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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 121/124 - APR 2018 Section CDS Students ONLY Instructor: A. Ableson 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 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 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.

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) At the point P on the contour diagram below, identify the sign of the two second-order partial derivatives. (A) f xx and f yy are both positive. (B) f xx is positive, and f yy is negative. (C) f xx is negative, and f yy is positive. (D) f xx and f yy are both negative. (2) Consider the function g(x, y) = x2 2 + exy. (A) g(x, y) does not have a global maximum, and does not have a global minimum. (B) g(x, y) has a global maximum, but does not have a global minimum. (C) g(x, y) does not have a global maximum, but has a global minimum. (D) g(x, y) has a global maximum, and has a global minimum. (3) On the contour diagram below, at which point (A, B, C or D) is the magnitude of the gradient vector the largest?

3 3 of 16 (4) What is the angle between the two vectors 2, 4, 1 and 4, 2, 1? (A) The angle is between 0 and π 2 radians. (B) The angle is between π and π radians. 2 (C) The vectors are perpendicular. (D) The vectors are parallel. 1 (5) Consider the two integrals, (1) dx and (2) 1 x5/2 Classify each integral as either divergent or convergent. 1 1 dx. x3/2 (A) (1) and (2) are both convergent. (B) (1) is convergent, and (2) is divergent. (C) (1) is divergent, and (2) is convergent. (D) (1) and (2) are both divergent. (6) How many distinct equilibria does the differential equation dl dt = L2 + 3L have? (A) The DE has no equilibria. (B) The DE has one equilibrium value. (C) The DE has two equilibrium values. (D) The DE has three or more equilibrium values. (7) Which of the following is a solution to the differential equation dy dt = y + 2t? (A) y = 2e t + 2t + 2 (B) y = 2e t 2t + 2 (C) y = 2e t + 2t 2 (D) y = 2e t 2t 2

4 of 16 (8) Consider the function z = sin(x2 + y 2 ) 1 + x 2 + y 2.

A B C D (9) The function f(x, y) = xy 2 x 2 y 2 has a critical point at the origin.

(A) The origin is a saddle point for f(x, y). (B) The origin is a local max for f(x, y).

(D) The origin is not a critical point of the types mentioned above.

4 4 of 16 (8) Consider the function z = sin(x2 + y 2 ) 1 + x 2 + y 2. Which of the surfaces is the graph of that function? A B C D (9) The function f(x, y) = xy 2 x 2 y 2 has a critical point at the origin. Knowing that f xx = 2, f yy = 2x 2, and f xy = 2y, classify this critical point. (A) The origin is a saddle point for f(x, y). (B) The origin is a local max for f(x, y). (C) The origin is a local min for f(x, y). (D) The origin is not a critical point of the types mentioned above. (10) Four functions are given below. For which function does doubling both x and y result in almost tripling 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

5 5 of 16 Section II. Volumes and Center of Mass [/9] 1. Consider the finite ( ) region R enclosed within the 1 x boundaries y = 4, x = 0, y = 0 and x = 3. 2 (a) On the axes to the right, sketch those boundaries, then shade in the region R. Clearly indicated the scales on your axes. y (b) Consider the solid generated by rotating the region R around the line y = 1. Write down a definite integral whose value is the volume of this solid. You do not need to evaluate this integral. x 2. Consider the finite region S under the graph of f(x) = 2 + x + 4 over the interval x = 0 and x = 5. (a) On the axes to the right, sketch f(x), then shade in the region S. Clearly indicated the scales on your axes. y (b) Write an expression using integral(s) for the y-center of mass of the region (ȳ). Assume that the density of the region is a constant ρ g/cm 2, and that all x, y lengths are in cm. You do not need to evaluate the integrals in this expression. x

6 6 of 16 Section III. Money Update [/10] Canada has 100 million toonies (2 dollar coins) in circulation currently, and that quantity remains constant over time. The government has identified a minor flaw in the coin s design that requires replacing all toonies with a new design. They decide that they will provide all banks with the new version of the toonie, and that every old toonie deposited to any bank will be replaced with the new version. Assume that a total of 0.5 million toonies are deposited per day, and 0.5 million are withdrawn per day across all the banks across Canada. If an old toonie is deposited, then it is replaced with a new toonie. If a new toonie is deposited, nothing changes. For simplicity, we assume this process is the same on every day (no differences for weekdays vs weekends, holidays, etc.). Let N(t) = number of new toonies in circulation in Canada at day t. (a) If N(t) is the number of new coins in circulation, how many old coins are in circulation? (b) If N(t) is the number of new coins in circulation, what fraction of the total circulation is in old coins? (c) Knowing N(t) on a given day, how many old coins will be deposited and replaced with new coins? (d) Use this information to write an equation for the rate of change of N(t), where dn dt coins per day: is in millions of dn dt =

7 7 of 16 Section III solving continued (e) Solve your differential equation from part (d) to find a formula for N(t). (f) How many days will it take for the proportion of new toonies in circulation to reach 90% of the total?

