A-level Mathematics/AQA/MPC3

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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}

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