Habergham Sixth Form Centre

Further Pure

Mathematics

Unit FP2

Partial Fractions

Complex Numbers

Geometry

Induction

Further Partial fractions

For every quadratic factor occurring in the denominator there will arise a partial fraction

ExampleExpress in partial fractions



Demoivre’s theorem

For all values of n, the value or one of the values in the case where n is fractional, of is

Let n be a positive integer A proof in product extensions using

A proof by induction might be required, as follows

Assume true for i.e

Then

Hence if true for it will be true for

Since it is true for ; hence it will be true for it will be true for and so on.

Let n be a negative integer equal to –m (m>0)

ExampleSimplify



Let n be a fraction equal to p/q where p and q are integers (q>0)

does not have a unique value, although one value is .

In general, application will be finding roots of complex numbers. It can be shown that (q a positive integer) has q different values. They are given by

where

ExampleFind the three cube roots of –i in the form , and display the points representing them on an argand diagram



=, ,

= , i , ; represent by , ,

The points lie on the perimeter of a circle of unit radius.

They are the vertices of an equilateral triangle.

Further Integration

ExampleFind (i) (ii)

(iii)

Derivative of a definite integral with respect to a variable limit

Let

Taking a to be a constant and t to be variable

Let

Taking b to be a constant and t to be variable

Example

Functions of real numbers

(1) Open and Closed intervals

Let and be end points of a finite interval in 0x

The open interval consists of all points of the interval with the exception of the end points so that .

Using set notation we can write interval =

The closed interval consists of all points of the open interval plus the end points so that i.e. interval =

Half open intervals =

(2) Continuity

(i) A function is said to be continuous where if

(ii) A function is said to be continuous everywhere if it is continuous

for every value in the domain.

(iii) A function is said to be continuous in the open interval so

long as it is continuous for all .

(iv) For a closed interval continuity at the end points will be d

determined by one sided limits.

i.e. continuity at so long as

continuity at so long as

Example for

for

Despite the change in direction at , by definition f is a continuous function.

Example for

for

A discontinuity here at .

Example

An apparent discontinuity here at , but is not with in the domain of f.

Hence f is continuous

Example

A discontinuity here at

Examplefor

for

Given that and are continuous for all x, find the values of b and c.

 for

for

(3) Bounded functions

(i) Examples of bounded functions

(greatest) (least)

lower boundupper bound

lowerupper

Note here that the upper bound is never attained. However for any value of where is a positive number as small as we please there will exist corresponding values of x

The following functions are not bounded

(ii) If f is a continuous function in the closed interval then it will be bounded i.e. there will be numbers m and M such that

m and M are the lower and upper bounds of the function and are attained at least once in the interval .

Examples

(4) Odd and Even functions

(i)If then f is said to be an even function the graph of will have reflective symmetry in 0y

Example ,

f is an even function

(ii)If then f is said to be an odd function. The graph of will have rotational symmetry about 0.

Example,

f is an odd function

(5) (Monotonic) Increasing and decreasing functions

If whenever then f is said to be (monotonic) increasing

If whenever then f is said to be (monotonic) decreasing

If the inequality sign is omitted from the definition,

f is strictly increasing.

f is strictly decreasing.

 f is strictly increasing f is strictly decreasing

Example for

f is strictly increasing

Examplefor

f is strictly decreasing

(6) (i) Image of a set under a function

Let be a closed interval in the domain of f

f increasing

The image of A under f is given by

f decreasing

The image of A under f is given by

f has a turning point in the interval

There are many possiblities

In this example with a max at

(ii) Inverse image of a set under a function

Let be a closed interval which is the image of a set under f

f increasing

The inverse image of B under f is given by

Similarly for f decreasing

When f is not one to oneMany possibilities arise

In this example

ExampleThe diagram shows a

Sketch of where for

(a)Find the coordinates of the maximum point

(b)Find where A is the interval

(c)Find where B is the interval

(a)

for SP  Max at

(b) and

It can be seen from the

Sketch that

 

(7) Rational Functions These will be of the form where p and q are polynomials graphical treatment will involve finding

(a)vertical asymptotes (if any) by solving

(b)other asymptotes by letting (note that these will be oblique when the degree of p exceeds the degree of q)

(c)stationary points (using either algebraic or calculus methods).

Example ,

Show algebraically that can not take values between 1 and 9. Deduce the coordinates of the turning points on the graph of

Find the equations of the asymptotes to the graph and sketch the graph.

Let 

For real values of x, “discriminant  0”



Critical values

It follows that and i.e. can not lie between 1 and 9.

Min point at *

Max point at *

Vertical asymptotes

Other asymptote As

is a horizontal asymptote

A few points; and

Example(i) Express in the form

(ii) Find the equation of asymptotes on the graph of . Also find the coordinates of turning points and sketch the graph.

(i)

(ii)Using

As i.e. is an oblique asymptote

Vertical asymptote

Using differentiation; for t.p.s.

 Min at

Trigonometry

Factorisation formula,

(i)

in words *

(ii)

in words *

(iii)

in words *

(iv)

in words *

* Learn, in words!

ExampleShow that

LHS = RHS

Reversing the formulae to express products as sums and differences

There are four formulae which do the job.

