For small angle $\theta$, measured in radians:
$$\sin \theta \space \approx \space \theta$$
$$\cos \theta \space \approx \space 1 - \frac{\theta^2}{2}$$
$$\tan \theta \space \approx \space \theta$$
Trigonometric Identities
$$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$$
$$\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$$
$$\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} (A \pm B \ne (k + \frac 1 2)\pi)$$
Differentiation
$$f(x) f'(x)$$
$$\tan x \sec^2 x$$
$$\csc x -\csc x \cot x$$
$$\sec x \sec x \tan x$$
$$\cot x -\csc^2 x$$
$$\frac{f(x)}{g(x)} \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$$
$$x = a^n \Leftrightarrow n = \log_a x \text{ for } a \gt 0 \text{ and } x \gt 0$$
$$\log_a x + \log_a y \equiv \log_a (xy)$$
$$\log_a x - \log_a y \equiv \log_a {x \overwithdelims ( ) y}$$
$$k \log_a x \equiv \log_a (x^k)$$
Coordinate Geometry
A straight line graph, gradient $m$ passing through $(x_1, y_1)$ has equation
$$y-y_1 = m(x-x_1)$$
Straight lines with gradients $m_1$ and $m_2$ are perpendicular when $m_1m_2 = -1$
Sequences
General term of an arithmetic progression:
$$u_n = a + (n - 1)d$$
General term of a geometric progression:
$$u_n = ar^{n-1}$$
Trigonometry
In the triangle ABC
$$\text{Sine rule: } \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
$$\text{Cosine rule: } a^2 = b^2 + c^2 - 2bc \cos A$$
$$\text{Area } = \frac{1}{2} ab \sin C$$
$$\cos ^2 A + \sin ^2 A \equiv 1$$
$$\sec ^2 A \equiv 1 + \tan ^2 A$$
$$\csc ^2 A \equiv 1 + \cot ^2 A$$
$$\sin 2A \equiv 2 \sin A \cos A$$
$$\cos 2A \equiv \cos ^2 A - \sin ^2 A$$
$$\tan 2A \equiv \frac{2 \tan A}{1 - \tan ^2 A}$$
Mensuration
Circumference and Area of circle, radius $r$ and diameter $d$:
$$C = 2 \pi r = \pi d A = \pi r^2$$
Pythagoras' Theorem: In any right-angled triangle where $a$, $b$ and $c$ are the lengths of the sides and $c$ is the
hypotenuse:
$$c^2 = a^2 + b^2$$
Area of a trapezium $= \frac{1}{2} (a + b)h$, where $a$ and $b$ are the lengths of the parallel sides and h is their
perpendicular separation.
Volume of a prism = area of cross section x length
For a circle of radius $r$, where an angle at the centre of $\theta$ radians subtends an arc of length $s$ and
encloses an associated sector of an area $a$:
$$s = r \theta a = \frac{1}{2} r^2 \theta$$
Calculus and Differential Equations
Differentiation
Function
Derivative
$x^n$
$nx^{n-1}$
$\sin kx$
$k \cos kx$
$\cos kx$
$-k \sin kx$
$e^{kx}$
$ke^{kx}$
$\ln x$
$\frac{1}{x}$
$f(x) + g(x)$
$f'(x) + g'(x)$
$f(x)g(x)$
$f'(x)g(x) + f(x)g'(x)$
$f(g(x))$
$f'(g(x))g'(x)$
Integration
Function
Integral
$x^n$
$\frac{1}{n+1} x^{n+1} + c, n \ne -1$
$\cos kx$
$\frac{1}{k} \sin kx + c$
$\sin kx$
$-\frac{1}{k} \cos kx + c$
$e^{kx}$
$\frac{1}{k} e^{kx} + c$
$\frac{1}{x}$
$\ln |x| x + c, x \ne 0$
$f'(x) + g'(x)$
$f(x) + g(x) + c$
$f'(g(x))g'(x)$
$f(g(x)) + c$
$$\text{Area under a curve: } \int_a^b y \space dx (y \ge 0)$$
Vectors
$$|x \textbf{i} + y \textbf{j} + z \textbf{k}| = \sqrt{(x^2 + y^2 + z^2)}$$
Mechanics
Forces and Equilibrium
Weight = mass x g
Friction: $F \le \mu R$
Newton's second law in the form: $F = ma$
Kinematics
For motion in a straight line with variable acceleration:
$$v = \frac{dr}{dt}$$
$$a = \frac{dv}{dt} = \frac{d^2r}{dt^2}$$
$$r = \int v \space dt$$
$$v = \int a \space dt$$
Statistics
The mean of a set of data:
$$\bar x = \frac{\sum x}{n} = \frac{\sum fx}{\sum f}$$
The standard Normal variable:
$$Z = \frac{X - \mu}{\sigma}$$
where
$$X \sim N (\mu , \sigma ^2)$$
Proof
Disproof by Counterexample
It's much easier to prove something is false than something is true. A counterexample is an example used to disprove a statement.
e.g. "For all positive integers $n, n^2 - n + 41$ is prime" has a counterexample of 41, so the statement is false.
