Math 155 Exam 2 Spring 2007 NAME: SECTION: TIME: INSTRUCTOR: Instructions: The exam is closed book and closed notes. You may use an approved calculator, but be sure to show your work on each problem for full credit. Work that is crossed out or erased will not be graded. Turn in any scratch paper that you use during the exam. You will have one hour and 45 minutes to work on the exam. Problem 1 2 3 4 5 6 7 Total Points 20 16 12 14 8 16 14 100 Score 1 1. (20 pts) Find the derivatives of the following functions. You do not have to simplify your answers. Be sure to use parentheses to indicate multiplication where appropriate. (a) f (x) = (1 + x2 )4 (b) f (t) = 2t · t2 (c) f (x) = ln(x) 3x + sin(ex ) 2 (d) f (x) = (cxb )(eax ) (e) f (t) = sin(cos(4 ln(t))) 2 2. (16 pts) Evaluate the following limits. Show all of your work. If you use L’Hospital’s Rule, justify why it can be applied each time you use it. If you use knowledge of leading behaviors, justify your work by explaining all of your steps. e5x + 2 x→∞ x2 + 1 (a) lim (b) lim x→∞ ln(2x) + 7x √ x2 + x + 4 √ (c) lim x→1 x−1 x−1 ln(e3x ) + 4x1/2 x→∞ x2 + 1 (d) lim 3 3. (14 points) Consider the discrete-time dynamical system: xt+1 = xt (xt − r) + r, 3+r r>0 (a) Verify that x∗ = r and x∗ = 3 + r are equilibria. (b) Show that the derivative of the updating function is 1 (2x − r). 3+r (c) Let r = 2. Use the Slope Criterion/Stability Test to determine the stability of each equilibrium. 4 4. (12 pts) Let f (x) = ax3 + x−1 , and a, c > 0. x2 + c (a) Find f∞ (x), the leading behavior of f (x) as x → ∞. (b) Find f0 (x), the leading behavior of f (x) as x → 0. (c) Let a = 2 and c = 4. Use the method of matched leading behaviors to sketch a graph of f (x) on the interval x ≥ 0. Label your axes and indicate where you have graphed f0 (x), f∞ (x), and f (x). 5 5. (8 points) For each item below, fill in the blank with the appropriate letter. Each blank gets one letter only, but letters can be used multiple times. Statement Response (a) f 0 (c) = 0 x = c is an equilibrium if x = c is a stable equilibrium if (b) f 0 (c) > 0 (c) f 0 (c) < 0 (d) f 0 (c) undefined x = c is a critical point or if if (e) f 00 (c) = 0 (f) f 00 (c) > 0 f (c) is a local maximum if and f (x) is increasing at x = c if f (x) is concave down at x = c if (g) f 00 (c) < 0 (h) f (c) = 0 (i) f (c) = c (j) |f 0 (c)| < 0 (k) |f 0 (c)| > 0 (l) |f 0 (c)| < 1 (m) |f 0 (c)| > 1 (n) |f 0 (c)| = 1 6 6. (16 points) Let f (x) = 2 . (x+2) (a) Find the critical points. (b) Write out the intervals of increase and decrease. Justify your answers with calculus. (c) Write the intervals of concave up and concave down. Justify your answers with calculus. (d) Use the information above to sketch a graph of f (x). 7 7. (14 pts) Consider the function f (x) on the interval [0, 10] where f (x) = x3 − 9x2 + 15x − 5 a) Find the critical points of f (x). b) Classify each critical point as a local minimum, local maximum or neither. Justify your answer by using either the first or second derivative test. c) Find the global maximum and global minimum on [0, 10] and give both the x and y coordinates of each.