8 8 of 16 Section IV. Ecology [/8] A scientist visits a number of apple orchards, where she records the concentrations, a and b, of two pesticides. She also counts the average number of visits by bees to apple trees in that orchard, V. Her observations are tabulated below. The concentrations a and b are in parts per million (ppm). # Bee Visits/day a (conc. in ppm) b (conc. in ppm) (a) Estimate the derivatives V V and at a = 0.4 ppm and b = 3.0 ppm based on the data. Be clear a b about how you are building your estimate. (Note: there are several ways to estimate these values; we will accept any appropriate method.) (b) Construct a linear approximation for the function V (a, b) around a = 0.4 ppm, b = 3.0 ppm. (c) Use your answer to Part (b) to estimate the number of visits by bees you would expect per day if the pesticide concentrations at an orchard were a = 1.0 ppm and b = 4.0 ppm.

9 9 of 16 Section V. Surfaces [/10] Consider the function z = h(x, y) = 3 y x, limited to the region 0 x 8, 0 y 8. 2 (a) On the axes below, draw and clearly label the contours at heights z = 1, 0 and 1. Remember the region is bounded to the domain shown (0 x 8, 0 y 8). y x (b) Evaluate the gradient at the point (2, 1), and draw the gradient vector on your contour diagram from part (a). (c) Compute the directional derivative at (2, 1) in the direction given by the vector 1, 4. (d) Without calculation, identify the (x, y) point where the global minimum of h occurs on the domain shown, 0 x 8, 0 y 8. Explain your answer briefly.

10 10 of 16 Section VI. Manufacturing [/7] A nation s annual steel output P (in thousands of tons) is modelled by the function where P (L, K) = 1.47L 0.6 K 0.4 L is the amount of labour (in thousands of hours) and K is the invested capital (in millions of dollars) used in manufacturing. (a) What are the units of P K? (b) Currently L = 30 and K = 8, but L is decreasing by 0.08 thousand hours per year, and K is increasing by 0.04 million dollars per year. Find the rate of change of national steel production per year. Include units in your answer.

11 11 of 16 Section VII. Container Optimization [/10] A closed rectangular box with an open with faces parallel to the coordinate planes has one bottom corner at the origin and the opposite top corner in the first octant on the plane 3x + 2y + z = 12 What are the dimensions of such a box with the largest possible surface area? (Any box with volume greater than or equal to zero is acceptable.) As part of your solution, verify that your answer is a local maximum for the surface area. z (x,y,z) x y

12 12 of 16 Section VIII. Differential Equations [/7] Consider the differential equation dy dt = 1 x2 y 2. (a) Of the following slope fields, circle the one that matches the differential equation (b) On the slope field you selected, sketch two solutions, using the entire domain shown: One solution passing through (0, 1), and a second solution passing through (1, 0). (c) Use Euler s method with step size t = 0.5 and starting value y(0) = 1 to estimate y(2).

13 13 of 16 Section IX. Lagrange Multipliers [/10] Consider the function f(x, y) = x 2 y 2, and the constraint y = x 2 3. Use the method of Lagrange multipliers to find the location of the maximum and minimum values of f(x, y), subject to the given constraint. Report your final values in the table at the bottom of the page, to 3 digits after the decimal. Min: x y f(x, y) Max:

14 14 of 16 Section X. Vectors [/9] 1. For what value(s) of t (if any) are the vectors 5, t, t 2 and 2, t, 5 perpendicular? 2. For what value(s) of s (if any) are the vectors 4, 6, s and 8, 3s, s 2 parallel? 3. A swimmer wants to cross north to south across a river. The river is flowing at 2 km/hr from the west; the swimmer can swim at 4 km/hr. (a) In which direction should the swimmer swim? Give your answer in a form like e.g. swim 5 degrees east of south. Swimmer Destination 300 m (b) If the river is 300 m across, how long will it take the swimmer to reach the south side?

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

MATH Exam 2 Solutions November 16, 2015

MATH Exam 2 Solutions November 16, 2015 MATH 1.54 Exam Solutions November 16, 15 1. Suppose f(x, y) is a differentiable function such that it and its derivatives take on the following values: (x, y) f(x, y) f x (x, y) f y (x, y) f xx (x, y)