A-level Mathematics (MEI)/C3

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[edit] 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

[edit] Differentiation

[edit] Standard Derivatives

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

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

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

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

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

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

[edit] 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:

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

[edit] 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.

[edit] Quotient Rule

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

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:

[edit] Implicit Differentiation

Implicit differentiation is used when a function is not a simple $y = s o m e t h i n g$ but contains a mixture of x and y parts. A typical example of this is to differentiate:

$y 2 + 2 y = 4 x 3$

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:

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

[edit] Integration

[edit] Integration by Substitution

[edit] 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:

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

[edit] Functions

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

[edit] Mappings

There are 4 different types of mapping. These are:

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

A-level Mathematics (MEI)/C3

From Wikibooks, the open-content textbooks collection

Contents

[edit] Introduction

[edit] Differentiation

[edit] Standard Derivatives

[edit] Chain Rule

[edit] Product Rule

[edit] Quotient Rule

[edit] Implicit Differentiation

[edit] Integration

[edit] Integration by Substitution

[edit] Integration by Parts

[edit] Functions

[edit] Mappings

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