A-level Mathematics/AQA/MPC3

Functions

Mappings and functions

We think of a function as an operation that takes one number and transforms it into another number. A mapping is a more general type of function. It is simply a way to relate a number in one set, to a number in another set. Let us look at three different types of mappings:

• one-to-one - this mapping gives one unique output for each input.
• many-to-one - this type of mapping will produce the same output for more than one value of $x$.
• one-to-many - this mapping produces more than one output for each input.

Only the first two of these mappings are functions. An example of a mapping which is not a function is $f(x)=\pm \sqrt{x}$

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

$|x|=\{\begin{array}{cc}x& \text{for }x\ge 0\\ -x& \text{for }x<0\end{array}$

Differentiation

Chain rule

The chain rule states that:

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

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

As you can see from above, the first step is to notice that we have a function that we can break down into two, each of which we know how to differentiate. Also, the function is of the form $f(g(x))$. The process is then to assign a variable to the inner function, usually $u$, and use the rule above;

Differentiate $y=2(x-1{)}^3$

We can see that this is of the correct form, and we know how to differentiate each bit.

Let $u=x-1$

Now we can rewrite the original function, $y=2u^3$

We can now differentiate each part;

$\frac{dy}{du}=6u^2$ and $\frac{du}{dx}=1$

Now applying the rule above; $\frac{dy}{dx}=\frac{dy}{du}*\frac{du}{dx}=6u^2*1=6u^2=6(x-1{)}^2$

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:

$(uv{)}^{\prime }=u{v}^{\prime }+u^{\prime }v$

Or in Lagrange notation:

If $k(x)=f(x)g(x)$,

then $k^{\prime }(x)=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$

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:

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

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

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

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

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

Standard trigonometric identities

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

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

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

Differentiation of sin x, cos x and tan x

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

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

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

Integration of sin(kx) and cos(kx)

In general,

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

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

Exponentials and logarithms

Differentiating exponentials and logarithms

In general,

$\text{when}y=e^{kx},\frac{dy}{dx}=k{e}^{kx}$

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

Natural logarithms

If $y=\ln x$, then

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

It follows from this result that

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

$\int \frac{f^{\prime }(x)}{f(x)}dx=\ln f(x)+c,\text{provided}f(x)>0$

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:

$V=\pi \underset{a}{\overset{b}{\int }}{y}^{2}dx$

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:

$V=\pi \underset{a}{\overset{b}{\int }}{x}^{2}dy$

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

$\underset{a}{\overset{b}{\int }}ydx\approx h[y_{\frac{1}{2}}+y_{\frac{3}{2}}+\dots +y_{n-\frac{3}{2}}+y_{n-\frac{1}{2}}]$

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

Simpson's rule

$\underset{a}{\overset{b}{\int }}ydx\approx \frac{h}{3}[(y_0+y_n)+4(y_1+y_3\dots +y_{n-1})+2(y_2+y_4+\dots +y_{n-2})]$

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