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1 1 of 18 HAND IN Answers recorded on exam paper. DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATH DEC 2016 Section CDS Students ONLY Instructor: A. Ableson INSTRUCTIONS: This examination is 3 HOURS in length. Only CASIO FX-991, or Blue Sticker 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 18 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) Find the solution to the equation ln(2x + 3) = 1. In which of the following intervals does the solution lie? (A) x < 1. (B) 1 x < 0. (C) 0 x < 1. (D) 1 x. (2) Given f(x) = 3 2x + ln(x 4 ), find the value of f (2). (A) f (2) is below 150. (B) f (2) is between 150 and 160. (C) f (2) is between 160 and 180. (D) f (2) is above 180. (2 + x) 4 (2) 4 (3) What is the value of the limit L = lim? x 0 x Tip: the limit definition of the derivative might be helpful to remember here. (A) The limit L < 4. (B) The limit 4 L < 16. (C) The limit 16 L < 32. (D) The limit 32 L.

3 3 of 18 (4) Consider the sinusoidal graph shown below. g(x) 1 π 2 π 2 1 Which of the following is a possible formula for this graph? (A) g(x) = sin (2x π) ( x ) (B) g(x) = sin 2 π ( (C) g(x) = sin 2x π ) ( 2 x (D) g(x) = sin 2 π ) 2 (5) Consider the equation e } 2x {{ x 3 } = 0. It is too complicated to be solved by hand. f(x) Starting at an initial solution estimate x 0 = 1.5, what would the next estimate of the solution, x 1, be if you used Newton s Method? (A) x 1 is below 0.5. (B) x 1 is between 0.5 and (C) x 1 is between 0.75 and 1. (D) x 1 is above 1. (6) Evaluate the limit lim x 0 1 cos(3x) x 2. (A) The limit value is above 3. (B) The limit value is below 3. (C) The limit value is exactly 3. (D) The limit value is undefined.

4 4 of 18 (7) If f(x) is an even function, which of the following is equal to (A) 1 2 (B) 1 2 (C) 2 (D) f(x) dx. f(x) dx. f(x) dx. 2 2 f(x) dx? (8) Consider the graph of h(t) shown below. h(t) A B C D t Which point on the graph produces the largest value for the ratio h(t)? t (A) Point A (B) Point B (C) Point C (D) Point D

5 5 of 18 (9) Consider the graph of g(x) shown below. g(x) x 1 x 2 x 3 x 4 x Which of the following graphs is the graph of an anti-derivative of g(x)? x 2 x 3 x x 2 x 3 x 1 x 4 x x 1 x 4 A B x 1 x 2 x 3 x 4 x x 1 x 2 x 3 x 4 x C D (10) What is the average of the function y = 4 (x 1) 2 over the interval x = ? (A) The average is below 2. (B) The average is between 2 and 4. (C) The average is between 4 and 8. (D) The average is above 8.

6 6 of 18 Section II. Families of Functions [/12] This section will refer to the family of functions defined by f(x) = 4x + a2 x where a is a positive constant, and we consider only the domain x > 0. Note that your answers to the following questions may depend on a. (a) At what x value(s) will f(x) = 0 (if any)? (b) Find both the x and y coordinates of the critical points of f(x) (if any). (c) Evaluate the limit lim x 0+ f(x). (d) Identify the intervals on the domain x > 0 where f(x) is concave up (if any), and then the intervals on which it is concave down (if any).

7 7 of 18 Families of Functions (cont.) f(x) = 4x + a2 x (e) Sketch the graph of f(x) for the following family members: (1) a = 2. (2) a = 4. (3) a = 8. Clearly indicate the origin, the scale on both axes, and choose your scaling so that the features indicated in (a)-(d) are visible.