Proof by Deduction
This is where a statement is proven by using well-known mathematical principles such as algebra or geometry.
e.g. "Show that for any integer $n, n^2 + n$ is always even."
If odd, then (odd $\times$ odd) + odd = odd + odd = even
If even, then (even $\times$ even) + even = even + even = even
Therefore, $n, n^2 + n$ is always even for any integer. QED Proofs are normally ended with QED, ■ or □
Laws of surds: $$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$$$$\sqrt{a} + \sqrt{b} \ne \sqrt{a+b}$$$$(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y}) = x - y$$$$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$$Rationalisation:
This is the process of removing surds from the bottom of a fraction. If in the form $\frac{x}{\sqrt{a}}$, multiply by $\frac{\sqrt{a}}{\sqrt{a}}$.
$$\text{e.g. } \frac{5}{\sqrt{7}} = \frac{5}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{5 \sqrt{7}}{7}$$
If the denominator is in the form $\frac{c}{b \pm \sqrt{a}}$, multiply by $\frac{b \mp \sqrt{a}}{b \mp \sqrt{a}}$.
$$\text{e.g. } \frac{1 + \sqrt{5}}{2 - \sqrt{3}} = \frac{1 + \sqrt{5}}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{(1 + \sqrt{5})(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})}$$
$$= \frac{2 + \sqrt{15} + \sqrt{3} + 2 \sqrt{5}}{4 - 2 \sqrt{3} + 2 \sqrt{3} - 3} = \frac{2 + \sqrt{15} + \sqrt{3} + 2 \sqrt{5}}{1}$$
$$= 2 + \sqrt{15} + \sqrt{3} + 2 \sqrt{5}$$
Quadratics
Positive $x^2$ coefficient $= \cup$ shape.
Negative $x^2$ coefficient $= \cap$ shape.
Completing the square: $$x^2 + bx + c \rightarrow (x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c$$ $$ax^2 + bx + c \rightarrow a(x + \frac{b}{2a})^2 - a(\frac{b}{2a})^2 + c$$
When $a = 1$:
$$\text{e.g. } x^2 - 6x + 4 = (x - 3)^2 - 9 + 4 = (x - 3)^2 - 5$$
Or when $a \ne 1$:
$$\text{e.g. } 2x^2 - 12x + 28 = 2(x^2 - 6x + 14) = 2((x - 3)^2 - 3^2 + 14)$$
$$ = 2((x - 3)^2 + 5) = 2(x - 3)^2 + 10$$
If you have $a(x + p)^2 + q = 0$, the turning point of the graph is $(-p, q)$. If $a$ is positive you will get a minimum point, if $a$ is negative you will get a maximum point.Quadratic Formula Proof:
$$ax^2 + bx + c = 0$$
$$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$
$$(x + \frac{b}{2a})^2 - \frac{b^2}{4a^2} + \frac{4ac}{4a^2} = 0$$
$$(x + \frac{b}{2a})^2 + \frac{-b^2 + 4ac}{4a^2} = 0$$
$$(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}$$
$$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$Inequalities:
$\lt$ and $\gt$ are strict inequalities. $\le$ and $\ge$ are inclusive inequalities. On a graph, strict inequalities are dotted lines and inclusive inequalities are solid lines.
When you multiply or divide an inequality by a negative number the direction of the inequality reverses.
In set notation, $\{x: x \gt a\}$ means all values of $x$ where $x$ is greater than $a$.
Also in set notation, $[x,y]$ is used to denote an inclusive interval between $x$ and $y$, whereas $(x,y)$ is exclusive.
Discriminant:
The discriminant tells you how many roots a quadratic has. It is sometimes represented by $\Delta$. If $b^2 - 4ac \lt 0$ there are no real roots.If $b^2 - 4ac = 0$ there are two equal real roots (or just one root).If $b^2 - 4ac \gt 0$ there are two distinct real roots.
Simultaneous Equations
Elimination:
Multiply each equation by a suitable number so that the two equations have the same leading coefficient.
Subtract the second equation from the first.
Solve this new equation for y.
Substitute y into either Equation 1 or Equation 2 and solve for x.
