A-level Mathematics (MEI)/C3

Introduction

This page covers the methods and techniques introduced during the C3 module. This assumes prior knowledge of C1 and C2 of the MEI Syllabus

Differentiation

Standard Derivatives

For the C3 module, there are a few standard results for differentiation that need to be learnt. These are:

$\frac{d}{dx}\ln x=\frac{1}{x}$


$\frac{d}{dx}{e}^{kx}=k{e}^x$


$\frac{d}{dx}\sin kx=k\cos kx$


$\frac{d}{dx}\cos kx=-k\sin kx$


$\frac{d}{dx}\tan kx=\frac{k}{co{s}^{2}kx}$

Chain Rule

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

The Chain Rule is used to differentiate when one function is applied to another function. A typical example of this is:

$y=\sin (x^2)$

One of the ways of remembering the chain rule is: Find the derivative outside, then multiply it by the derivative inside. In the example above, this becomes:

$\frac{dy}{dx}=2x\cos (x^2)$

Product Rule

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

The product rule is used when two functions are multiplied together.

Quotient Rule

$\frac{d}{dx}\frac{u}{v}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$

The quotient rule is used when one function is divided by another. It is a specific case of the product rule. A typical example of this is:

Implicit Differentiation

Implicit differentiation is used when a function is not a simple $y=something$ but contains a mixture of x and y parts. A typical example of this is to differentiate:

$y^2+2y=4x^3$

When differentiating the y components of the expression you differentiate as normal, and then multiply by $\frac{dy}{dx}$. So differentiating both sides of the above expression it becomes:

$2y\frac{dy}{dx}+2\frac{dy}{dx}=12x^2$

The by factorising the left hand side and cancelling, this becomes:

$\frac{dy}{dx}=\frac{6x^2}{y+1}$

Integration

Integration by Parts

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

Integration by parts is used when you have two functions multiplied together, such as ln x and a simple polynomial, where 1 function is not the derivative of the other. As an example:

$\int x\ln x$

In this expression use the substitutions: $u=\ln x$ and $\frac{dv}{dx}=x$. In almost all other expressions, the polynomial is taken as u. After substituting, the expression in the example becomes:

$\int x\ln x=\ln x\int x-\int x\frac{d}{dx}\ln x$

After integrating and differentiating the respective parts of the expression, this becomes:

$\int x\ln x=\frac{1}{2}{x}^2\ln x-x.$

Functions

All functions can be talked about in terms of their domain (x axis) and co-domain or range (y-axis).

Mappings

There are 4 different types of mapping. These are:

Many to Many
One to Many
Many to One
One to One