8 8 of 18 Section III. LCD Panel Design [/10] An LCD panel (like TV) has a 4:3 ratio of width to height (so x y = 4 3 using x and y as defined in the diagram to the right.) The sale price p in dollars of a panel is determined by the length of its diagonal (in inches): p(d) = 20d. The manufacturing cost m in dollars, however, is determined by the area a of the panel (in square inches): d x y m(a) = a2 What size of panel makes the most profit (i.e., maximizes p m)? Report your final values in the table at the bottom of the page. x y = 4 3 a = xy Optimal x = y = d = Profit =

9 9 of 18 Section IV. Integration [/10] Evaluate the following indefinite integrals. (a) ln(x)x 2 dx (b) Determine by differentiating whether or not your answer to question (a) is correct. State your conclusion clearly for the grader. (c) t 4 3 t t dt (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 18 Section V. Magnetism [/10] Consider a wire with constant radius R. The strength of a magnetic field B generated by a current in this wire depends on the distance r from the center of the wire. B 0 is a constant related to the general strength of the field. 1 r R B 0 for 0 r < R B(r) = R r B 0 for r R (i) Find the lim B(r) using one-sided limits ( lim and lim ) r R r R + r R (ii) Sketch the graph of B(r) on the axes below. The transition point, r = R, is marked for you already. B(r) 0 R r (iii) Is the function B(r) continuous at r = R? Explain your answer in a sentence or two. (iv) Is the function B(r) differentiable at r = R? Explain your answer in a sentence or two. 1 You do not need to know any physics to answer this question.

11 11 of 18 Section VI. Vase Volume [/8] A vase is 20 cm tall and has square cross-sections. The perimeter of each cross-section is measured at intervals of 5 cm starting from its base giving the following table: Distance from base (cm) Perimeter (cm) Use the Trapezoidal rule to estimate the volume of the vase. Vase Volume Estimate:

12 12 of 18 Section VII. Approximations [/10] You are told two facts about a function g(x): g(1) = 2, or in other words, the point (1, 2) is on the graph of g(x), and the derivative of g(x) is given by g (x) = x Note that you will not be able to integrate to find g(x). (a) Find the equation of the tangent line to g(x) at the point (1, 2). (b) Use your linear approximation to estimate the value of g(1.05). (c) Is the true value of g(1.05) larger or smaller than your estimate from part (b)? Explain your answer in a sentence or two. (d) Find the second degree Taylor polynomial of g(x) centered at the point x = 1.

13 13 of 18 Section VIII. Related Rates [/10] An oil refining facility has sprung a leak, and is now spilling oil onto a nearby pond. The oil is leaking out a constant rate of 5 liters per second, forming a slick that is roughly circular in shape and meters thick. Answer the following questions, and remember to include units in all your answers. Reminder: there are 1000 liters in a cubic meter. (a) Assuming that at t = 0 there was no oil in the slick, how long does it take for the radius of the slick to reach 70 meters? (b) At what rate is the radius of the slick increasing when the radius is 70 meters? (c) What is the total volume of oil that spilled onto the pond at the moment the radius of the slick is increasing at a rate of 0.2 m/s?

14 14 of 18 Section IX. Chemistry [/10] Francine is just starting a unit on catalysts in her Chemistry course. She has read the definition: Catalyst: A substance that helps a reaction to go faster without being used up in the reaction. The particular reaction she is studying generates oxygen (O 2 ) at a rate given by R(C) = C 2 g/min where C is the amount of catalyst present in milligrams. In her lab, Francine starts her reaction at t = 0 minutes. For two minutes, she lets the reaction run with no catalyst present. Starting at t = 2 minutes, she begins to add catalyst, at a rate of 0.75 mg/min. At t = 5 minutes, she stops the reaction. Reminder: You must include units in your answers whenever appropriate. (a) Give an expression for the amount of catalyst present during{ the 5 minutes of the experiment. You will need to give your answer in piece-wise form, e.g. C(t) =.... (b) During the first two minutes, what is the rate of O 2 production? (c) How much O 2 is produced in the first two minutes of the reaction?

15 15 of 18 (d) Using your answer to part (a), write out the rate of O 2 production over the interval t = 2 to t = 5. Your final rate should be written as a function of t only, and not a function of C. (e) Compute the total amount of O 2 produced between t = 0 and t = 5.

16 16 of 18 Space for additional work. space. Indicate clearly which Section you are continuing if you use this

17 17 of 18 Space for additional work. space. Indicate clearly which Section you are continuing if you use this

18 18 of 18 Space for additional work. space. Indicate clearly which Section you are continuing if you use this