A polynomial is a function of the form:
$$P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0$$
$x$ is called the variable.
$a_0$ is called the constant term.
$a_r$ is called the coefficient of $x^r$.
$a_n$ is called the leading coefficient.
$n$ is the degree.
A polynomial of degree 1 is called linear, degree 2 is called quadratic, degree 3 is called cubic, and degree 4 is called quartic.
Polynomials to Linear Graphs:
$y = ax^n$
$\log y = \log (a x^n)$
$\log y = n \log x + \log a$
$y = mx + c$
Simplifying Polynomials:
Factorise the numerator and the denominator to cancel as many terms as possible.
$$\text{e.g. } \frac{x^2 + 5x + 6}{x^2 + 6x + 8} = \frac{(x + 2)(x + 3)}{(x + 2)(x + 4)} = \frac{x + 3}{x + 4}$$
Factor Theorem:For expression $f(x)$, if $f(a) = 0$ then $(x - a)$ is a factor of $f(x)$.
This is used to help find common factors.
e.g. If $f(x) = x^3 + 6x^2 + 11x + 6$, show that $(x + 2)$ is a factor.
$f(-2) = (-2)^3 + 6(-2)^2 + 11(-2) + 6 = -8 + 24 -22 + 6 = 0$
$f(-2) = 0$ so $(x + 2)$ is a factor.
Polynomial Multiplication:
Multiplying polynomials requires multiplying each term in the first polynomial by each term in the second, and then simplifying
e.g. $(x^2+2)(3x^2+4x+1) = 3x^4 + 4x^3 + x^2 + 6x^2 + 8x + 2$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 3x^4 + 4x^3 + 7x^2 + 8x + 2$
Polynomial Division:
Dividing a polynomial by an expression in the form $(ax \pm b)$ where $a \ge 1$: Method 1
e.g. Divide $x^3 + 6x^2 + 11x + 6$ by $(x + 2)$ and then fully factorise the expression.
$$ \begin{array}{rll} x + 2 \enclose{longdiv}{x^3 + 6x^2 + 11x + 6}\kern-.2ex \\[-3pt] \end{array} $$
Work out the remainder by subtracting $x^3 + 2x^2$ from $x^3 + 6x^2$. Carry down the next part of the expression.
$$ \begin{array}{rll} x^2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\[-3pt] x + 2 \enclose{longdiv}{x^3 + 6x^2 + 11x + 6}\kern-.2ex \\[-3pt] \underline{x^3 + 2x^2}~~~~~~~~~~~~~~~~~~ \\[-3pt] \color{red}{0 + 4x^2 + 11x}~~~~~~~ \\[-3pt] \end{array} $$
Work out the remainder by subtracting $4x^2 + 8x$ from $4x^2 + 11x$. Carry down the next part of the expression.
$$ \begin{array}{rll} x^2 + 4x~~~~~~~~~~~~~~~~~~~~ \\[-3pt] x + 2 \enclose{longdiv}{x^3 + 6x^2 + 11x + 6}\kern-.2ex \\[-3pt] \underline{x^3 + 2x^2}~~~~~~~~~~~~~~~~~~ \\[-3pt] 0 + 4x^2 + 11x~~~~~~~ \\[-3pt] \underline{4x^2 + ~~8x}~~~~~~~ \\[-3pt] \color{red}{3x + 6} ~\\[-3pt] \end{array} $$
Multiply $x^2$ by 2 and subtract this from $6x^2$ to find out how many more $x^2$ terms are needed. Divide this by $x$.
$$x^3 + 6x^2 + 11x + 6 = (x + 2)(x^2 \color{red}{+ 4x}$$
Multiply $4x$ by 2 and subtract this from $11x$ to find out how many more $x$ terms are needed. Divide this by $x$.
$$x^3 + 6x^2 + 11x + 6 = (x + 2)(x^2 + 4x \color{red}{+ 3)}$$
Multiply 3 by 2 and subtract this from 6 to find the remainder.
$$x^3 + 6x^2 + 11x + 6 = (x + 2)(x^2 + 4x + 3) \color{red}{+ 0}$$
Summary: Find how many $x^n$ terms you have in right bracket, find how many you have on left side, divide the difference by $x$ and add on, repeat.
