Name: Date: Unit 4 Exam Unit Quiz: Derivatives Formulas Here is a short list of area and volume formulas that you might need: For a circle of radius r: Area =TTr2, Circumference =2Tr. For a sphere of radius r: Volume =(4/3) Tir3, Surface Area =4TTr2. For a right circular cylinder with radius r and height h: Volume =TTr2h, Surface Area =2Trrh+2TTr2. For a right circular cone with radius r and height h: Volume =(1/3) TTrah. Multiple Choice/Short Answer, Section 1 1. What is the value of g '(0) if y = g(x) has a graph given by: 2 - 0 -2 0 -2- 4moon2. If h(x) has a graph given by: Provide a graph of the derivative on the axis to the right (or copy/paste one from geogebra). 3. If g(x) = csc (x3), then g'(x) = (show work finding derivative) A. 3x2 csc2(x3 ) B. cSC2(x3) C. -3x2csc(x3)cot(x3) D. 3csc2(x3) E. 3x2csc2(x3) 4. The inflection point of the curve y = x3 - 9x2 + 4 is (show work finding derivatives): A. (-1, -6) B. (0, 4) C. (1, -4) D. (3, -50) E. There is no point of inflection. 5. Suppose the functions f and g and their derivatives have the following values at x = 1. Let h(x) = f(x)*g(x). Evaluate h'(1) X f (x ) g(x) f' (x ) g'(x) 2 13 -1 -25 -15 moow -1 11 266. The slope of the tangent line to the graph f (x) = _ at the point (2, 4) is (show work finding derivative): A. -1 B. 0 C. 2 D. -2/3 E. undefined 7. The derivative of: It = x3 + 3x2y -3y2 (show work finding derivative) 8. Which of the following functions is continuous but NOT differentiable at x = 1? 1. f (x) = |x -1/ 1. f(x ) = *+1 xs1 (2x -1 x>1 Ill. f (x) = (x -1)-/3 A. I only B. II only C. I and III only D. II and Ill only E. All of these functions are continuous but not differentiable at x = 1. 9. If f (x) = cos *-"), find f'(n) (show work finding derivative):10. Given the graph of x), sketch the graph of f(x). Use the axes provided at the right or copyipaste a sketch from geogebra. Free Response Question 1 1. Use the function: y = x2 + 2 to answer the following questions". Remember to justify your answers. A. Find the derivative of the function using the limit process (difference quotient). The answer should include all work in solving the I B. Verify your answer from (A) above by using the power rule to find the derivative.\fFree Response Question 2 2. Suppose a math book is thrown down from the top of the tallest tree in California, which is 378 feet tall with an initial velocity of -10 his The position function for tree-falling objects is: 3\") = -16 + vot+ so. A. Determine the position and velocity functions for the book. B. Determine the average velocity of the book on the interval [0, 1]. C. Find the instantaneous velocities when t = 0 and t = 1. D. At what time is the instantaneous velocity of the coin equal to the average velocity of the book found in part B? E. What is the name of the theorem that says there must be at least one solution to part D? F. Find the velocity of the book just before it hits the ground