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

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1 Daily Practice 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). Today we are going to learn about recurrence relations. Homework Due tomorrow! Q3. Show that the roots of 2x(x 1) + 1 = 6x 7 are equal and find x. Recurrence relations are sequences in which each term is a function of the previous terms, where the terms are labelled u 0, u 1, u 2... They are very useful for calculating long term patterns. So we can say in general terms u n+ 1 = au n where a is 1 + interest rate as a decimal For example: A house worth increases in value by 5% per annum. What is it's value each year over 3 years 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

2 Daily Practice Q1. Line l1 has equation 2y - x = 0. (a) Line l2 is perpendicular to l1. Find the gradient of l2 A car designer has calculated that water escapes from an engine cooling system If 2 litres is added each month, (b) Calculate the angle l2 makes with the positive direction of the x - axis (b) Calculate the volume of water in the engine after 3 months Ex. 5C Pg 72,73 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 Q1. State the nature of the roots of the quadratic function 6x x - 5 Q2. Express 2x x + 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

3 Linear Example: u n + 1 = 1.5u n + 4, (i) Calculate the value of u 3 when u 0 = 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 (Limits) Linear (Limits) The limit of a recurrence relation: If -1 < a < 1 then u n tends to a limit. The limit is L = b 1 - a Example: Find the first three terms and the limit of the sequence u n + 1 = 0.25u n + 7 where u 0 = -2 as n -> Page 78 Q1 b, d, e, g, i Daily Practice 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, Today we will be continuing to learn about the limits of recurrence relations. Homework Due Tuesday. passing through A and B with centre C

4 Linear (Limits) Example 2: Daily Practice 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 x 2 + y 2 + 4x + 2y 20 = 0. Solving to find a and b Example: Today we will be learning how to solve recurrence relations for a and b. Homework due Tuesday Pg. 79 Q1 a, c, g, j Q3, 4 Daily Practice 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 x 2 + y 2-2x - 15 = 0 and Today we will be working out questions on linked recurrence relations & practising mixed questions. x 2 + y 2-14x - 16y + 77 = 0 touch externally

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7 Daily Practice 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 u n + 1 = 0.4u n + 16 and u 0 = 8, find the values of u 1 and u 2 Q4. State the centre and radius of the circle x 2 + y 2 + 2x - 6y= 18