s56 (5.3) recurrence relations.notebook september 09,...
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![Page 1: S56 (5.3) Recurrence Relations.notebook September 09, 2015missdeely.weebly.com/uploads/2/1/3/0/21307444/rr_notes_higher.pdf · Linear Recurrence Relations Example: un + 1 = 1.5u n](https://reader030.vdocuments.us/reader030/viewer/2022041117/5f2c721ef16c755e6a07e808/html5/thumbnails/1.jpg)
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
![Page 2: S56 (5.3) Recurrence Relations.notebook September 09, 2015missdeely.weebly.com/uploads/2/1/3/0/21307444/rr_notes_higher.pdf · Linear Recurrence Relations Example: un + 1 = 1.5u n](https://reader030.vdocuments.us/reader030/viewer/2022041117/5f2c721ef16c755e6a07e808/html5/thumbnails/2.jpg)
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
![Page 3: S56 (5.3) Recurrence Relations.notebook September 09, 2015missdeely.weebly.com/uploads/2/1/3/0/21307444/rr_notes_higher.pdf · Linear Recurrence Relations Example: un + 1 = 1.5u n](https://reader030.vdocuments.us/reader030/viewer/2022041117/5f2c721ef16c755e6a07e808/html5/thumbnails/3.jpg)
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.
![Page 4: S56 (5.3) Recurrence Relations.notebook September 09, 2015missdeely.weebly.com/uploads/2/1/3/0/21307444/rr_notes_higher.pdf · Linear Recurrence Relations Example: un + 1 = 1.5u n](https://reader030.vdocuments.us/reader030/viewer/2022041117/5f2c721ef16c755e6a07e808/html5/thumbnails/4.jpg)
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.
![Page 5: S56 (5.3) Recurrence Relations.notebook September 09, 2015missdeely.weebly.com/uploads/2/1/3/0/21307444/rr_notes_higher.pdf · Linear Recurrence Relations Example: un + 1 = 1.5u n](https://reader030.vdocuments.us/reader030/viewer/2022041117/5f2c721ef16c755e6a07e808/html5/thumbnails/5.jpg)
S56 (5.3) Recurrence Relations.notebook September 09, 2015
![Page 6: S56 (5.3) Recurrence Relations.notebook September 09, 2015missdeely.weebly.com/uploads/2/1/3/0/21307444/rr_notes_higher.pdf · Linear Recurrence Relations Example: un + 1 = 1.5u n](https://reader030.vdocuments.us/reader030/viewer/2022041117/5f2c721ef16c755e6a07e808/html5/thumbnails/6.jpg)
S56 (5.3) Recurrence Relations.notebook September 09, 2015
![Page 7: S56 (5.3) Recurrence Relations.notebook September 09, 2015missdeely.weebly.com/uploads/2/1/3/0/21307444/rr_notes_higher.pdf · Linear Recurrence Relations Example: un + 1 = 1.5u n](https://reader030.vdocuments.us/reader030/viewer/2022041117/5f2c721ef16c755e6a07e808/html5/thumbnails/7.jpg)
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