Difference of Two Squares
$a^2 - b^2$ can be factorised as $(a-b)(a+b)$
Trigonometry
General Rules
SOHCAHTOA (for right angled triangles):
$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite}}{\text{adjacent}}$$
Sin Rule (for triangle ABC):
$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \frac{a}{sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Cosine Rule (for triangle ABC):
$$a^2 = b^2 + c^2 - 2bc \cos A$$
Length of any arc of a circle (where $\theta$ is in radians):
$$l = r \theta$$
Area of a sector of a circle (where $\theta$ is in radians):
$$A = \frac{1}{2} r^2 \theta$$
Area of a triangle:
$$A = \frac{1}{2} ab \sin C$$
Identities:
$$\cos ^2 \theta + \sin ^2 \theta \equiv 1 1 + \tan ^2 \theta \equiv \sec ^2 \theta \cot ^2 \theta + 1 \equiv \csc ^2 \theta$$
$$a \cos \theta + b \sin \theta \equiv R \cos (\theta - \alpha) \text { where } R = \sqrt{a^2 + b^2}, \space \cos \alpha = \frac{a}{R} \text{ and } \sin \alpha = \frac{b}{R}$$
$$a \cos \theta + b \sin \theta \equiv R \sin (\theta + \beta) \text { where } R = \sqrt{a^2 + b^2}, \space \sin \beta = \frac{a}{R} \text{ and } \cos \beta = \frac{b}{R}$$
$$\sin(-\theta) = -\sin(\theta)$$
$$\cos(-\theta) = \cos\theta$$
$$\csc^{-1}(\theta) = \sin^{-1}(\frac{1}{\theta})$$
$$\sec^{-1}(\theta) = \cos^{-1}(\frac{1}{\theta})$$
$$\cot^{-1}(\theta) = \tan^{-1}(\frac{1}{\theta})$$
Radians
One radian is equal to about 57.3° and is written as 1 rad or 1$^c$.
$$180° = \pi \space rads$$
$$90° = \frac{\pi}{2} \space rads$$
$$360° = 2\pi \space rads$$
To convert from degrees to radians multiply by $\frac{\pi}{180}$.
To convert from radians to degrees multiply by $\frac{180}{\pi}$.
Sine, Cosine and Tangent
Table of values:
To help remember some values (use SOHCAHTOA on these triangles):
When solving a trig equation such as $\cos(3x) = -\frac{1}{2}$ for $0 \le x \le 360^o$
$\tan ^{-1}(\theta) = x, 180 + x, 360 + x, 540 + x, 720 + x ...$
$\color{#f5f5f5}{\tan ^{-1}(\theta)} = x + 180n$ for all $n \in \mathbb N$
Domain: $\{x ∈ \mathbb R\}$
Range: $\{y ∈ \mathbb R, y \ne \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \pm \frac{5\pi}{2} ...\}$
$ = \{y ∈ \mathbb R, y \ne \pm (\frac{\pi}{2} + k\pi)\} \text{ where } k ∈ \mathbb Z$
Secant
$$\sec x = \frac{1}{\cos x}$$ Domain: $\{x ∈ \mathbb R, x \ne k \pi + \frac{\pi}{2}\} \text{ where } k ∈ \mathbb Z$
Range: $\{|y| \ge 1\} = {|y| ∈ [1, \infty)}$
Cosecant
$$\csc x = \frac{1}{\sin x}$$ Domain: $\{x ∈ \mathbb R, x \ne \pm k \pi\} \text{ where } k ∈ \mathbb Z$
Range: $\{|y| \ge 1\} = {|y| ∈ [1, \infty)}$
Cotangent
$$\cot x = \frac{1}{\tan x}$$ Domain: $\{x ∈ \mathbb R, x \ne \pm k \pi\} \text{ where } k ∈ \mathbb Z$
Range: $\{y ∈ \mathbb R\}$
Arcsec $\rightarrow \sec ^{-1}()$
$\sec ^{-1}(\theta) = x, 360 - x, 360 + x, 720 - x, 720 + x ...$
$\color{#f5f5f5}{\sec ^{-1}(\theta)} = x + 360n, 360n - x$ for all $n \in \mathbb N$
Domain: $\{|x| \ge 1\} = {|x| ∈ [1, \infty)}$
Range: $\{y ∈ \mathbb R, y \ne k \pi + \frac{\pi}{2}\} \text{ where } k ∈ \mathbb Z$
Arccosec $\rightarrow \csc ^{-1}()$
$\csc ^{-1}(\theta) = x, 360 - x, 360 + x, 720 - x, 720 + x ...$
$\color{#f5f5f5}{\csc ^{-1}(\theta)} = x + 360n, 360n - x$ for all $n \in \mathbb N$
Domain: $\{|x| \ge 1\} = {|x| ∈ [1, \infty)}$
Range: $\{y ∈ \mathbb R, y \ne \pm k \pi\} \text{ where } k ∈ \mathbb Z$
Arccot $\rightarrow \cot ^{-1}()$
$\cot ^{-1}(\theta) = x, 180 + x, 360 + x, 540 + x, 720 + x ...$
$\color{#f5f5f5}{\cot ^{-1}(\theta)} = x + 180n$ for all $n \in \mathbb N$
Domain: $\{x ∈ \mathbb R\}$
