A-level Mathematics/AQA/MPC3

Functions

Domain and range of a function

In general:

• f(x) is called the image of x.
• The set of permitted x values is called the domain of the function
• The set of all images is called the range of the function

Modulus function

The modulus of x, written | x | , is defined as

$|x| = \begin{cases} x & \mbox{for } x \ge 0 \\ -x & \mbox{for } x < 0 \end{cases}$

Differentiation

Chain rule

The chain rule states that:

If y is a function of u, and u is a function of x,

$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$

Product rule

The product rule states that:

If y = uv, where u and v are both functions of x, then

$\frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx}$

An alternative way of writing the product rule is:

$(uv)' = uv' + u'v \,\!$

Quotient rule

The quotient rule states that:

If $y = \frac{u}{v}$, where u and v are functions of x, then

$\frac{d}{dx} \left ( \frac{u}{v} \right ) = \frac{ v \frac{du}{dx} - u \frac{dv}{dx} }{v^2}$

An alternative way of writing the quotient rule is:

$\left ( \frac{u}{v} \right )' = \frac{u'v - uv'}{v^2}$

x as a function of y

In general,

$\frac{dy}{dx} = \frac{1}{ \frac{dx}{dy} }$

Trigonometric functions

The functions cosec θ, sec θ and cot θ

$\operatorname{cosec}{\theta} = \frac{1}{\sin{\theta}}$

$\sec{\theta} = \frac{1}{\cos{\theta}}$

$\cot{\theta} = \frac{1}{\tan{\theta}}$

Standard trigonometric identities

$\cot{\theta} = \frac{ \cos{ \theta } } {\sin{ \theta } }$

$\sec^2{\theta} = 1 + \tan^2{\theta} \,\!$

$\operatorname{cosec}^2{\theta} = 1 + \cot^2{\theta}$

Differentiation of sin x, cos x and tan x

$\frac{d}{dx} \left ( \sin{x} \right ) = \cos{x}$

$\frac{d}{dx} \left ( \cos{x} \right ) = -\sin{x}$

$\frac{d}{dx} \left ( \tan{x} \right ) = \sec^2{x}$

Integration of sin(kx) and cos(kx)

In general,

$\int \cos{kx}\ dx = \frac{1}{k} \sin{kx} + c$

$\int \sin{kx}\ dx = - \frac{1}{k} \cos{kx} + c$

Exponentials and logarithms

Differentiating exponentials and logarithms

In general,

$\mbox{when}\ y = e^{kx},\ \frac{dy}{dx} = ke^{kx}$

$\int e^{kx}\ dx = \frac{1}{k} e^{kx} + c$

Natural logarithms

If y = ln x, then

$\frac{dy}{dx} = \frac{1}{x}$

It follows from this result that

$\int \frac{1}{x}\ dx = \ln{x} + c$

$\int \frac{f'(x)}{f(x)}\ dx = \ln{f(x)} + c,\ \mbox{provided}\ f(x) > 0$

Integration

Integration by parts

$\int u \frac{dv}{dx}\ dx = uv - \int v \frac{du}{dx}\ dx$

Standard integrals

$\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1}{ \left ( \frac{x}{a} \right ) } + c$

$\int \frac{dx}{\sqrt{(a^2 - x^2)}} = \sin^{-1}{ \left ( \frac{x}{a} \right ) } + c$

Volumes of revolution

The volume of the solid formed when the area under the curve y = f(x), between x = a and x = b, is rotated through 360° about the x-axis is given by:

$V = \pi \int_a^b y^2\ dx$

The volume of the solid formed when the area under the curve y = f(x), between y = a and y = b, is rotated through 360° about the y-axis is given by:

$V = \pi \int_a^b x^2\ dy$

Numerical methods

Iterative methods

An iterative method is a process that is repeated to produce a sequence of approximations to the required solution.

Numerical integration

Mid ordinate rule

$\int_a^b y\ dx \approx h \lbrack y_{\frac{1}{2}} + y_{\frac{3}{2}} + \ldots + y_{n-\frac{3}{2}} + y_{n-\frac{1}{2}} \rbrack$

$\mbox{where}\ h = \frac{b-a}{n}$

Simpson's rule

$\int_a^b y\ dx \approx \frac{h}{3} \lbrack \left ( y_0 + y_n \right ) + 4\left ( y_1 + y_3 \ldots + y_{n-1} \right ) + 2\left ( y_2 + y_4 + \ldots + y_{n-2} \right ) \rbrack$

$\mbox{where}\ h = \frac{b-a}{n}\ \mbox{and}\ n\ \mbox{is even}$