AP Calculus Unit 2 Exam 2012 PDF

AP CALCULUS TEST #2 Unit #2 – Conceptualizing the Derivative Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQU...

AP CALCULUS TEST #2 Unit #2 – Conceptualizing the Derivative Name______________________________________________________Date_____________________ 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 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 173 Mark Sparks 2012  xh  x 2. Find lim. h 0 h A. x B. – x x 1 C. D. 2 2 x 2 E. x 3. If g ' (1)  3 , then which of the following could be the equation for g(x)? x3  2 x 2  4 x I. g ( x)  2 x 2  7 x  3 II. g ( x)  4 x  5x III. g ( 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. Graph of f(x) 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. I and III only B. I only C. III only D. I and II only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 174 Mark Sparks 2012  x2  2x 5. Which of the following would represent f ' ( x) if f ( x)  ? x 3x  1 3x  2 A. B. x 2 x 3 x 1 1 C. D. 2 x 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 C. III only D. I, II, and III E. II and 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 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 175 Mark Sparks 2012  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? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 176 Mark Sparks 2012 c. Using your graphing calculator, sketch a graph of f ' ( x) on the axes below on the interval 0 < x < 2π. Then, determine the value(s) of x at which the graph of f(x) reaches a relative maximum or minimum. Justify your answers. d. Based on the graph of f ' ( x) , on what open interval(s) within the interval 0 < x < 2π is f(x) increasing? Decreasing? Justify your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 177 Mark Sparks 2012 AP CALCULUS *TEST #2* Unit #2 – Conceptualizing the Derivative Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (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 Not Permitted Section The derivative, g ' , of a polynomial function g is continuous and has exactly two zeros. Selected values of g ' are given in the table below. Use the table to answer questions 1 – 2. 8. On which of the following intervals is the graph of g(x) increasing? I. x < −2 II. −2 < x < 2 III. x > 2 A. I only B. II only C. III only D. I and III only E. I and II only 9. At what value(s) of x does the graph of g(x) reach a relative maximum? A. 2 only B. −2 and 2 only C. −2 only D. 0 and 2 only E. g(x) does not have a relative maximum. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 178 Mark Sparks 2012 10. The graph of a function, h(x) is shown to the right. Which of the following conclusions can be made about the derivative of h, h ' ( x) ? A. h ' ( x) > 0 when x = −3. B. h ' ( x) = 0 when x = −2. C. h ' ( x) < 0 when x = 2 D. Both A and B are valid conclusions. E. Both B and C are valid conclusions. 11. Let f be the function defined by f ( x)  4 x3  5x  3. Which of the following is an equation of the line tangent to the graph of f at the point where x = –1? A. y = 7x – 3 B. y = 7x + 5 C. y = 7x + 11 D. y = –5x – 1 E. y = –5x – 5 12. The function f is defined on the closed interval [0, 8]. The graph of its derivative, f ' , is pictured below. If f(3) = 5, then what is the equation of the tangent line to the graph of f when x = 3? A. y = 2 B. y = 5 C. y – 5 = 2(x – 3) D. y + 5 = 2(x – 3) E. y + 5 = 2(x + 3) 3x 13. If h(x) = 3 , then what is the slope of the normal line to the graph of h when x = –8? x 1 A. 4 B. –4 C. –1 D. 12 E. 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 179 Mark Sparks 2012 14. The graph of f ( )    2 sin  , for 0 < θ < 2π, has a relative maximum at… A.    only 3 B.   4 only 3 C.   2 only 3 D.   2 and 4 3 3 E.   5 and 7 6 6 Free Response Consider the function f(x) = 3x2 – x3. Determine each of the following properties of the graph of f(x). a. Determine the interval(s) where f(x) is increasing or decreasing. Justify your answers. b. Determine the coordinates of any relative maximums or minimums of f(x). Justify your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 180 Mark Sparks 2012 c. If g(x) = (2 − x2), then what is the derivative of the function h( x)  f ( x)  g ( x). Show your work. d. Lillian questions if the derivative of the product of two functions is equivalent to the product of the derivatives of the two functions. In other words, if h( x)  f ( x)  g ( x) then is h ' ( x)  f ' ( x)  g ' ( x) ? Using f(x) and g(x), show and tell her the answer to her question. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 181 Mark Sparks 2012