In (i) you will see

and there are three more for , and . None of these are given in the booklet and you can make do without them anyway by reversing the factorisation formulae.

Example

ExampleFind in exact form.

‘t’ formulaewhere

(iii) this is not!

Proof for (iii)

Let 

There are several ways of proving the other two.

E.g. “double” angle formula

These can be adapted for other half angles

Eg where

ExampleShow that

ExampleUse ‘t’ formulae to solve the equation for





 

Solution set

General Solutions of trig equations [Formulae to be learnt]

In each case  is the principle value for the equation, and , etc..

NB, ,

General solution or

In each case  is the principle value corresponding to the positive square root.

Special cases.

ExampleFind the general solutions of the equations

(i) (ii)

(iii) where is in radians.

(i)



(ii) 

p.v. for is

General solution set given by



(iii)Here, use

General solution set



Check the answers to the last example in the first section by substituting etc.

Geometry

The parabola

(i) Defined as the locus of a point which moves such that its distance from a fixed point (the focus) is always equal to its distance from a fixed line (the directrix).

Denoting the distance of the focus from the directrix by 2a, taking the origin 0 to be midway between them and axes 0x, 0y perpendicular to a parallel to the directrix as shown in the diagram, we have:

Recall vertex (0,0) and the axis (of symmetry) ox from previous work (P1)

(ii) Other standard parabolae will be

(iii) Translated form

Example Show that the equation represents a parabola. Find the vertex, focus and directrix and sketch a graph of the parabola.



By completing the square.

Now C.F. with

Vertex i.e. i.e.

Focus i.e. i.e.

Directrix i.e.

(iv) Tangent to at



gradient at =

Equation

Now

This can be learnt and quoted, but you might be asked to obtain it (as above)

It can also be adapted for tangents to translated forms, but it isn’t worth the hassle. Use first principles!

(v) Parametrics for

Equations

Coordinates

Tangent and normal at to

Starting with



Tangent

Normal

(vi) Parametrics and translated form

ExampleShow that the locus of is a parabola finding its vertex, focus and directrix and sketch its graph.



C.F.

Vertex

Focus

Directrix

(vii) Gradient form of tangent to

Let the gradient be m, put into the equation of the tangent in (v) to give and point of contact

The Ellipse

(i) Defined as the locus of a point which moves such that the ratio of its distance from a fixed point (the focus) to that from a fixed line (directrix) is a constant e (eccentricity) which is in the range

The ellipse generated by focus S and directrix is identical to that generated by focus S’ and directrix . So we talk of two foci and two directrices.

For a>bAA’ is called the major axis; length 2a

BB’ is called the minor axis; length 2b

e is the eccentricity where

EquationFor SP = ePN

etc.. reducing to

This Proof is required.

ExampleAn ellipse has equation . Find the eccentricity, foci and directrices.

 

Also

Foci are at i.e. at

Directrices are i.e.

(ii) Translated form

ExampleAn ellipse has centre (1,2), major axis of length 4 parallel to 0y and minor axis of length 3 parallel to 0x. To find the equation, eccentricity, foci and directrices.

Bearing in mind that the major axis is in the direction parallel to 0y..

For e

 , ,

 Foci at Directrices at

(iii) Tangent to at



 at gradient =

Equation



 but the RHS = 1



-as for parabola, could be learnt.

(iv) Parametrics for

Equations coordinates

The tangent at can be found by substituting , in the result of (iii) giving

Starting with the parametric equations

 etc

For the normal, gradient =

You might obtain the equation in the form

(v) Parametrics and translated form

ExampleShow that the locus of is an ellipse.

This is an ellipse centre (–1 , 2) with semi major axis of length 2 and semi minor axis of length 1.

(vi) Chords and diameters(similar to circle)

The Hyperbola

(i) Defined as the locus of a point which moves such that the ratio of its distance from a fixed point (focus) to that from a fixed line (directrix) is a constant e (eccentricity) which is in the range

Note the oblique asymptotes , vertices foci directrices and

Equation again, proof could be required

SP = ePN

etc…

(ii) Briefly, through similarities with the ellipse

Tangent at

Parametrics

(iii) The rectangular hyperbola is one in which the asymptotes are perpendicular. That being the case the gradient of each asymptote will be equal to one and hence a = b

Equation

Eccentricity

The rectangular hyperbola is usually dealt with by using the asymptotes as axes.

The transformation leads to

Tangent and normal

,

Tangent at

Normal

Another standard rectangular hyperbola would be

(iv) Translated rectangular hyperbola

Any rational linear function graphs to rectangular hyperbolae

ExampleA curve has equation . Show that the curve is a rectangular hyperbola finding its centre and asymptotes; sketch the graph.

First use long division



Cfcentre at

 centre at

Horizontal asymptote

Vertical asymptote

Check point

Parametric equations for the curve would be give by

i.e.

For this curve the tangent or normal at say point (2, 5) would be found by using:

etc..

Given parametrics, use parametric differentiation.

In the knowledge that is a rectangular hyperbola the asymptotes can be found as in the following

Example

Vertical asymptotedenominator = 0i.e.

Horizontal asymptote let

 is horizontal asymptote

One point

By long division

Notes