DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATH 121/124 - APR 2014 A. Ableson, T. Day, A. Hoefel

Size: px

Start display at page:

Download "DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATH 121/124 - APR 2014 A. Ableson, T. Day, A. Hoefel"

Cameron Newton
5 years ago
Views:

1 Page 1 of 18 DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATH 121/124 - APR 2014 A. Ableson, T. Day, A. Hoefel INSTRUCTIONS: Answer all questions, writing clearly in the space provided. If you need more room, there are blank pages at the end of the exam. If you use these pages, you must provide clear directions to the marker, e.g. Continued on page 17. 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. HAND-IN Answers recorded on question paper. 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 I II III IV V Possible Grade Section VI VII VIII IX/X Total Possible Grade 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 Page 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) Consider the differential equation dp dt differential equation? = 100(50 P). What are the equilibria for this (A) P = 0 (B) P = 50 (C) P = 0 and P = 50 (D) This differential equation has no equilibria. (2) A surface is defined by z = (x 2 + 1)sin(y) + xy 2. By setting one variable constant, we generate a plane that intersects the surface. Which of the following planes will intersect the surface in a sine curve? (A) x = 0. (B) x = 1. (C) y = 0. (D) y = 1. (3) Four functions are given below. For which function does doubling both x and y result in a doubling of f? (A) f(x,y) = x 0.25 y 0.25 (B) f(x,y) = x 0.5 y 0.5 (C) f(x,y) = x 0.75 y 0.75 (D) f(x,y) = x 1.00 y 1.00

3 Page 3 of 18 (4) Consider the function f(x,y) = 7 x 2 +y 2. Which of the following is a graph of this function? A B C D (5) Consider the region between y = x 3 and the x-axis on the interval from x = 0 to x = 1. Which of the following integrals represents the volume generated by rotating this region around the y axis? (A) (B) (C) (D) π(x 3 ) 2 dx π(1 2 (x 3 ) 2 ) dx π( 3 y) 2 dy π(1 2 ( 3 y) 2 ) dy

4 Page 4 of 18 (6) The productivity of a wheat field (P) is measured under various levels of fertilizer (F) and rainfall (R). Results are shown in the table below. R F Which of the following is a possible linear approximation for the wheat field productivity for environments similar to F = 200, R = 12? (A) P(F,R) F 100R. (B) P(F,R) F 50R. (C) P(F,R) (F 200) 100(R 12). (D) P(F,R) (F 200) 50(R 12). (7) Consider the contour diagram for f(x,y) shown below, with the point P indicated. In which direction is the gradient at P directed? P y 5 4 x (A) Roughly in the direction of 1, 1 (B) Roughly in the direction of 1, 1 (C) Roughly in the direction of 1, 1 (D) Roughly in the direction of 1, 1

5 Page 5 of 18 (8) A large ship is being towed by two tug boats. The larger tug exerts a force which is 50% greater than the smaller tug; the larger tug is pulling at an angle of 25 degrees north of east. Which direction must the smaller tug pull to ensure that the ship travels due east? (A) Between 0 and 20 degrees south of east. (B) Between 20 and 30 degrees south of east. (C) Between 30 and 40 degrees south of east. (D) Between 30 and 50 degrees south of east. (9) Which of the following functions is a solution to the differential equation dy dx = x y? (Assume y > 0.) (A) y = x (B) y = x+1 (C) y = x 2 (D) y = x 2 +1 (10) Which of the following describes the second derivatives at the point Q on the contour diagram below? Q (A) f yy is positive; f xy is positive. (B) f yy is positive; f xy is negative. (C) f yy is negative; f xy is positive. (D) f yy is negative; f xy is negative. y x

6 Page 6 of 18 Section II. Consider the region between the x axis and the function y = e x/3, on the interval 0 x 3. All x and y values are in cm. (a) Sketch the region described, clearly indicating the vertical scale (b) Imagine this region is cut out of sheet of metal, with constant density 7 g/cm 2. Find the total mass of the region. (c) Find the x coordinate of the center of mass the piece of metal. (You do not need to find the y coordinate of the center of mass.)

7 Page 7 of 18 Section III. A mathematical painter is imagining her next composition. She will be painting the region under the graph f(x) = 1 between x = 0 and x = 1. All x and y measurements are xp in meters. (a) For her to be able to paint the region, it must have a finite area. For what values of p does the region have a finite area? Support your answer with an appropriate calculation. (b) After looking at her stock of paint, she finds she has enough paint to cover 32 square meters. If she wants to use up all her paint on this project, what value of p should she choose for her design? (Assume that so long as the area is 32 square meters, she can somehow paint the area in finite time.)