Range: $\{y ∈ \mathbb R, y \ne \pm k \pi\} \text{ where } k ∈ \mathbb Z$
Double Angle Formulae
$\cos 2x = \cos ^2 x - \sin ^2 x = 2 \cos ^2 x - 1 = 1 - 2 \sin ^2 x$$\sin 2x = 2 \sin x \cos x$$\tan 2x = \frac{2 \tan x}{1 - \tan ^2 x}$
Compound Angle Formulae
$\cos (A \pm B) \equiv \cos A \cos B \mp \sin A \sin B$$\sin (A \pm B) \equiv \sin A \cos B \pm \cos A \sin B$$\tan (A \pm B) \equiv \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
Small Angle Approximations
For small angle $\theta$, measured in radians: $\sin \theta \space \approx \space \theta$$\cos \theta \space \approx \space 1 - \frac{\theta^2}{2}$$\tan \theta \space \approx \space \theta$
Exponentials and Logarithms
Exponentials
An exponential has $x$ in the power. The graph of $a^x$ is:
Exponentials to Linear Graphs
$y = ab^x$
$\log y = \log (ab^x)$
$\log y = (\log b)x + \log a$
$y = mx + c$
Logarithms
$\log_a x$ is the inverse function of $a^x$, where $a$ is positive and $x \ge 0$.
$\ln x$ is the inverse function of e$^x$. The graph of $\ln x$ is:
Laws of Logarithms
$\log_a x + \log_a y \equiv \log_a (xy)$$\log_a x - \log_a y \equiv \log_a {x \overwithdelims ( ) y}$$k \log_a x \equiv \log_a (x^k)$
Gradient
The gradient of $e^{kx}$ at $(x, y) = ke^{kx}$.
Differentiation
To differentiate, multiply by the power, then subtract one from the power, and repeat for each term.$$y = x^a \frac{dy}{dx} = ax^{a-1}$$$y$ can be written as $f(x)$ and $\frac{dy}{dx}$ can be written as $f'(x)$.
Set $f'(x) = 0$. Solve for x coordinates. Plug back into $f(x)$ to get y coordinates. If the second derivative, $\frac{d^2y}{dx^2}$ or $f''(x)$, is 0, it is a point of inflection. If it is positive, it is a minimum point. If it is negative, it is a maximum point.
Tangents and Normals
The gradient of the tangent to a curve is the differential of the equation of the curve.The gradient of the normal is the negative reciprocal of the gradient of the tangent.
Differentiation by Inspection
$$f'(g(x)) = f'(x) \times g'(x) \space du$$
Treat the function $g(x)$ as an $x$, differentiate, and then multiply by the differential of $g(x)$.
e.g. $\frac{d}{dx} (2x+3)^3$
$$\frac{d}{dx}e^x = e^x \frac{d}{dx}e^{kx} = ke^{kx} \frac{d}{dx}e^{f(x)} = f'(x)e^{f(x)}$$$$\frac{d}{dx}\ln x = \frac{1}{x}$$$$\frac{d}{dx}a^{kx} = k (\ln a) a^{kx}$$$$\frac{d}{dx}\sin (ax) = a \cos (ax)$$$$\frac{d}{dx}\cos (ax) = -a \sin (ax)$$$$\frac{d}{dx}\tan (ax) = a \sec^2 (ax)$$$$\frac{d}{dx}\sec (ax) = a \sec (ax) \tan (ax)$$$$\frac{d}{dx}\csc (ax) = -a \csc (ax) \cot (ax)$$$$\frac{d}{dx}\cot (ax) = -a \csc^2 (ax)$$$$\frac{d}{dx}\sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}$$$$\frac{d}{dx}\cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}}$$$$\frac{d}{dx}\tan^{-1} x = \frac{1}{1 + x^2}$$$$\text{If } x \text{ and } y \text{ are defined in terms of parameter } t \text{, then } \frac{dy}{dx} = \frac{dy}{dt} + \frac{dx}{dt}$$
Integration
To integrate, add one to the power, then divide by the new power, and repeat for each term. Add $c$ on the end as a constant.
$$\int x^a \space dx = \frac{x^{a+1}}{a+1} + c$$
Fundamental Theorem of Calculus
$$\frac{d}{dx} \int_a^x f(t) dt = f(x)$$
Integration by Inspection
$$\int f(g(x)) = \frac{\int f(x)}{g'(x)} \space du$$
Treat the function $g(x)$ as an $x$, integrate, and then divide by the differential of $g(x)$.
e.g. $\int (2x+3)^3$
For a definite integral, when you change the function (e.g. change $dx$ to $du$), you must change the limits as well. The new limits can be found by plugging the old limits into the equation you've substituted in (e.g. $u = x + 1$).
