Math 124 Final Examination Winter 2015 !!! READ...INSTRUCTIONS...READ!!!

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1 1 Math 124 Final Examination Winter 2015 Print Your Name Signature Student ID Number Quiz Section Professor s Name TA s Name!!! READ...INSTRUCTIONS...READ!!! 1. Your exam contains 7 problems and 11 pages; PLEASE MAKE SURE YOU HAVE A COMPLETE EXAM. 2. The entire exam is worth 100 points. Point values for problems vary and these are clearly indicated. You have 2 hours and 50 minutes for this final exam. 3. Make sure to ALWAYS SHOW YOUR WORK; you will not receive any partial credit unless all work is clearly shown. If in doubt, ask for clarification. 4. There is plenty of space on the exam to do your work. If you need extra space, use the back pages of the exam and clearly indicate this. 5. You are allowed one sheet of handwritten notes (both sides). 6. NO CALCULATORS are permitted for this exam. Problem Total Points Score Total 100

2 2 1. (15 points) (a) Let f(t) = cos(3t 2 7t). Find f (t) (b) Let g(x) = x 4x e 3+ln(5x). Find g (x) (c) Find ln(sin(x)) lim x π/2 cos(x)

3 2. (14 points) The Scrambler is a popular carnival ride consisting of two sets of rotating arms, which rotate in opposite directions. The position of a person riding the scrambler is the sum of the two circular motions. The coordinates (x,y) of a rider at time t are given by parametric equations: x = 2cos(t)+sin(2t) y = 2sin(t)+cos(2t) (a) Compute the equation of the tangent line to the path of the rider at t = 0. (b) Write down a formula for the rider s speed at time t. No need to simplify yet. 3

4 4 (problem 2 continued) (c) Simplify your answer to part b) and determine all values of t, 0 < t < 2π, at which the speed of the rider is zero. Two helpful formulas: sin 2 (t) + cos 2 (t) = 1; sin(t) cos(2t) + sin(2t) cos(t) = sin(3t) (d) The curve at the right traces out the rider s path. Circle the positions of the rider at the times when the speed is zero. No justification is necessary.

5 5 3. (14 points) The lateral surface area of a cone of radius r and height h (that is, the surface area excluding the base) is: A = πr r 2 +h 2 (a) Find a formula (in terms of r and h) for dr dh A = 15π cm 2. for a cone with lateral surface area (b) Evaluate the derivative of part (a) when r = 3 cm and h = 4 cm. (c) Suppose that the height of the cone decreases 0.1 cm (from 4.0 to 3.9 cm). Use differentials to approximate how much the radius must increase in order keep the lateral surface area of the cone constant.

6 6 4. (16 points) Suppose that f(x) is a continuous function on the interval ( 4, ). The graph of its derivative, f (x), is shown in the picture below. Use it to answer the following questions. Note: This is not the graph of f(x)! It s the graph of f (x). The graph of f (x) (a) List the interval(s) on which the function f(x) is increasing. (b) Determine all x (if any) for which the function f(x) has a local maximum. (c) At x = 6, is the graph of f(x) concave up, concave down or neither? (d) lim x 0 f (x) x = (e) Let g(x) = f(sin(3πx)+2). Compute g (0). (f) Suppose that f(2) = 7. Using the tangent line approximation at x = 2, estimate f(1.9). (g) Is your estimate in the previous part an overestimate or underestimate? Why? (h) Sketch the graph of f (x)

7 7 5. (14 points) Two carts, A and B, are connected by a rope 28 ft long that passes over a pulley P (see the figure). P The point Q is on the floor 12 feet directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 7 ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q? A Q 12 B

8 8 6. (14 points) You wish to make a rectangular enclosure like the one in the figure, with two partitions which separate the enclosure into three sections. The fencing (dotted lines) costs $5 per foot, and the material for the partitions (solid lines) costs $7 per foot. You want the enclosure to have an area of 540 square feet. What should the dimensions of the enclosure be to minimize the cost of construction? Be sure to explain why your result yields the minimum cost. Partitions Fencing

9 9 7. (13 points) Let f(x) = e x cos(x) on the domain D = [ π ] 2,2π. (a) Find the subinterval of D on which f is decreasing. (b) Find the subinterval of D on which f is concave down. There is more space for this part on the next page.

10 10 (problem 7 continued) (c) Find every critical number of f in D and determine whether it corresponds to a local minimum, a local maximum, or neither. (d) Find the global minimum and global maximum values of f in D? Explain.

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