear differential equations}$	$\displaystyle \mathbf{x} = Ae^{\lambda_1 t}\mathbf{p}_1 + Be^{\lambda_2 t}\mathbf{p}_2$