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1 1 of 16 HAND IN Answers recorded on exam paper. DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON DEC 2015 MATH 121 Instructors: A. Ableson, S. Greenhalgh, D. Rorabaugh INSTRUCTIONS: Answer all questions, writing clearly in the space provided. If you need more room, continue your answer on the blank pages at the back of the exam, providing clear directions to the marker. For full marks, you must show all your work and explain how you arrived at your answers, unless explicitly told to do otherwise. Only CASIO FX-991, Gold Sticker or Blue Sticker calculators are permitted. Write your student number clearly at the top of each page. You have three hours to complete the examination. Wherever appropriate, include units in your answers. When drawing graphs, add labels and scales on all axes. PLEASE NOTE: Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer questions as written. I II III IV V VI VII VIII IX Total This material is copyrighted and is for the sole use of students registered in MATH 121 and writing this examination. This material shall not be distributed or disseminated. Failure to abide by these conditions is a breach of copyright and may also constitute a breach of academic integrity under the University Senate s Academic Integrity Policy Statement.

2 2 of 16 Section I. Multiple Choice (10 questions, 2 marks each) Each question has four possible answers, labeled (A), (B), (C), and (D). Choose the most appropriate answer. Write your answer in the space provided, using UPPERCASE letters. Illegible answers will be marked incorrect. You DO NOT need to justify your answer. (1) Consider the graph of f(x) below. 2 f(x) x 2 On which interval below is the average value of f(x) the highest? (A) The interval 2 x 1. (B) The interval 1 x 1. (C) The interval 1 x 4. (D) The interval 2 x 4. (2) Consider the function f(t) = 14 2 ( ) 7πx sin. What is the period of f(t)? 3 (A) 7π 3 (B) 7π 6 (C) 5 6 (D) 6 7 (3) If h(t) is an even function, and (A) 0 (B) 5 (C) 5 2 (D) h(t) dt = 5, then what is the value of h(t) dt?

3 3 of 16 (4) The Northern California Reservoir Authority tracks the rate of water inflow and water outflow (in millions of gallons per day) in its system. Below is the graph of those inflow and outflow rates over a full year; t = 0 is January 1st. Flow rate Inflow Rate 1 0 Outflow Rate t (months) Out of the times listed below, at what time is the quantity of water in the reservoir the lowest? (A) At t = 4. (B) At t = 5. (C) At t = 7.5. (D) At t = 10. (5) For the function g(x) = x + tan(x), where is its global minimum on the interval 0 x π? (A) x = 0. (B) x = π 4. (C) x = 3π 4. (D) The function does not have a global minimum on the interval. (6) What is the value of the limit lim x 0 cos(5x) 1 cos(3x) 1? (A) 3 5. (B) 5 3. (C) (D) The given limit does not exist.

4 4 of 16 (7) At which of the following points would the curve defined by x 2 + xy + y 2 = 27 have a vertical tangent line? (A) (-3,6) (B) (-6,6) (C) (6,-6) (D) (6,-3) (8) Which of the following is the Taylor polynomial of degree 3 for y = ln(x + 1), centered at x = 0? (A) P 3 (x) = 1 + x x 2 + 2x 3 (B) P 3 (x) = 1 + x 1 2 x x3 (C) P 3 (x) = x x 2 + 2x 3 (D) P 3 (x) = x 1 2 x x3 b, if x < 2 (9) Let f(x) = 3 b x, if x 2 For what value(s) of b will this function be continuous for all real x values? (A) b = 1 (B) b = 3 (C) b = and b = 3. (D) No choice of b will make f(x) continuous for all real x values. (10) A scientist places 16 cells in a petri dish. She knows the cells grow at an exponential rate, doubling in number every three hours. How many hours will it take for there to be 8192 cells in the dish? (A) 81 hours. (B) 27 hours. (C) 9 hours. (D) 3 hours.

