S56 (5.3) Recurrence Relations.notebook September 09,...

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S56 (5.3) Recurrence Relations.notebook September 09, 2015

Daily Practice 31.8.2015

Q1. Write down the equation of a circle with centre (-1, 4) and radius 5

Q2. Given the circle with equation (x – 4)2 + (y + 5)2 = 40.

Find the equation of the tangent to this circle at

the point P(2,1).

Q3. Show that the roots of 2x(x – 1) + 1 = 6x – 7 are equal and find x.

Today we are going to learn about recurrence

relations.

Homework Due tomorrow!

Recurrence Relations

Recurrence relations are sequences in which each term is a function of the previous terms, where the terms are labelled u0 , u1 , u2 ...

They are very useful for calculating long term patterns.

For example: A house worth £128 000 increases in value by 5% per annum . What is it's value each year over 3 years

So we can say in general terms un+ 1 = aun where a is 1 + interest rate as a decimal

Example:

Example 2:

A patient is injected with 75ml of medicine. Every 4 hours, 20% of the medicine passes out of his bloodstream. To compensate, a further 10ml dose is administered every 4 hours.

i) Write a recurrence relation for the amount of medicine in the bloodstream

ii) Calculate the amount of medicine remaining after 24 hours

A car designer has calculated that water escapes from an engine cooling system

If 2 litres is added each month,

(b) Calculate the volume of water in the engine after 3 months

Ex. 5C Pg 72,73

Q1. Line l1 has equation √2y - x = 0.

(a) Line l2 is perpendicular to l1. Find the gradient of l2

(b) Calculate the angle l2 makes with the positive direction of

the x - axis

Q2. (a)AB is a line parallel to the line with equation y + 3x = 25. A

has coordinates (-1, 10). Find the equation of AB.

(b) 3y = x + 11 is the perpendicular bisector of AB. Find the

coordinates of B

Today we will be continuing work on recurrence relations.

Homework due!

Q1. State the nature of the roots of the quadratic function 6x2 + 10x - 5

Q2. Express 2x2 + 12x + 1 in the form a(x + b)2 + c

Today we will be continuing to learn about

recurrence relations and their limits.

Homework Online due 8.9.15

Linear Recurrence Relations

Example:

un + 1 = 1.5un + 4,

(i) Calculate the value of u3 when u0 = 6

Limits

If a > 1 or a < -1 then the sequence will be divergent (increasing or decreasing forever).

If -1 < a < 1, then the sequence coverges towards a limit and is known as a convergent sequence.

Linear Recurrence Relations (Limits)

The limit of a recurrence relation:

If -1 < a < 1 then un tends to a limit. The limit is L = b

Example: Find the first three terms and the limit of the sequence

as n -> ∞

un + 1 = 0.25un + 7 where u0 = -2

Q1 b, d,

e, g, i

Q1. Points A(-1, -1) and B(7, 3) lie on the circumference of a

circle with centre C

(a) Find the equation of the perpendicular bisector of AB.

CB is parallel to the x - axis.

(b) Find the equation of the circle,

passing through A and B with centre C

Today we will be continuing to learn about the

limits of recurrence relations.

Homework Due Tuesday.

Linear Recurrence Relations (Limits)Example 2:

Q1. In triangle ABC, A is (-2,-3), B is (2,-2) and C is (-4,4).

(a) Find the equation of AD the altitude from A.

(b) Find the equation of AP, the median through BC

Q2. Find the points of intersection of the line y =2x + 8 and

the circle with equation x2+ y2 + 4x + 2y – 20 = 0.

Today we will be learning how to solve recurrence

relations for a and b.

Homework due Tuesday 8.9.15

Solving Recurrence Relations to find a and b

Example:

Pg. 79 Q1 a, c, g, j Q3, 4

Q1. State the centre and the radius of the circle

x² + y² - 6x - 18y = -62

Q2. Find the equation of the circle with centre (0, 0) that

passes through (3, 8)

Q3. Show that the circles x2 + y2 - 2x - 15 = 0 and

x2 + y2 - 14x - 16y + 77 = 0 touch externally

Today we will be working out questions on linked

recurrence relations & practising mixed questions.

Q1. State the gradient of the line parallel to 4x - 2y + 10 = 0

Q2. State the equation of the perpendicular bisector of A(3, 1) and B(5, -3)

Q3. Given un + 1 = 0.4un + 16 and u0 = 8, find the values of u1 and u2

Q4. State the centre and radius of the circle x2 + y2 + 2x - 6y= 18

S56 (5.3) Recurrence Relations.notebook September 09,...

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