Integration by Parts
$$\int u \frac{dv}{dx} \space dx = uv - \int v \frac{du}{dx} \space dx$$
e.g. $\int x \sin x \space dx$
$u = x \frac{dv}{dx} = \sin x$
$\frac{du}{dx} = 1 v = -\cos x$
$\int x \sin x \space dx = x(-\cos x) - \int (1)(-\cos x) \space dx = -x \cos x + \int \cos x \space dx$
$\int x \sin x \space dx = -x \cos x + \sin x + c$
Binomial expansion is only valid for $|\frac{bx}{a}| \lt 1$.
Sigma Notation
$$\sum_{n=b}^a f(n)$$
where a is the highest term and b is the lowest term of the function $f(n)$.
This means the sum of all terms of f(n) between b and a.
e.g.$$\sum_{n=1}^3 n^2 = 1 + 4 + 9 = 14$$
Series Notation
The sum of a sequence is called a series, denoted $S_n$ for the sum of the first n terms. $$S_n = \sum_{i=0}^n U_i$$
Arithmetic Progression
Arithmetic progressions have a common difference, denoted by d. $U_n = U_1 + (n-1)d$
An arithmetic series can be found using:
$$S_n = \frac{n}{2}(2U_1 + (n-1)d)$$
$$S_n = \frac{n}{2}(U_1 + U_n)$$
Geometric Progression
Geometric progressions have a common ratio, denoted by r. $U_n = U_1 r^{n-1}$
A geometric series can be found using: $$S_n = \frac{U_1(r^n - 1)}{r-1}$$
Sum to Infinity
For a geometric series, if $|r| \lt 1$, as $n$ gets larger, $r^n$ gets closer to 0. As $n \rightarrow \infty$, $r^n \rightarrow 0 \therefore S_n \rightarrow \frac{a}{1-r}$. This can be written as $S_\infty = \frac{a}{1-r}$.
Functions
For all of the graphs below, $f(x) = x^2$, and $c = 5$. The blue line is $f(x)$ and the orange line is the changed equation.
$y = f(x + c)$:
Graph moves to the left by c.
Translate graph by $\begin{pmatrix}-c\\0\end{pmatrix}$
$y = f(cx)$:
Graph has been stretched by a factor of $\frac{1}{c}$ in the x-direction.
Note: $f(-x)$ is a reflection in the y-axis.
$y = f(x) + c$:
Graph moves up by c.
Translate graph by $\begin{pmatrix}0\\c\end{pmatrix}$
$y = cf(x)$:
Graph is stretched by a factor of c in the y-direction.
Note: $-f(x)$ is a reflection in the x-axis.
If two translations are applied in the same direction, if they're both outside the function, use BIDMAS for the order; if they're if both inside the function, use BIDMAS backwards.
Inverse Functions
To get the inverse function:
Write $y = f(x)$
Swap $x$ and $y$
Rearrange to get $y$
Substitute $y = f^{-1}(x)$
Any function and its inverse are symmetrical about the line $y=x$.
A population is the whole set of items that are of interest
A sample is some subset of the population intended to represent the population
Each individual thing in the population that can be sampled is known as the sampling unit
Often sampling units of a population are individually named or numbered to form a list called the sampling frame
Data collected from the entire population is known as a census
A parameter is a characteristic of a whole population such as the mean or variance, sometimes unknown
A statistic is a value based on a sample, and will most likely not have the same value as the parameter it is estimating
Samples or Census
Sampling reduces accuracy, but it is often impossible to analyse the whole population. You must avoid bias in the sample so it is as representative of the population as possible.
A census gives a completely accurate result, but is time consuming and expensive. It could also require destroying the product (e.g. counting the number of beans in a tin). It means there is a large volume of data to process.
A sample is cheaper, quicker and means there is less data to process. However, the data may not be accurate and it may not be large enough to represent small sub groups.
The larger the sample size the more accurate your results are likely to be.
Types of Sampling
Simple Random Sampling
Each item in the sampling frame has an identifying number. A random number generator picks random numbers, and the corresponding sampling unit is used.
This is bias free and easy and cheap. However, it is not suitable when you have a large population as you need the whole sampling frame.