5 5 of 16 Section II. Related Rates [/10] A student team is launching a model rocket. The rocket goes straight up from its launch site, and is tracked by the team, which is standing 20 m away from the launch site. When the rocket reaches the point where the angle between the rocket appears to be at an angle of π 3 radians above the horizon, the students measure that the angle is increasing at a rate of 0.25 rad/s. At what rate, in m/s, is the rocket rising at this moment? Note: you must define any new variables you introduce, either in words on on the diagram. Rocket Trajectory Rocket Students θ 20 m Launch Site

6 6 of 16 Section III. Optimization [/10] (a) Draw the graph of y = 12 x 2, over the entire domain shown. Clearly indicate the vertical scale. y x (b) Now imagine that we can inscribe a rectangle under this graph, with the rectangle s base on the x axis and its upper corners on the parabola y = 12 x 2. What are the dimensions of such a rectangle with the greatest possible area? Optimal values Width = Height = Area =

7 7 of 16 Section IV. Integration (a) Evaluate the integral x ln(x) dx [/10] (b) Determine by differentiating whether or not your answer to question (a) is correct. (c) Evaluate the integral x 2 x 3 1 dx. (d) Determine by differentiating whether or not your answer to question (c) is correct.

8 8 of 16 Section V. Families of Functions [/12] Consider the family of functions defined by a(1 e bx ) where a and b are positive constants, and we consider only the domain x 0. Note that your answers to the following questions may depend on a and b. (a) At what x value(s) will f(x) = 0? (b) What is the value of f (0)? (c) Find the critical points (if any) of f(x). (d) Evaluate the limit lim x f(x).

9 9 of 16 Families of Functions (cont.) f(x) = a(1 e bx ) (e) Sketch the graph of f(x) for the following family members: (1) a = 1, b = 1. (2) a = 3, b = 1. (3) a = 3, b = 10. Clearly indicate the scale on both axes, and choose your scaling so that the features indicated in (a)-(d) are visible. y x

10 10 of 16 Section VI. Derivatives [/8] (a) Find the derivative of cos(x). (b) Starting with the relation: cos(arccos(x)) = x, use implicit differentiation to derive the standard formula for d dx arccos(x). Note: the trig identity sin 2 (x) + cos 2 (x) = 1 may be useful.

11 11 of 16 Section VII. Sky-Diving [/8] The downward velocity, v, of a sky-diver jumping out of a plane is given by v = mg k (1 e kt/m ) where g is the acceleration due to gravity, and k is a constant related to the shape of the object. (a) Find a formula for the height of the body, h, above the surface of the earth as a function of time. Assume the body starts at height h 0. Keep all the constants (m, k and g) as letters during this calculation. (b) A sky-diver jumps out of a plane at a height of 2500 m, but then has trouble reaching the handle that opens her parachute: she has to search for 15 seconds after her jump before she can open her parachute. A parachute should be opened at least 1000 m above the ground for a safe landing: does this sky-diver succeed in opening her parachute in time? You can use the values k = 4, g = 9.8 m/s 2 and m = 75 kg for your calculations.

12 12 of 16 Section VIII. Integrals as Areas [/12] (a) Sketch the graph of f(x) = 3 x 4 over the interval 0 x 2. Clearly indicate the vertical scale. y (b) Based on your sketch, do you expect the sign of 2 0 (3 x 4) dx to be positive or negative? Explain your answer in a sentence or two. x 1 2 (c) Estimate the value of the integral 2 0 (3 x 4) dx using the Trapezoidal rule with 4 intervals. (d) Compute the exact value of the integral 2 0 (3 x 4) dx.

13 13 of 16 Section IX. Discrete Time System [/10] A dynamic system follows the updating function q will be limited to the interval 0 q 2. q t+1 = 2.5q t q t (a) Assuming that the initial q value is q 0 = 0.1, predict the population level for the next 4 generations (up to and including q 4 ). Report all figures to 3 digits after the decimal. (b) Using your answer to part (a), sketch a graph of q t vs. time. Be sure to label your axes. (c) Sketch the graph of q t+1 vs. q t, and draw a cobweb diagram starting at q 0 = 0.1.

14 14 of 16 q t+1 = 2.5q t q t + 0.5, 0 q t 2 (d) Use the updating formula to find all equilibrium value(s) of q t. (e) Classify each of the equilibrium level(s) you found in part (d) as either stable or unstable, using the slope test for equilibria.

15 15 of 16 Space for additional work. space. Indicate clearly which Section you are continuing if you use this

16 16 of 16 Space for additional work. space. Indicate clearly which Section you are continuing if you use this