(Will not be accepted without staple) [First Name] [Last Name] Math-2417, thq 05, 17-09-16, 14:59 [Net ID] [Problem Section] Instructions: 1. Fill in the requested information on the line above. 2. This handout is due at the beginning of problem section. 3. This handout must be printed out and stapled. You may print it double sided. 4. Your work must be hand written on this handout. 5. You must show all work. You may receive zero or reduced points for insufficient work. 6. Your work must be neatly organized and written. You may receive zero or reduced points for sloppy work. 7. Only a subset of these questions will be graded. You will not be told which questions will be graded in advance. Page 2 of 8 Math-2417, thq 05, 17-09-16, 14:59 Questions 1 2 3 4 5 6 7 Total Points 10 10 10 10 10 10 10 30 Score 1. (10 points) (a) Show that the equation 2x3 + 5x 4 = 0 has exactly one real root. (b) Suppose that f (x) is differentiable for all x, that f (3) = 2 and that f 0 (x) 7. Find the maximum value of f (5). Page 3 of 8 Math-2417, thq 05, 17-09-16, 14:59 2. (10 points) A trucker handed in a ticket at a toll booth showing that in 2.5 hours he had covered 213 miles on a toll road with speed limit of 65 mph. The trucker was cited for speeding. Why? Page 4 of 8 Math-2417, thq 05, 17-09-16, 14:59 3. (10 points) Consider the graph of the function f (x) = x2 x + 6. f (x) 6 4 2 5 3 1 1 3 x 2 4 6 (a) Find the equation of the secant line joining points (-2, 4) and (2, 0). (b) Use the Mean Value Theorem to determine a point c in the interval [-2, 2] such that the tangent line is parallel to the secant line. (c) Find the equation of the tangent line through c, and then sketch it in the graph. Page 5 of 8 Math-2417, thq 05, 17-09-16, 14:59 4. (10 points) Use the First Derivative test to find and classify each of the critical points of the function f in the open interval (-3, 3). f (x) = cos x x sin x Page 6 of 8 Math-2417, thq 05, 17-09-16, 14:59 5. (10 points) Find the intervals on which the function f (x) = x3 + 6x2 10 is increasing and decreasing. Sketch the graph of f (x) and identify any local maxima and minima. Any global extrema should also be identified. Page 7 of 8 Math-2417, thq 05, 17-09-16, 14:59 6. (10 points) Find the exact coordinates of the inflection points and critical points marked on the graph of 2 (a) f (x) = x3 2x2 6x 3 (b) f (x) = 4x5 100x3 5 Page 8 of 8 Math-2417, thq 05, 17-09-16, 14:59 7. (10 points) Apply the Second Derivative Test to find the local and global extrema and the points of inflection of f (x) = cos(2x), x . Then determine the open intervals on which the graph is concave upward or concave downward. \f\f\f\f\f\f\f\f\f\f\f\f\f\f