Systematic Sampling
The sampling frame is ordered, and then sampling units are chosen at regular intervals. e.g. Take every $k^{th}$ element where $k = \frac{\text{Population size (N)}}{\text{Sample size (n)}}$ starting at a random item between 1 and $k$.
This is simple and quick to use, and suitable for large samples. However, you need the whole sampling frame and it can introduce bias if sampling frame is not ordered randomly.
Stratified Sampling
Population is divided into groups (strata) and a simple random sample is done on each. The same proportion - $\frac{\text{n}}{\text{N}}$ - is sampled from each strata.
This is used when the sample is large and the population naturally divides into groups (e.g. a school).
This represents population structure and guarantees proportional representation of groups within the population. However the population must be clearly classified into distinguishable strata, which some populations won't do. Selection within each stratum suffers from the same disadvantages as simple random sampling.
Cluster Sampling
Population is divided into strata, and then either simple random sampling or systematic sampling is used to select which groups are used. A simple random sample or systematic sample is then done on the selected groups.
This is simpler and cheaper than stratified sampling as not all of the groups are used. Has same disadvatanges as stratified sampling, plus sampling units within a cluster/strata are more likely to have similar characteristics, introducing bias.
Quota Sampling
This is not random so it can be used without a sampling frame.
The population is divided into groups according to characteristic. A quota of items/people in each group is set to try and reflect the group's proportion in the whole population. The interviewer then selects sampling units until the quota for each group is filled.
This allows a small sample to still be representative of population, needs no sampling frame and is quick, easy and inexpensive. It also allows for easy comparision between different groups in the population.
However, non-random sampling can introduce bias. Population must be divided into groups, which can be costly or inaccurate. Increasing scope of study increases number of groups, adding time and expense. Also non-responses are not recorded.
Opportunity Sampling
A variant of quota sampling is opportunity sampling, where we find people who meet the critera at the same time the survery is being carried out rather than selecting a quota in advance. A common example is asking people as they walk past in supermarkets.
This is easy to carry out and inexpensive, but unlikely to provide a representative sample as people in a specific area are likely to have certain characteristics. It is also highly dependent on individual researcher.
Data Presentation and Interpretation
Measures of Location and Spread
In stats, $x$ represents the value of multiple objects. $\sum x$ is the sum of all values represented by $x$. $\bar{x}$ is the mean of $x$.
Measures of location are single values which describe a position in a data set.
Measures of central tendency are measures of location concerned with the centre of the data.
Measures of spread are to do with how data is spread out.
Mean $\bar{x} = \frac{\sum x}{n}$ where $\sum x$ is the sum of all the observations of the sample and $n$ is the number of observations in the sample$\bar{x}$ represents the sample mean, and $\mu$ represents the population mean. $\bar{x}$ is an estimator for $\mu$Frequency Tables The mean of a sample with given frequencies is $\bar{x} = \frac{\sum (f x)}{\sum f}$Median
To get the median, put the numbers in numerical order, and get the $\frac{n-1}{2}^{th}$ position where $n$ is the amount of numbers. If this is halfway between two numbers then get the mean of the two positions.
Measures of spread
Interquartile range and range are both examples of measures of spread. We often prefer the IQR as it eliminates outliers.
Variance
Variance is a measure of spread that takes all values into account. By definition it is the average squared distance from the mean. $$\sigma^2 = \frac{\sum (x - \bar{x})^2}{n}$$
$$\sigma^2 = \frac{\sum x^2}{n} - \bar{x}^2$$
This can be remembered as the mean of the squares minus the square of the mean.
To get the best estimate of the population variance you need to divide by $n-1$ instead of $n$ in the variance formula.
Tip: If the question says "Find the variance/standard deviation of these values" then divide by $n$. If the question says "Find an estimate of population variance from this sample" then divide by $n-1$. Otherwise, either will get the marks.
Standard Deviation
S.D. ($\sigma$) is the square root of variance.
Representing Data
Box Plots
Skewness
A distribution is positively skewed if Q$_3$ - Q$_2$ $\gt$ Q$_2$ - Q$_1$.
A distribution is negatively skewed if Q$_3$ - Q$_2$ $\lt$ Q$_2$ - Q$_1$
Scatter Diagrams
A linear correlation is when you plot measurements on a scatter diagram and the points lie on or near a straight line. Correlation does not imply causation
Bar Charts
These are for discrete data, and are plotted against frequency.
Histograms
These are used for continuous data, and are plotted against frequency density. Frequency is represented by the area of the bar. Frequency = class width $\times$ frequency density Outliers
Outliers are values which don't fit into the trend of the other data.
