STUDENT NUMBER: Page 1 of 15

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1 STUDENT NUMBER: Page 1 of 15 INSTRUCTIONS: DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATHEMATICS 121 DECEMBER EXAM VERSION 1 Alan Ableson Answer all questions, writing clearly in the space provided. If you need more room, continue your answer on the back of the previous page, providing clear directions to the marker. 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 exam questions as written. FOR MARKER S USE ONLY Section Possible Received I 20 II 15 III 15 IV 10 V 15 VI 10 VII 15 TOTAL 100

2 STUDENT NUMBER: Page 2 of 15 Section I: Multiple Choice (10 questions, 2 marks each) Each question has four possible answers, labelled (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) Given f(x) = 4+ 5 ( ) 3πx 5 2 cos, what is the period of f(x)? 2 (A) 3π 2 (B) 4π 3 (C) 4 3 (D) 5 2 (2) The integral arcsin(x) dx equals (A) 1 1 x 2 (B) 1 2 (arcsin(x))2 (C) x arcsin(x) x 1 x 2 (D) xarcsin(x)+ 1 x 2 (3) Let A be the average value of the function f(x) = x 2 over the interval x [ 2,2]. Which of the following statements is true? (A) A < 0 (B) 0 A < 2 (C) 2 A < 4 (D) 4 A < 8

3 STUDENT NUMBER: Page 3 of 15 (4) Imagine a cube that is gradually increasing in size. If the lengths of the sides of the cube are growing at 2 cm/s, then the volume of the cube will be growing at (A) 8 (current side lengths) cm 3 /s (B) 6 (current side lengths) 2 cm 3 /s ( ) 4 (C) 3 π 8 cm 3 /s (D) 8 cm 3 /s (5) The graph of the derivative, f (x) is shown below. f (x) x On the interval x [0,10], the original function f(x) has its global minimum closest to (A) x = 0 (B) x = 8 (C) x = 9 (D) x = 10 3x (6) The value of the limit lim x 0 2e is x (A) 0 (B) 1 (C) 3 2 (D)

4 STUDENT NUMBER: Page 4 of 15 (7) Consider the graphs of f(x) and g(x) shown below f(x) 2 g(x) x x We now define h(x) = f(g(x)). The value of h (3) is (A) 2 (B) -1 (C) -2 (D) undefined (8) The following equation is from your integral table. 1 x 2 +a dx = 1 ( x ) 2 a arctan +C a 1 The integral dx equals (2x 3) (A) 2x 3 arctan(2x 3)+C (B) arctan(2x 3)+C (C) 1 2 arctan(2x 3)+C (D) 1 3 arctan(2x 3)+C

5 STUDENT NUMBER: Page 5 of 15 (9) 3 functions, f 1,f 2,f 3, all pass through the point (0,1). The functions have the following quadratic Taylor approximations. f 1 (x) = 1+x+x 2 f 2 (x) = 1 x+x 2 f 3 (x) = 1 x x 2 Which of the sets of graphs shows the functions f 1,f 2 and f 3? A B (10) If 8 4 C (3f(x)+1) dx = 16, then 8 4 D f(x) dx equals (A) 5 3 (B) 4 (C) 5 (D) 13 3

6 STUDENT NUMBER: Page 6 of 15 Section II: Family of Functions Consider the family of functions g(t) = t 2 e kt, with t 0, and where k is a positive parameter (or constant). Wherever necessary, express your answers to the questions below in terms of k. (i) Find all point(s) where g(t) = 0. (ii) Find all the critical points of g(t) = 0, and identify them as local maxima, minima, or neither. (iii) Find value of g(t) at the critical point(s) you found above.

7 STUDENT NUMBER: Page 7 of 15 (iv) Find lim t g(t). (v) Use the answers from parts (i) to (iv) to sketch the graphs of g(t) for the specific k values k = 1 and k = 3. Clearly label and scale the axes. Labelallthefeaturesofthegraphsyouusedtoconstruct yoursketch. E.g. g (t) = 0, or from (iii).

8 STUDENT NUMBER: Page 8 of 15 Section III: Maximizing Profit A company making q thousand units of its product has found the following relationships. Cost(q) = 500+3q 2 Revenue(q) = 1000(1 e 0.5q ) (i) Write an expression for the profit as a function of q. (ii) This profit function has one critical point, which is a global maximum. Set up the equation for the critical points of the profit function. (Do not try to solve the equation once you have it; see next part.) (iii) The equation from the previous part cannot be solved by hand. Instead, use 2 steps of Newton s Method to find an approximate value for q at the critical point. (Hint: q = 5 is fairly close to the critical point.) In recording your calculations, keep all values to 2 digits after the decimal.

9 STUDENT NUMBER: Page 9 of 15 (more space for Newton s method if necessary) (iv) Find the marginal revenue and marginal cost functions, MR(q) and MC(q). (v) Try to confirm your answer to part (iii) by verifying that the marginal revenue and marginal costs are equal at the critical point you found. If they are not equal, explain why their values differ.

10 STUDENT NUMBER: Page 10 of 15 Section IV: Magnetism 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 B(r) = r B R 0 R r B 0 for r < R 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) 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 STUDENT NUMBER: Page 11 of 15 Section V: Catalysts 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. (i) 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. (ii) During the first two minutes, what is the rate of O 2 production? (iii) How much O 2 is produced in the first two minutes of the reaction?

12 STUDENT NUMBER: Page 12 of 15 (iv) Using your answer to part (i), 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. (v) Compute the total amount of O 2 produced between t = 0 and t = 5.

13 STUDENT NUMBER: Page 13 of 15 Section VI: Integration (i)sketchthegraphoff(x) = xsin(πx)ontheaxes below. Clearly indicate the vertical scale. (ii) Referring only to your sketch, state whether 2 0 xsin(πx) dx is positive or negative. Explain your reasoning in a sentence or two (iii) Estimate the value of the integral 2 0 x sin(πx) dx using the trapezoidal rule with 4 intervals. (iv) Find the exact value of the integral 2 0 x sin(πx) dx using anti-derivatives.

14 STUDENT NUMBER: Page 14 of 15 Section VII: Optimization The graph and equation of an oval are shown below x 2 +16y 2 = Now consider a rectangle inscribed within the oval. Two examples are shown below. 40 (x, y) (x, y) (i) If we identify the right-hand corner of the rectangle as the point (x,y), what will the area of the rectangle be? (ii) Write the area of the rectangle as a function of y only (or, if you prefer, of x only).

15 STUDENT NUMBER: Page 15 of 15 (iii) Find the values of x and y that give the rectangle with the largest area. Show that the values you find are at least a local maximum for the area.