8 Page 8 of 18 Section IV. A titration buret (a tall, narrow cylinder used to measure liquids) has a hole in the bottom, out of which the liquids flow. If the buret is filled with water, the height of the water in the buret drops at a rate proportional to the square root of the current water height. (a) Let h be the height of water in buret. Write the differential equation for the rate of change of h with respect to the current height. (b) Your differential equation in (a) has a constant of proportionality in it. Is that constant positive or negative? Explain your answer. (c) At what height h does the water level become constant? Explain why this height is reasonable in the context of the problem. (d) If water starts to flow out of the buret at t = 0 seconds, at what time t will the water level be dropping the most rapidly? Explain your answer by referring to the differential equation.

9 Page 9 of 18 (e) Find the family of functions that satisfies the differential equation. I.e., solve for h(t). (f) The buret originally is filled up with water to a height of 16 cm, and after 10 seconds the water level has dropped to a height of 4 cm. Determine the time at which the buret will be empty.

10 Page 10 of 18 Section V. The temperature on a flat surface is given by T(x,y). At the point (2,1), the temperature is T(2,1) = 5 degrees; also, T x (2,1) = 4 degrees/cm, and T y (2,1) = 3 degrees/cm. (a) Use a linear approximation to estimate the temperature at (x, y) = (2.2, 1.7). (b) A bug is walking on the surface, with its position given by x(t) = 1+t and y(t) = 2 t 3 with x,y measured in cm and t in minutes. At what rate is the temperature around the bug changing at t = 3? Give units in your answer. (Note that at t = 3, the bug is at coordinates (2,1).)

11 Page 11 of 18 Section VI. Consider the function z = h(x,y) = y 2 4x, limited to the region 0 x 4, 0 y 4. (a) On the axes below, draw and clearly label the contours at heights z = 0, 4 and 8. Remember the region is bounded to the domain shown (0 x 4, 0 y 4) (b) Evaluate the gradient at the point (1, 2), and draw the gradient vector on your contour diagram from part (a). (c) Compute the directional derivative at (1, 2) in the direction given by the vector 4, 3. (d) Without calculation, identify the (x, y) point where the global maximum of h occurs on the domain shown, 0 x 4, 0 y 4. Explain your answer briefly.

12 Page 12 of 18 Section VII. Find the point on the plane z = 1+3x+2y that is closest to the origin. Confirm that your answer is a local minimum for the distance.

13 Page 13 of 18 Section VIII. Let f(x,y) = x 2 +3y 2. (a) Findtheminimum valueoff(x,y), andits(x,y)location, subject totheconstraint x+y = 4, using the method of Lagrange multipliers. Min occurs at x = and y = Minimum value of f(x,y) = (b) Find the value of Lagrange multiplier, λ, for the optimum from part (a). (c) Use the Lagrange multiplier value to estimate the minimum value of f(x,y) on the modified constraint x+y = 3.8.

14 Page 14 of 18 Section IX. Based on Biology Tutorial - Answer only one of Section IX or Section X. When a population is being harvested or culled at a continuous rate of H individuals per year, but the natural (unharvested) population would grow according to a logistic model with carrying capacity K, the resulting differential equation for the population N is given by: dn dt = r ( K N K ) N H (a) If r = 5, K = 1000, and H = 1200, find the equilibrium population(s), if any. (b) Using the same constants as in part (a), use the sign of dn dt population of 500 will grow or shrink over time. to predict whether an initial (c) If r = 5, K = 1000, and H =1300, find the equilibrium population(s), if any. (d) If the harvesting level is kept at H = 1300 (with r = 5 and K = 1000 again), what will happen to the population in the long run? Support your answer.

15 Page 15 of 18 Section X. Based on Economics Tutorial - Answer only one of Section IX or Section X. Cigarette consumption in the US was approximately 2.9 billion cartons this year, and a study indicated that the demand elasticity was The price per carton is currently $60. (a) Estimate the number of cartons that would be sold if an additional $4/carton tax were added. (b) If the current $60 price has the current tax rate of $12/carton already embedded in the price, what would the increase or decrease in the government s revenue be if the $4 additional tax were imposed? (c) Construct a linear approximation for demand D for cigarettes as a function of the price p, for prices near $60 per carton. (d) Use your previous linear approximation for demand to build a the function that computes the government s tax revenue R based on the total carton price p, assuming that price changes will only be due to changes in the tax per carton.

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

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

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

DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATH 121/124 - APR 2014 A. Ableson, T. Day, A. Hoefel

Page 1 of 18 STUDENT NUMBER: DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATH 121/124 - APR 2014 A. Ableson, T. Day, A. Hoefel INSTRUCTIONS: Answer all questions, writing clearly