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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 APR 2017 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 X 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) Which of the following differential equations is not separable? (A) dy = x(y + 1) dx (B) dy = (x + 1)(y + 1) dx (C) dy dx = x + 1 (D) dy dx = xy + 1 (2) Consider the surface defined by z = (x 2 + 1) sin(y) + xy 2. Which of the following planes produces a parabola where it intersects this surface? (A) x = 0 (B) y = 0 (C) z = 0 (D) None of the above. (3) An airplane is flying at an airspeed of 400 km/h, while being pushed by a wind that is blowing at 50 km/h from the southwest. If the plane orients itself so as to be flying due east, what will the airplane s speed be, relative to the ground? (A) Ground speed will be 450 km/h or more. (B) Ground speed will be 425 km/h or more, and less that 450 km/h. (C) Ground speed will be 400 km/h or more, and less that 425 km/h. (D) Ground speed will be less than 400 km/h.

3 3 of 16 (4) Which of the following statements about vectors is not necessarily true for all choices of vectors a and b? (A) a + b = b + a (B) b + a = a + b (C) a + b = a + b (D) a b = a b cos(θ), where θ is the angle between a and b. (5) Which of the following contour diagrams represents the contours of the surface z = y 2 + x? (A) (B) (C) (D)

4 4 of 16 (6) For what value(s) of t are the vectors parallel? (A) t = 1. (B) t = 0. (C) t = 1. (D) All real values of t. < 6, 4, 3 > and < 2, (t t + 1), t) > 3 (7) A thin 2 m long rod has lineal density δ(x) = 4 x g/m, where x, in meters, is measured from the left end of the rod. Where is the center of mass of the rod? (A) Undefined/not enough information. (B) In the range 0 x < 1. (C) At x = 1. (D) In the range 1 < x 2. (8) The contour diagram for the function f(x, y) is shown below. Assuming the shape of function is well described by the contour diagram, how many critical points does f(x, y) have in region shown above? (A) 6 or more (B) 5 (C) 4 (D) 3 or fewer

5 5 of 16 (9) Consider the differential equation dy = y + 2x. If we start approximating the solution to the equation dx at x = 1 and y = 3, using Euler s method with an interval size of x = 0.5, our next predicted point on the solution curve will be (A) (x, y) = (1.5, 4) (B) (x, y) = (1.5, 3.5) (C) (x, y) = (1.5, 1) (D) (x, y) = (1.5, 0.5) (10) You are at the point (2, -1, -1), standing upright and facing the xz plane. The positive z axis is upwards. You take two steps forward, turn left, and take four steps forward. What are the coordinates of your final position? (A) (0, -5, -1) (B) (0, 3, -1) (C) (-2, 1, -1) (D) (-2, 3, -1)

6 6 of 16 Section II. Areas and Volumes 1. Consider the finite region R enclosed within the boundaries y = ln(x), x = 1, and y = 3. (a) On the axes to the right, sketch those boundaries, then shade in the region R. Clearly indicated the scales on your axes. [/1] (b) Consider the solid generated by rotating the region R around the x axis. Write down a definite integral whose value is the volume of this solid. [/3] 2. Find the area enclosed between the graphs of y = sin(πx) and y = x on the interval x = A sketch may help. [/4]

7 7 of 16 Section III. Improper Integrals For each of the following integrals, use the definition of an improper integral to determine whether it converges or diverges. If converges, find the value of the integral. (a) 0 te kt dt, where k is a positive constant. [/5] (b) π/2 0 cos(x) sin(x) dx. [/5]

8 8 of 16 Section IV. Learning As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between: the material currently remembered, and some positive constant, a. (a) Let y(t) be the fraction of the original material remembered t weeks after the course has ended. Set up a differential equation for y. Your equation will contain two constants; the constant a is less than y for all t. [/3] (b) Find the general solution to the differential equation. [/5] (c) Describe the practical meaning (in terms of the amount remembered) of the constant a in your solution to part (b). [/2]

9 9 of 16 Section V. Slope Fields Consider the slope field shown to the right. (a) Circle the formula below for differential equation that corresponds to the slope field. No explanation is required. [/1] y dy dx = 2x + y dy dx = x + 2y 1 dy dx = 2x y dy dx = x 2y x (b) For each of the following points, sketch the solution that passes through it. Each graph sketch should extend to boundary of the slope field shown. [/3] 1 (0,-2) (-1,0) (0,1) 2 (c) Most of the solutions to the DE are fairly complicated functions. However, by studying the slope field you can identify one simple linear function that is a solution. What is the formula for that linear solution? [/2] (d) Show that your solution from part (c) satisfies the differential equation you picked in part (a). [/2]

10 10 of 16 Section VI. Functions of Two Variables 1. Find a formula for the linear function z = g(x, y) described by the table below. [/4] y x The Dubois formula s(w, h) relates a person s skin surface area, s (in m 2 ) to weight w (in kg), and height h (in cm). A contour diagram of this function is shown to the right. (a) Find the value s(70, 210) and then express the meaning of the value in a sentence. [/1] (b) Estimate the values s w (70, 210) and s h (70, 210), being clear about how you are using the contour diagram to arrive at your answer. (Note: there are several ways to estimate these derivatives; we will accept any appropriate method.) [/3] s w (70, 210) = s h (70, 210) = (c) Write the meaning of each value in sentence form. [/2] s w (70, 210) : s h (70, 210) :

11 11 of 16 Section VII. Optimization Consider the surface defined by f(x, y) = x 3 + y 2 12x 6y 24. (a) Find the critical point(s) of f(x, y). [/3] (b) Classify each of the critical points as a local min, local max, or saddle point. [/3] (c) Does this surface have a global maximum? Support your answer with a brief explanation. [/2]

12 12 of 16 Section VIII. Surfaces Consider the function z = h(x, y) = x y, 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, 2 and 1. Remember the region is bounded to the domain shown (0 x 4, 0 y 4). [/3] y x (b) Evaluate the gradient at the point (3, 1), and draw the gradient vector on your contour diagram from part (a). [/2] (c) Compute the directional derivative at (3, 1) in the direction given by the vector 4, 3. [/2] (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. [/1]

13 13 of 16 Section IX. A hiker is moving over a mountain whose height is given by h(x, y) = x y 2 where x, y and z are measured in meters. (a) If the hiker starts at (x, y) = (30, 50), in which direction should they walk if they want to go in the steepest uphill direction? [/3] (b) What is the slope in the direction found in (a)? [/2] (c) The hiker instead decides to follow a marked trail. Their path on the map is defined by the functions x(t) = 70 20t and y(t) = t, where t is measured in minutes. At what rate is the hiker ascending or descending at t = 2? Include units in your answer. (Note that at t = 2, the hiker is at coordinates (30, 50).) [/3]

14 14 of 16 Section X. Optimizing Production A firm manufactures a commodity at two different factories. The total cost of manufacturing depends on the quantities, q 1 and q 2, supplied by each factory, and is expressed by the joint cost function, C = f(q 1, q 2 ) = q q 1 q 2 + 2q The company s objective is to produce 400 units, while minimizing production costs. Use the method of Lagrange multipliers to determine how many units should be supplied by each factory. [/10]

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

I II III IV V VI VII VIII IX Total

I II III IV V VI VII VIII IX Total 1 of 18 HAND IN Answers recorded on exam paper. DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATH 121 - DEC 2016 Section 700 - CDS Students ONLY Instructor: A. Ableson INSTRUCTIONS: