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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 MATH DEC 2017 Section 002, 003 (Main Campus Sections) Instructor: A. Ableson, D. Rorabaugh INSTRUCTIONS: This examination is 3 HOURS in length. Only CASIO FX-991 calculators are permitted. Answer all questions, writing clearly in the space provided. If you need more room, continue your answer on one of the blank pages at the back, 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. Wherever appropriate, include units in your answers. When drawing graphs, add labels and scales on all axes. Put your student number on all pages, including this front page. PLEASE NOTE: Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer exam 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 121/124 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) The sale price of a used car can be approximated by the function S(t) = 25(0.85) t where t is in years from the date of manufacture, and S is the sale price in thousands of dollars. What does the value S (8) 1.1 mean? (A) The sale price is decreasing by $1100 per year when the car is 8 years old. (B) The sale price decreased by $1100 over the 8 years after its manufacture. (C) The sale price decreased by $1100 from its original price when the car is 8 years old. (D) The sale price is decreasing by $1100. (2) The function g(x) is an odd function. If value of 2 (A) ( 2 (B) 6 (C) 0 (D) ( 2 ( 2 ( 2 g(x) dx? ) g(x) dx < ) g(x) dx < 0 ) g(x) dx < 6 ) g(x) dx 3 0 g(x) dx = 4, then which of the following is true about the (3) Consider the graph of f(x) shown on the right, and the related integral 4 0 f(x) dx. How does the exact integral value relate to the LEFT(n) and MID(n) integral estimates? f(x) (A) LEFT(n) and MID (n) will both overestimate the integral. (B) LEFT(n) and MID (n) will both underestimate the integral. (C) LEFT(n) will under- and MID (n) will overestimate the integral. (D) LEFT(n) will over- and MID (n) will underestimate the integral x

3 3 of 16 (4) The total cost to manufacture q units of a product, T (q), is shown on the graph to the right. Out of the four points shown on the graph, which choice of q will minimize the average production cost per unit, T (q)? q (A) Point A. Cost T(q) B C D (B) Point B. (C) Point C. (D) Point D A q (5) The graph of h(x) is shown to the right. Which of the following is true about the average value of this function over the interval t = 2 to t = 5? (A) Avg. h(x) on [2, 5] is less than 1. (B) Avg. h(x) on [2, 5] is greater than or equal to 1, and less than 2. (C) Avg. h(x) on [2, 5] is greater than or equal to 2, and less than 4. (D) Avg. h(x) on [2, 5] is greater than or equal to 4. h(x) x (6) Compute the derivative of f(t) = t cos(t 2 + 1). (A) f (t) = cos(t 2 + 1) + t cos(t 2 + 1)(2t + 1) (B) f (t) = cos(t 2 + 1) t sin(t 2 + 1)(2t) (C) f (t) = cos(t 2 + 1) + t cos(t 2 + 1)(t 2 + 1) (D) f (t) = cos(t 2 + 1) t sin(t 2 + 1)(t 2 + 1)

4 4 of 16 (7) A population model is developed to predict a population level over time, and one formulation is given by P (t) = P 0 e 0.3t, with t measured in years, and P 0 representing the population at time t = 0. Which of the following formulas is most closely equivalent to the function P (t)? (A) Q(t) = P 0 t 0.74 (B) Q(t) = P t (C) Q(t) = P 0 t 1.35 (D) Q(t) = P t (8) A voltage V across a resistance R generates a current of I = V amps. If a constant voltage of 10 volts R is put across a resistance that is increasing at a rate of 2 ohms per second when the resistance is 7 ohms, at what rate is the current changing? (A) The current is decreasing at more than 0.4 amps/sec. (B) The current is decreasing, but at less than 0.4 amps/sec. (C) The current is increasing at more than 0.4 amps/sec. (D) The current is increasing, but at less than 0.4 amps/sec. (9) Find the value of the integral I = describes that value. 2 2 (x 2 + 1) dx, and identify which of the following statements (A) I = 0 (B) 0 < I 3 (C) 3 < I 5 (D) 5 < I (10) To the right is a graph of the derivative f (x), for some function f(x). At which of the following x values does the original f(x) function have a local minimum? f (x) (A) f(x) has a local minimum at x = 4. (B) f(x) has a local minimum at x = 8. (C) f(x) has a local minimum at x = 10. (D) f(x) has a local minimum at x = x

