Math 124 Final Examination Winter 2016 !!! READ...INSTRUCTIONS...READ!!!

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1 Math 124 Final Examination Winter 2016 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 9 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. If you use a trial and error (or guess and check) method when an algebraic method is available, you will not receive full credit. 5. 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. 6. You are allowed one sheet of handwritten notes (both sides). 7. Only TI30X-IIS CALCULATORS are permitted for this exam. Problem Total Points Score Total 100 1

2 1. (15 points) Compute the following limits, showing your work and/or explaining your answers. x (a) lim x 0 sin(πx) (b) lim x 3 e x 3 1 x 2 4x+2 (c) lim x 4x 16x 2 +2x+1 Compute the derivatives of the following functions. You need not simplify your answers. (d) f(x) = x π +π x +cos(πx) e π (e) f(x) = (tanx) x2 2

3 2. (14 points) The parametric equations for the path of a particle are given by for all values of t. x(t) = cos(πt)+t 2, y(t) = 2(t 1)sin((t+1)π), (a) The particle passes through the origin when t = 1. Find the equation of the line tangent to the particle s path at that time. (b) Is the horizontal velocity of the particle increasing or decreasing when t = 1? (c) Find a second time t 2 when the particle passes through the origin again. (d) Is the particle s speed greater at t = 1 or at t = t 2? 3

4 3. (14 points) The drawing shows a region R bounded by four semicircles. The dashed sides of the x y rectangle are the diameters of the semicircles. Suppose that at first x = 4 cm and y = 6 cm. If you decrease x to 3.8 cm, what is the new value you should choose for y in order for the area of R to remain the same? Use implicit differentiation and linearization (also called the tangent line approximation and the method of differentials ). Give your answer to the nearest cm. You must use implicit differentiation and linearization to do this problem. x y 4

5 4. (14 points) Suppose f(x) is a continuous function in the interval [-4,8]. The graph of the DERIVATIVE function y = f (x) is shown. Use the graph, and the information below to answer the following questions y = f (x) f(0) = 0 f(1) = 4/ No partial credit for this problem. Each part is worth two points (a) Find all critical numbers for f(x). (b) Find all local maxima forf (x values). If there are no local maxima, write NONE. (c) Circle the larger value : f(1) or f(7) (d) For what x in the interval [0,8] does f(x) have an absolute maximum? (e) Determine all inflection points for f(x). (f) Using the linear approximation of f at x = 1, estimate of f(1.1). (g) If G(x) = sin(πf(x)), calculate G (1). 5

6 5. (14 points) Tank A has the shape of an inverted cone. It has height 8 meters and the radius at the top is 2 meters. It is full of water. Tank B has the shape of a cylinder with circular base of radius 3 meters. It is empty. The water is to be pumped from Tank A into Tank B. The water level in Tank A is dropping at a rate of 15 centimeters/minute when the height of the water is 5 meters. How fast is the water rising in Tank B at that time? 2.0 meter radius 3.0 meter radius Recall that the volume of a right circular cone is 1 3 πr2 h. 6

7 6. (14 points) You are using metal rods to form a frame for a rectangular solid with square top and bottom. The rods that form the vertical edges cost $2/foot, and the rods that form the horizontal edges of the top and bottom cost only $1/foot. Find the dimensions of the rectangular solid of greatest volume that you can construct if the budget for the frame is $40. Be sure to verify that your answer has maximum volume. h w w 7

8 7. (15 points) Let f(x) be the function y = f(x) = 1+4xe 1 8 x2 (a) Find all intervals over which f(x) is decreasing. (b) Find all intervals over which f(x) is concave up. This problem continues on the next page. 8

9 Continuing from the previous page y = f(x) = 1+4xe 1 8 x2. (c) Give the equation(s) for any vertical or horizontal asymptote(s) for f(x). (d) Sketch the graph of f(x) using the grid below. Clearly label both the x and y coordinates of all critical points and all points of inflection. (You may label the coordinates with approximations accurate to two digits after the decimal)