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
Recurrence Relations
So we can say in general terms un+ 1 = aun where a is 1 + interest rate as a decimal
Recurrence Relations
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
Recurrence Relations
S56 (5.3) Recurrence Relations.notebook September 09, 2015
Recurrence Relations
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
Daily Practice 1.9.2015
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!
Daily Practice 2.9.2015
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
Q3.
Today we will be continuing to learn about
recurrence relations and their limits.
Homework Online due 8.9.15
S56 (5.3) Recurrence Relations.notebook September 09, 2015
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)
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 -> ∞
1 - a
un + 1 = 0.25un + 7 where u0 = -2
Page 78 Q1 b, d,
e, g, i
Daily Practice 3.9.2015
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.
S56 (5.3) Recurrence Relations.notebook September 09, 2015
Linear Recurrence Relations (Limits)Example 2:
Daily Practice 4.9.2015
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
Daily Practice 7.9.2015
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.
S56 (5.3) Recurrence Relations.notebook September 09, 2015
S56 (5.3) Recurrence Relations.notebook September 09, 2015
S56 (5.3) Recurrence Relations.notebook September 09, 2015
Daily Practice 8.9.15
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