5 5 of 16 Section II. Function Analysis [/15] This section will refer to the function defined by f(x) = f (x) = x + 4 (x 2)(x 2 + 2x 8) and f (x) = (a) What is the domain of f(x)? 2(x + 4) (x 2) 2 (x 2 + 2x 8) x x 2. You may take for granted that + 4x 16 (b) What are the x and y intercepts of f(x)? (c) Evaluate the limit lim x 2 f(x). x intercept(s): y intercept(s): (d) Evaluate the limit lim x 4 f(x). (e) Evaluate the limits lim f(x) and lim f(x). x + x

6 6 of 16 Section II continued. x 2 16 f(x) = 2x 2 + 4x 16 f (x) = (f) Identify all the critical points of f(x). x + 4 (x 2)(x 2 + 2x 8) f (x) = 2(x + 4) (x 2) 2 (x 2 + 2x 8) (g) Identify all the intervals where f(x) is increasing, and all the intervals where f(x) is decreasing. (h) Sketch a graph of f(x) on the axes below. Ensure that you choose your vertical scale so that the features you identified in parts (b)-(g) are visible. f(x) x

7 7 of 16 Section III. Taylor Polynomials [/7] (a) Construct the second-degree Taylor polynomial approximation to f(x) = ln(3x) at the point x = 1. ln(3x) ln(3) (b) Evaluate the limit lim. You can use your answer from part (a), or l Hopital s rule if it x 1 1 x applies, to assist you.

8 8 of 16 Section IV. Consider the sideways parabola defined by the relation x = y [/10] (a) Sketch the parabola on the axes to the right. Clearly indicate the vertical scale. y (b) Find a formula for the slope of the tangent line at any point (x, y) on the parabola. x (c) Find a formula for a tangent line drawn to the parabola, at the point with y coordinate y = b. (d) Find the (x, y) coordinates of the two points on the parabola where such tangent lines also pass through the point (0, 0).

9 9 of 16 Section V. Integration [/10] Evaluate the following indefinite integrals. (a) x sin(5x) dx (b) Determine by differentiating whether or not your answer to question (a) is correct. State your conclusion clearly for the grader. (c) x 2 cos(x 3 ) dx (d) Determine by differentiating whether or not your answer to question (c) is correct. Again state your conclusions clearly for the grader.

10 10 of 16 Section VI. Dosage Response [/10] For some positive constant k, a patient s temperature change, T, in degrees Celsius, due to a dose of D milligrams of medication, is given by T (D) = ( k 3 D ) D 2. 4 (a) What dosage maximizes the temperature change? T (D). Note: your dosage value may depend on k. Show that your answer is a local maximum for (b) The sensitivity of the body to the drug is defined as S(D) = dt. What dosage level is associated dd with the maximum sensitivity? (You do not need to show that your answer is a local maximum this time.) Note: your dosage value may depend on k.

11 11 of 16 Section VII. A UFO (unidentified flying object) appeared on a military radar tracker for a brief 8 second time period. The UFO was moving erratically, and its acceleration towards and away from the radar station was given by the function below; a is in m/s 2 and t is in seconds. [/10] t 0 t < 4 a(t) = 5 4 t < t 8 Recall that acceleration is the derivative (with respect to time) of velocity. We define velocity or acceleration towards the radar station as positive, and velocity or acceleration away as negative. At t = 0, the UFO had a velocity of v 0 = 70 m/s (positive, so towards the radar station). (a) Sketch a graph of a(t) on the axes below. Be sure to indicate the scale clearly on the vertical axis. Please use as much of the grid/graph space as possible for ease of reading. a(t) (b) Find the exact velocity of the UFO at t = 8. Explain or show by calculations how you arrived at your answer. Note that the velocity of the UFO is a continuous function of time. t (c) Sketch the graph of the velocity of the UFO, v(t), over time. Indicate the scale on the vertical axis. Indicate the coordinates of v(0), v(4), v (6) and v(8) on the graph. v(t) t

12 12 of 16 Section VIII. The equation arctan(x) = x 0.5 is impossible to solve by hand. [/8] (a) The value x = 1 is the closest integer solution to this equation. Use three iterations Newton s method to improve this initial estimate of the solution. Report 4 digits after the decimal in each calculation. (b) Determine whether your final approximate solution found in part (a) is close to an exact solution. State your conclusion clearly for the grader.

13 13 of 16 Section IX. Population Model [/10] A population reproduces according to a discrete-time updating function, defined by ( q t+1 = 2q t 1 q ) t 10 where p is the population in thousands, t is in generations. For this population, 0 q 10. (a) Assuming that the initial population level is q 0 = 9, predict the population level for the next 4 generations (up to and including q 4 ). (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 = 9.

14 14 of 16 Section IX continued. ( q t+1 = 2q t 1 q ) t 10 (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