10 Questions in Exam 2 - Calculus | MATH 1120, Exams of Calculus

Download 10 Questions in Exam 2 - Calculus | MATH 1120 and more Exams Calculus in PDF only on Docsity! Math 1120 Calculus Test 2. October 18, 2001 Your name The multiple choice problems count 4 points each. In the multiple choice section, circle the correct choice (or choices). You must show your work on the other problems 5 through 10. The total number of points available is 131. 1. Questions (a) through (e) refer to the graph of the function f given below. −1−2−3−4 1 2 3 4 1 2 3 4 −1 −2 −3 −4 ◦ ◦• • .... ..... .... ... ... .... .... .... ................................................................................ ..... .... ..... ..... ............................... (a) lim x→3 f(x) = (A) 0 (B) 1 (C) 2 (D) 4 (E) does not exist (b) lim x→2− f(x) = (A) 0 (B) 1 (C) 2 (D) 4 (E) does not exist (c) A good estimate of f ′(−2) is (A) −1 (B) 0 (C) 1 (D) 2 (E) there is no good estimate (d) A good estimate of f ′(−1) is (A) −1 (B) 0 (C) 1 (D) 2 (E) there is no good estimate (e) A good estimate of f ′(3) is (A) −1 (B) 0 (C) 1 (D) 2 (E) there is no good estimate 2. The line tangent to the graph of a function f at the point (2, 3) on the graph also goes through the point (−1, 9). What is f ′(2)? (A) −2 (B) −1 (C) 0 (D) 1 (E) 2 1 Math 1120 Calculus Test 2. 3. What is the slope of the tangent line to the graph of f(x) = (3x)−1 at the point (1,1/3)? (A) −2/9 (B) −1/3 (C) −2/27 (D) −2/81 (E) 0 4. True-false questions. These count 2 points each. (a) True or false. If f ′(x) > 0 for each x in the interval (−1, 1), then f is increasing on (−1, 1). (b) True or false. If f(a) < 0, f(b) > 0, and f ′(x) > 0 for each x in (a, b), then there is one and only one number c in (a, b) such that f(c) = 0. (c) True or false. The graph of a function cannot touch or intersect a hori- zontal asymptote to the graph of f . (d) True or false. If f ′(c) = 0, then f has a relative maximum or a relative minimum at x = c. (e) True or false. If f has a relative maximum or a relative minimum at x = c, then f ′(c) = 0. (f) True or false. If f ′(c) = 0 and f ′′(c) < 0, then f has a relative maximum at x = c. (g) True or false. If f and g are differentiable, then d dx [f(x)g(x)] = f ′(x)g′(x). (h) True or false. If f and g are differentiable, then d dx [f(x) g(x) ] = f ′(x) g′(x) . (i) True or false. If f and g are differentiable and h(x) = f ◦ g, then h′(x) = f [g(x)]g′(x). (j) If f and g are differentiable and a and b are constants, then d dx [af(x) + bg(x)] = a d dx f(x) + b d dx g(x). 2 Math 1120 Calculus Test 2. 8. (15 points) (a) State the hypothesis of the Intermediate Value Theorem (IVT). (b) State the conclusion of the Intermediate Value Theorem. (c) Does the function f(x) = √ x + 4 satisfy the hypothesis of IVT over the interval [0, 12]. If so, find a whole number M between f(0) and f(12), and then find a number c in the interval (0, 12) such that f(c) = M . 9. (12 points) Suppose the functions f and g and their derivatives are given by the table of values shown. Complete the table by calculating the values of the derivatives of both f ◦g(x) and g ◦f(x) for each of the values of x in the table. x f(x) f ′(x) g(x) g′(x) df◦g(x) dx dg◦f(x) dx 0 2 3 1 3 1 3 4 5 2 2 2 1 1 4 3 5 3 4 1 4 4 1 3 2 5 2 0 0 4 5 Math 1120 Calculus Test 2. 10. (30 points) Compute the following derivatives. (a) Let f(x) = x2 − (1/x). Find d dx f(x). (b) Let g(x) = √ 3x3 + 4. What is g′(x)? (c) Find d dx ((2x + 1)3 · (3x2 − 1)) (d) Find d dx 2x+1 x2+2 (e) Find d dt (t−3 + t−2)3. 6