Conceptualizing the Derivative | Amherst College

University:
Amherst College
Course:
MATH 111 | Introduction to the Calculus
Academic year:

2022
Views:

202

Pages:

4
Author:

5prinzg1s

AP CALCULUS TEST #2 Unit #2 – Conceptualizing the Derivative A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice − Calculator Permitted Section 1. If f ( x)  3x( x  2) 2 , then what is the slope of the tangent line to the graph when x = –1? A. 2 B. –3 C. 1 D. 3 E. –2 xh  x . h h 0 2. Find lim A. x B. – x C. x 2 D. E. 2 x 1 2 x 3. If g ' (1)  3 , then which of the following could be the equation for g(x)? I. g ( x)  2 x 2  7 x  3 II. g ( x)  4 x  5x III. g ( x)  x3  2 x 2  4 x x2 A. I only B. I and II only C. II only D. II and III only E. I, II and III 4. The graph of a polynomial function, f(x), is pictured to the right. Which of the following statements is/are true about f ' ( x) ? I. f ' ( x) < 0 on the interval (–1, 2). II. f ' ( x) changes from negative to positive when x = –1. III. There are two values of x such that f ' ( x) = 0. A. B. C. D. E. I and III only I only III only I and II only I, II, and III Graph of f(x) 5. Which of the following would represent f ' ( x) if f ( x)  A. 3x  1 x B. C. 3 x 1 2 x D. x2  2x ? x 3x  2 2 x 1 x E. 4 x 3  4 x 6. The graph pictured to the right is the graph of f ' ( x) , the derivative of a polynomial function, f(x). Which of the following statements is/are true? I. f(x) is increasing on the interval (–5, 2). II. f(x) has a relative minimum when x = –5. III. The slope of the normal line drawn to f(x) at x = –4 is  1 . 3 A. II only B. I and II only D. I, II, and III E. II and III only C. III only 7. If g ' ( x)  3x( x  2) 2 , then the graph of g(x) has a relative maximum at what value(s) of x? A. 0 only B. –2 and 0 only C. –2 only D. –2 and  2 only 3 E. g(x) never reaches a relative maximum Free Response Consider the function f ( x)  2 x  3 cos x on the interval 0 < x < 2π. The graph of f(x) is shown to the right. Answer the following questions rounding all values of x to three decimal places. a. Show, algebraically, that f ' ( x)  1 3 x sin x . Make sure you show each step of your work. x b. Based on the graph of f(x), will the slope of the normal line drawn to the graph of f at x = 4 be positive or negative?

Description

Related Documents

Almost There!

Free up your schedule!

Take 5 seconds to unlock