# 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 ${\displaystyle x}$.
• 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 ${\displaystyle f(x)=\pm {\sqrt {x}}}$

### Domain and range of a function

In general:

• ${\displaystyle f(x)}$ is called the image of ${\displaystyle x}$.
• The set of permitted ${\displaystyle x}$ 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 ${\displaystyle x}$, written ${\displaystyle |x|}$, is defined as

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

## Differentiation

### Chain rule

The chain rule states that:

If ${\displaystyle y}$ is a function of ${\displaystyle u}$, and ${\displaystyle u}$ is a function of ${\displaystyle x}$,

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

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 ${\displaystyle f(g(x))}$. The process is then to assign a variable to the inner function, usually ${\displaystyle u}$, and use the rule above;

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

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

Let ${\displaystyle u=x-1}$

Now we can rewrite the original function, ${\displaystyle y=2u^{3}}$

We can now differentiate each part;

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

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

### Product rule

The product rule states that:

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

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

An alternative way of writing the product rule is:

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

Or in Lagrange notation:

If ${\displaystyle k(x)=f(x)g(x)}$,

then ${\displaystyle k'(x)=f'(x)g(x)+f(x)g'(x)}$

### Quotient rule

The quotient rule states that:

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

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

An alternative way of writing the quotient rule is:

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

### x as a function of y

In general,

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

## Trigonometric functions

### The functions cosec θ, sec θ and cot θ

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

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

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

### Standard trigonometric identities

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

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

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

### Differentiation of sin x, cos x and tan x

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

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

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

### Integration of sin(kx) and cos(kx)

In general,

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

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

## Exponentials and logarithms

### Differentiating exponentials and logarithms

In general,

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

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

### Natural logarithms

If ${\displaystyle y=\ln {x}}$, then

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

It follows from this result that

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

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

## Integration

### Integration by parts

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

### Standard integrals

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

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

### Volumes of revolution

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

${\displaystyle V=\pi \int _{a}^{b}y^{2}\ dx}$

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

${\displaystyle V=\pi \int _{a}^{b}x^{2}\ dy}$

## 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

${\displaystyle \int _{a}^{b}y\ dx\approx h\lbrack y_{\frac {1}{2}}+y_{\frac {3}{2}}+\ldots +y_{n-{\frac {3}{2}}}+y_{n-{\frac {1}{2}}}\rbrack }$
${\displaystyle {\mbox{where}}\ h={\frac {b-a}{n}}}$

Simpson's rule

${\displaystyle \int _{a}^{b}y\ dx\approx {\frac {h}{3}}\lbrack \left(y_{0}+y_{n}\right)+4\left(y_{1}+y_{3}\ldots +y_{n-1}\right)+2\left(y_{2}+y_{4}+\ldots +y_{n-2}\right)\rbrack }$
${\displaystyle {\mbox{where}}\ h={\frac {b-a}{n}}\ {\mbox{and}}\ n\ {\mbox{is even}}}$