Differentiation Application Task - Calculus Past Paper 2016 - PDF

12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK Mathematical Methods Unit 3 SAC 1 Functions and Calculus Technology Active...

12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK Mathematical Methods Unit 3 SAC 1 Functions and Calculus Technology Active Open Book Duration: 120 + 3 x 60 = 300 Minutes Marks: 41 + 3 x 22 = 110 Marks Teacher:____________________________ Class:_______________________ Name:______________________________________________________________ Wendy’s Wedding Question 1 (44 marks) Wendy and Peter decided that they wanted to get married. To highlight the importance of this event in their lives, they determined to hold the wedding ceremony at the local cathedral. On the day of the wedding, after exhausting encounters with a beautician and a hairdresser, Wendy was running late. Upon finally disembarking from the wedding car near the cathedral, Wendy was located at point A (see Diagram 1) and wanted to rendezvous with her father standing at the main cathedral door located at point B and anxiously waiting to give her away. In spite of the fact that the gardens around the cathedral were considerable, she wanted to meet up with her father in the least possible time so that the ceremony could start as promptly as possible. Wendy’s speed on the smooth concrete path, 𝑣" , was 2 ms-1 whilst her speed on the grass, 𝑣# , was 1 ms-1. Leaving A, Wendy headed for point X on BC which was x m away from B. Diagram 1 (AX = d, XB = x, XC = 200 – x) Page 1 of 15 page 1 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK Using the measurements given in Diagram 1, determine the following: (a) The distance, d m, travelled by Wendy on the grass in term of x. (2 marks) (b) The time taken to travel from A to X in seconds in terms of x. (2 marks) (c) The total time, T(x), taken by Wendy to rendezvous with her father in terms of x. (2 marks) (d) Using interval notation, state the domain of T(x). (1 mark) Page 2 of 15 page 2 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK (e) dT i. Find. (1 mark) dx ii. Determine the value of x to two decimal places which corresponds to T(x) being minimized. (2 marks) (f) Determine the minimum value of T(x) in seconds to two decimal places. (2 marks) (g) Use a gradient table to confirm that this value of x results in T(x) being the minimum value. (2 marks) Page 3 of 15 page 3 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK d ⎛ dT ⎞ (h) Using a calculator, evaluate ⎜ ⎟ at the value of x resulting in T(x) being dx ⎝ dx ⎠ minimized. (ie. Differentiate again and sub in the x value from (e) part ii) (2 marks) (i) How long would it have taken Wendy had she followed the path ACB? (1 mark) (j) How long would it have taken Wendy had she gone from A to B directly? (2 marks) Page 4 of 15 page 4 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK (k) How long would it have taken Wendy if she had travelled to a point midway between B and C, to two decimal places? (2 marks) (l) i. Sketch the graph of y = T(x) and give the endpoints and the turning point in co-ordinate form. (3 marks) ii. State, correct to the nearest whole number, the range of T(x). (1 mark) Page 5 of 15 page 5 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK (m) Using interval notation, i. Over what set of values does T(x) have an inverse? ii. Over what set of values is T(x) continuous? iii. Over what set of values is T(x) differentiable? (3 marks) (n) i. What path would Wendy need to take if her speed on the grass was the same as her speed on the concrete? (1 mark) ii. Explain your reasoning to part i. (1 mark) Page 6 of 15 page 6 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK (o) In Diagram 1, O = O(0,0), A = A(200,0), B = B(0,150) and C = C(200,150). How far along the route AX, to two decimal places, did Wendy travel before she was closest to the point C, given that she was following the pervious path of minimum time? (4 marks) Diagram 1 (AX = d, XB = x, XC = 200 – x) Hint 1: The closest point along AX, call it point D, occurs when AX and DC meet at right angles. Hint 2: Find the equations of the lines AX and DC. Page 7 of 15 page 7 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK Diagram 2: (AX1 = x, X2B = x, X2C = 200 – x ) (p) If Wendy had decided to travel along the path AX1X2B as indicated in Diagram 2: i. What would be the new time function T(x) for this case? (2 marks) ii. What would be the domain of this function? (1 mark) iii. What would be the exact value of x for which this time function is a minimum? (2 marks) Page 8 of 15 page 8 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK iv. What would be the exact minimum value of T(x)? (2 marks) Page 9 of 15 page 9 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK Question 2 (22 marks) After meeting her father at the cathedral door, Wendy slowly walked with him along the main aisle heading towards the altar to the sound of Mendelssohn's wedding march. Just as the organist was reaching a crescendo, Wendy noticed immediately to her right a magnificent leadlight stained glass window which had the upper perimeter given by the following equation: 𝑓(𝑥 ) = 𝑎𝑥 + + 𝑏𝑥 + 𝑐 This window is shown in the following diagram. Diagram 3 Page 10 of 15 page 10 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK (a) Using Diagram 3, show that the values of a, b and c are -1/4, 0 and 16 respectively. (3 marks) (b) Express f(x) with numerical coefficients. (1 mark) (c) What are the domain and range of f(x)? (2 marks) Page 11 of 15 page 11 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK (d) What is the gradient of f(x) at x = 0, 1, 2? (3 marks) (e) What do you observe about the gradient as you move further away from the axis of symmetry of the window? (2 marks) Page 12 of 15 page 12 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK Question 3 (22 marks) After advancing further down the aisle, Wendy looked upwards observing that the shape of the right hand side of the main arches supporting the roof structure above the aisle was given by the following equation: 𝑔(𝑥 ) = 𝑎𝑥 0 + 𝑏𝑥 + + 𝑐𝑥 + 𝑑, 𝑥 ≥ 0 where x is the horizontal distance from the centre of the arch. 05 +0 55 (a) Given that 𝑔(0) = + , 𝑔(2) = 14, 𝑔(4) = + ,g(7) = + , determine the values of a, b, c, d in rational form. Then express g(x) as a cubic polynomial with known rational coefficients. (b) Give an expression for the reflected image of g(x) in the y-axis. (c) Express the y-intercept of y = g(x) in co-ordinate form. Page 13 of 15 page 13 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK (d) Discuss the differentiability of: 𝑔(−𝑥), 𝑥 < 0 𝑦=; 𝑔(𝑥), 𝑥 ≥ 0 (e) Draw the graph of: 𝑔(−𝑥), 𝑥 < 0 𝑦=; 𝑔(𝑥), 𝑥 ≥ 0 expressing the y-intercept and the roots in co-ordinate form. It is worth noting that a pointed arch or cusp is a characteristic of Gothic architecture. Page 14 of 15 page 14 12MM Differentation Application Task Revision #3 - Questions MAV SACs 2016, MM APPLICATION TASK (f) What is the gradient of g(x) when (x ,y) = (8.6380,0) to four decimal places? Page 15 of 15 page 15