AP Calculus AB Semester 1 Page 3 of 6 Semester Final Version B Full Name: Instructor Name: Date: (print dearly) Free Response Section 1 (with calculator) You may use your calculator for these questions. 1. The radius, r, of a sphere is increasing at a constant rate of 0.05 meters per second. A. At the time when the radius of the sphere is 12 meters, what is the rate of increase in its volume? (12 points) B. At the time when the volume of the sphere is 36: cubic meters, what is the rate of increase in its surface area? (15 points) C. Express the rate at which the volume of the sphere changes with respect to the surface area of the sphere (as a function of r). (6 points) Copyright @ 2021 Apex Learning. See Terms of Use for further information. Images of the TI-04 calculator are used with the permission of Texas Instruments Incorporated. Copyright @ 2011 Texas Instruments Incorporated.AP Calculus AB Semester 1 Page 4 of 6 Semester Final Version B Full Name: Instructor Name: Date: [print dearly) 2. A particle moves along the x-axis so that its velocity at any time t 2 0 is given by v(t) = (2n - 5)t - sin(nt) A. Find the acceleration at any time t. (6 points) B. Find the minimum acceleration of the particle over the interval [0, 3]. (12 points) C. Find the maximum velocity of the particle over the interval [0, 2]. (12 points) Copyright @ 2021 Apex Leaming. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright @ 2011 Texas Instruments Incorporated.AP Calculus AB Semester 1 page 5 of 6 Semester Final Version B Full Name: Instructor Name: Date: [print dearly) Free Response Section 2 (without calculator) You may not use your calculator for these questions. 3. Consider the function h(x) = a(-2x + 1)5 - b, where a * 0 and b * 0 are constants. A. Find h'(x) and h"(x). (6 points) B. Show that h is monotonic (that is, that either h always increases or remains constant or h always decreases or remains constant). (12 points) C. Show that the x-coordinate(s) of the location(s) of the critical points are independent of a and b. (12 points) Copyright @ 2021 Apex Learning. See Terms of Use for further information. Images of the TI-04 calculator are used with the permission of Texas Instruments Incorporated. Copyright @ 2011 Texas Instruments Incorporated.AP Calculus AB Semester 1 Page 6 of 6 Semester Final Version B Full Name: Instructor Name: Date: (print clearly) ( asinx + b, if x 2n 4. Consider g(x) = (x2 _ xx + 2, if x > 2n A. Find the values of a and b such that g(x) is a differentiable function (12 points). B. Write the equation of the tangent line to g(x) at x = 2x. (9 points) C. Use the tangent line equation from part B to write an approximation for the value of g(6). Do not simplify. (6 points) Copyright @ 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright @ 2011 Texas Instruments Incorporated