A common definition is values which are 1.5 IQRs or more beyond either $Q_1$ or $Q_3$. A different definition may be given in an exam.
Cumulative Frequency
Cumulative Frequency is the total frequency up to a certain number.
On a cumulative frequency graph, points should be plotted at the highest end of the range, and the lowest end of the first range is plotted at 0. When estimating from a cumulative frequency graph, you can use $\frac{n}{4}$ and $\frac{3n}{4}$ for $Q_1$ and $Q_3$, and $\frac{n}{2}$ for the median.
This can be used to find the number of values greater than or smaller than a value. For example, to get the number of times less than 1.15s in the graph above, go up from 10.15 on the x-axis until you hit the line, then go across until you hit the y-axis, giving a result of 26.
Correlation
Product Moment Correlation Coefficient (PMCC)
PMCC is a measure of correlation of data and is denoted $r$.
$-1 \le r \le 1$
Probability
An experiment is a repeatable process which gives rise to a number of outcomes.
An event is a set of one or more of these outcomes.
A sample space is the set of all possible outcomes.
Set Notation
A and B
The intersection of A and B
A ∩ B
A or B
The union of A and B
A ∪ B
Not A
The complement of A
A$'$
Probability Equations$$P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$$Conditional Probability is defined by $P(B|A) = \frac{P(A ∩ B)}{P(A)}$Events are Mutually Exclusive if $P(A ∩ B) = 0$, $P(A ∩ B') = P(A)$ or $P(A ∪ B) = P(A) + P(B)$Events are Independent if $P(A ∩ B) = P(A) \times P(B)$
Distribution
$\bar{x} = $sample mean, $\mu = $population mean $s = $sample SD, $\sigma = $population SD
A random variable is a quantity which can vary for different members of a population. The sum of the probabilities of each outcome occurring for a random variable is 1
Random variables can be discrete, e.g. rolling a dice, or continuous, e.g. measuring height.
A random variable is usually represented by $X$, and a specific outcome which $X$ can take is represented by $x$. So $P(X = x)$ means the probability that the random variable has the outcome $x$.
Probability Distribution Function
Also known as a probability mass function, this maps outcomes to probabilities for a discrete random variable. It can either be a table or a rule.
To be a probability distribution, all probabilities for outcomes must be between 0 and 1, and the sum of the probabilities is 1.
Binomial Distribution
The conditions for a binomial distribution are FICT:
A fixed number $n$ trials
The trials are independent
A constant probability $p$ of success in each trial
There are two outcomes in each trial, 'success' and 'failure'
For $X \sim B(n, \space p)$,
where $n$ = number of trials and $p$ = probability of success,
$$P(X = x) = {n\choose p} p^x (1-p)^{n-x}$$
$P(X = x)$ can also be found with the Binomial PD function on a calculator.
To find $P(X \le x)$ use the Binomial CD on a calculator. To get into the form of $P(X \le x)$:
$P(X \lt x) = P(X \le x-1)$
$P(X \ge x) = 1 - P(X \le x-1)$
$P(X \gt x) = 1 - P(X \le x)$
Normal Distribution
$X \sim N(\mu, \space \sigma^2)$,
where $\mu$ = mean and $\sigma^2$ = variance
Variable must be continuous.
Total area under the graph is 1.
$P(x \lt X \lt y) =$ area under the curve between $x$ and $y$. $P(X = x) = 0 \space \therefore \space \le = \lt$
On the calculator, Normal CD for $P(x \lt X \lt y)$. If a bound is undefined, use (-)9999999.
Inverse Normal works out $x$ when you have $P(X \lt x) = Area$.
Standard Normal (Z) Distribution: $Z \sim N(0, 1)$$z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}$
Points of inflection occur at $x = \mu \pm \sigma$
Hypothesis Testing
A hypothesis is a statement made about the value of a population parameter.
The null hypothesis, $H_0$, is that nothing has changed from the original value/measurement.
The alternative hypothesis, $H_1$, is that there has been some change in the population parameter.
Binomial Distribution
If the distribution of a population is $X \sim N(\mu, \sigma^2)$ then the distribution of the mean of a random sample of $n$ of those things will be $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$
The p-value, or calculated probability, is the probability of finding the observed results when $H_0$ of a study question is true. It's calculated using the cumulative normal.
If the p-value is less than the significance level of the test then there is significant evidence to reject the null hypothesis.
If the p-value is greater than the significance level of the test then there isn't significant evidence to reject the null hypothesis.
The critical value is the cut off value to be within the significance level. It's calculated using the inverse normal.
The critical region is the area in which the null hypothesis will be rejected. The acceptance region is where the null hypotheses will be accepted.