Math 124 Final Examination Autumn Turn off all cell phones, pagers, radios, mp3 players, and other similar devices.

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1 Math 124 Final Examination Autumn 2016 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name Turn off all cell phones, pagers, radios, mp3 players, and other similar devices. This exam is closed book. You may use one sheet of handwritten notes (both sides OK). Do not share notes. No photocopied materials are allowed. Give your answers in exact form, for example π 3 or 5 3. You can use only Texas Instruments TI-30X IIS calculator. In order to receive credit, you must show all of your work. Ifyoudonotindicatethewayinwhich you solved a problem, you may get little or no credit for it, even if your answer is correct. Place a box around your answer to each question. If you need more room, use the backs of the pages and indicate that you have done so. Raise your hand if you have a question. This exam has 9 pages, plus this cover sheet. Please make sure that your exam is complete. Question Points Score Question Points Score Total :28:53 1/16 FinalFall16 (#38)

2 Math 124, Autumn 2016 Final Examination Page 1 of 9 1. (12 total points) Find the derivative of the following functions. (a) (4 points) g(x)=e x2 arctanx (b) (4 points) Suppose that f (0)=π/4 and f (0)=3. Let h(x)=ln(tan( f (x))). Computeh (0). (c) (4 points) y =(3 + 2sinx) 3x :28:53 2/16 FinalFall16 (2/16)

3 Math 124, Autumn 2016 Final Examination Page 2 of 9 2. (12 total points) Compute the following limits. If you apply L Hôpital s rule then you must show that you have checked the hypotheses. ( (a) (4 points) lim x 4 + 7x 2 x 2) x (b) (4 points) lim t 1 1 t + lnt 1 + cos(πt) x (c) (4 points) lim x 2x sinx :28:53 3/16 FinalFall16 (3/16)

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5 Math 124, Autumn 2016 Final Examination Page 3 of 9 3. (12 total points) An object is moving along an ellipse. Its location is given by the parametric equations x(t)=1 + 2cost y(t)=2 + 4sint In this problem we take 0 t 2π. (a) (3 points) Find a formula that gives the slope of the tangent line to the path at time t as a function of t. (b) (4 points) Find the equation of the tangent line at t = π 3. (c) (5 points) Find all the values of t when the tangent line is perpendicular to the line x 2y = 3. 5/16 FinalFall16 (5/16)

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7 Math 124, Autumn 2016 Final Examination Page 4 of 9 4. (12 total points) For this problem, refer to the pictured graph of the function y = f (x) on the interval [-2,12]. f (x) f (7) (a) (2 points) lim = x 7 x 7 (b) (2 points) lim x 2 f (x)= (c) (2 points) lim x 2 f (x)= f (x) (d) (2 points) lim = x 2 x (e) (2 points) Circle the smallest number in this list: f (0) f (1) f (7) f (9) f (11) (f) (2 points) Give an interval (a,b) on which f (x) is increasing. 7/16 FinalFall16 (7/16)

8 Math 124, Autumn 2016 Final Examination Page 5 of 9 5. (12 total points) Consider the plane curve 2x 4 4xy + y 2 = 16. (a) (6 points) Use the tangent line approximation at (0,4) to estimate the value of y when x = (b) (6 points) Find the second derivative d2 y at (0,4) and use this to decide if the tangent line dx 2 approximation is an overestimate or an underestimate near (0,4). Explainyourreasoning. 8/16 FinalFall16 (8/16)

9 6. (12 points) An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is $300,000/km over land to a point P on the north bank and $500,000/km under the river to the tanks. refinery 6km river 2km storage tanks (a) To minimize the cost of the pipeline, where should P be located? Be sure to justify you have found the minimum. (b) What is the resultant minimum cost? 7 9/16 FinalFall16 (9/16)

10 7. (16 points) Consider the function f(x) =x 5 e 2x (a) Determine all horizontal or vertical asymptotes of f(x), if any. (b) Compute f (x). List both the x and y coordinates of all critical points. Determine all the intervals on which f(x) is increasing. This problem continues on the next page. 9 10/16 FinalFall16 (10/16)

11 (c) Compute f (x). List both the x and y coordinates of all inflection points. Determine all intervals on which f(x) is concave up. (d) Sketch the graph of f(x). Label every point on the graph that corresponds to a critical number or an inflection point /16 FinalFall16 (11/16)

12 rocket 5. (14 points) A stationary camera is filming the blast-o of a rocket into space from a safe distance of 1 km. As the rocket rises vertically, the camera s lens rotates upward in order to keep the rocket in sight. When the rocket s elevation reaches 2 km, its velocity is 3/10 km per second. camera α (a) How fast is the angle between the ground and the camera s line of sight (the line from the camera to the rocket) changing at that instant? Include units in your answer. (b) How fast is the distance between the camera and the rocket changing at that instant? Include units in your answer. 7 12/16 FinalFall16 (12/16)

13 3. (9 points) The length L of a rectangle increases by 3 ft/min while the width W decreases by 2 ft/min. When the length is 15 ft and the width is 8 ft, what is the rate at which the following are changing. (Make sure to state whether the rate is increasing or decreasing and include units.) (a) The area A. (b) The length of the diagonal D. 6 13/16 FinalFall16 (13/16)

14 3. (19 points) In this problem you should use the law of cosines: a 2 + b 2 2ab cos θ = c 2, where θ is an angle in the triangle, c is the opposite side, and a and b are the adjacent sides. Two straight roads intersect at point P at a 60 angle. Car A is traveling away from P on one road, and car B on the other road. You re in car A, which has a device that can measure distance from car B and also the rate at which that distance is increasing. At acertainmomentyouhavetraveled3kmawayfrompandaremoving at 80 km/hr. At that time the device shows that your distance from car B is 7 km, and this distance is increasing at 100 km/hr. car B b c P θ a car A At that instant find (a) (8 pts) the distance car B has traveled from P; (b) (11 pts) the speed at which car B is moving. 4 14/16 FinalFall16 (14/16)

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