1.1Functions and mappings

Transformation of graphs

Example 1

Example 2Combined Transformations

1.2Composite functions

1.3Inverse functions

The inverse trig functions

The inverse trig functions only exist when we restrict the domain of the original functions. So that they become 1:1 mappings

1.4The modulus function and inequalities

Curve Sketching

2.1Definitions, equations & graphs

Laws of Logarithms

Definitions & Graphs

2.2Modelling with exponential functions

3.1The Reciprocal Functions

3.2The Compound & Double Angle Formulae

Double Angle Formula

3.3The forms r cos(x +/- y), r sin(x +/- y)

4.1The Chain Rule

4.2The Product Rules

4.3The Quotient Rules

Inverse Functions

Exponentials & Logs

Further example

5.1Integration by substitution

Further example

Integration by inspection

Integration of Exponentials & Logarithms

5.2Volumes of revolution

6.1Numerical Methods

Change of Sign Methods

To find the root of a function f(x), if

f(a) > 0

f(b) < 0

and the function is continuous, then there is a root between a and b.

We can narrow this down by successively halving the size of the interval, until we get an answer to the required accuracy.

Example

We can continue this method indefinitely, to whatever accuracy we require

This method is useful for confirming the existence of a root between two given values. It can be unreliable however, sometimes failing to find a root(s) when they are very close together, or ‘finding’ a root that doesn’t exist.

Iterative Methods

(there are many other re-arrangements you could have done)

Then start at any value (it will be given in the question; here we use x=0, and show 3 iterations)

This method continues until it converges on the solution x = 0.276 (to 3 dp)

Iterative methods will often fail to find the root for given re-arrangements. If this is the case, you must try a different re-arrangement.

6.2Numerical Integration

In this example, yn = y8

1.1Transformations 1

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(5)

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1.2Composite Functions 1

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1.3Inverse Functions 1

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1.4Modulus function & inequalities 1

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2.1Log Equations & Graphs 1

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2.2Modelling 1

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3.1The Reciprocal Functions 1

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3.2Compound & Double-angle formulae 1

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3.3 The forms rcos(x+/-y), rsin(x+/-y) 1

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4.1The Chain Rule 1

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4.2The Product Rule 1

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4.3The Quotient Rule 1

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(3)

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5.1Integration by Substitution 1

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5.2 Volumes of revolution1

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6.1Numerical Methods 1

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6.2Numerical Integration 1

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1.1Transformations 1Solutions

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(5)

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1.2Composite Functions 1Solutions

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1.3Inverse Functions 1Solutions

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1.4Modulus function & inequalities 1 Solutions

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2.1Log Equations & Graphs 1Solutions

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2.2Modelling 1 Solutions

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3.1The Reciprocal Functions 1Solutions

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3.2Compound & Double-angle formulae 1Sols

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3.3 The forms rcos(x+/-y), rsin(x+/-y) 1Solutions

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4.1The Chain Rule 1Solutions

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4.2The Product Rule 1 Solutions

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4.3The Quotient Rule 1 Solutions

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(3)

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5.1Integration by Substitution 1Solutions

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=3.5 + 4ln2

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5.2 Volumes of revolution1 Solutions

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6.1Numerical Methods 1Solutions

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6.2Numerical Integration 1Solutions

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