AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be on the final eam. The eam will be administered on Thursday December 1 during the two hour block period. The eam will consist of multiple choice questions. This is a closed notes and non-calculator eam. 1. Determine the following limit. (Hint: Use the graph of the function.) lim Page 1
. Let, 1 f( ). 1, 1 Determine the following limit. (Hint: Use the graph of the function.) lim f( ) 1. Let f ( ) 1 and g( ). Find the limits: (a) lim f( ) (b) lim g ( ) 1 (c) lim g( f ( )) 4 4. Let f and g( ). Find the limits: ( ) + 4 (a) lim f( ) 1 (b) lim g ( ) 1 (c) lim g( f ( )) Page
. Find the limit: lim sin 6. Find the limit: lim cos 1 6 7. Find the limit: lim tan 4 8. Find the following limit (if it eists). Write a simpler function that agrees with the given function at all but one point. 1 lim 9. Find the following limit (if it eists). Write a simpler function that agrees with the given function at all but one point. lim 9 + 16 9 10. Determine the limit (if it eists): sin 1 cos lim 0 8 11. Determine the limit (if it eists): 4 1cos lim 0 Page
1. Find the -values (if any) at which the function f ( ) 4 1 11 is not continuous. Which of the discontinuitites are removable? 1. Find the -values (if any) at which the function f( ) is not continuous. Which + 4 of the discontinuitites are removable? 14. + 8 Find the -values (if any) at which the function f( ) + 4 is not continuous. Which of the discontinuitites are removable? 1. Find the limit: + 1 lim 16. Find the limit: lim + 0 1 17. Find the derivative of the following function using the limiting process. f ( ) 8 1 18. Find the derivative of the following function using the limiting process. f ( ) +6 7 19. Find the derivative of the following function using the limiting process. f ( ) +8 1 Page 4
0. Find the derivative of the following function using the limiting process. f( ) 4 +10 1. Find the derivative of the following function using the limiting process. f( ) 1. Find the derivative of the following function using the limiting process. f ( ) 7 7. Find the slope of the graph of the function at the given value. f ( ) 4 when 4. Find the slope of the graph of the function at the given value. when 4 f ( ) 4 8. Find the slope of the graph of the function at the given value. f ( ) ( 4) when 6. Find the slope of the graph of the function at the given value. 4 7 7 8 h(z) z z when z 7. Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y ( ) 6 7 Page
8. Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y 4 ( ) 7 9. Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y ( ) 8 9 0. Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y ( ) 6 4 1. Use the product rule to differentiate. f ( t) t 7 t 6. Use the product rule to differentiate. R( t) t cos t. Use the product rule to differentiate. g( v) v sin v 4. Use the product rule to differentiate. 4 g( s) s cos s Page 6
. Use the quotient rule to differentiate. R ( ) 10 6. Use the quotient rule to differentiate. 7 t Pt () t 7. Use the quotient rule to differentiate. g ( ) sin 8. Find the second derivative of the function. f ( ) 7 9. Find the second derivative of the function. Hs () s 4s s 40. Find the second derivative of the function. Q( t) t sect 41. Find dy/d by implicit differentiation. y 16 4. Find dy/d by implicit differentiation. 1y y 16 Page 7
4. Find dy/d by implicit differentiation. 6 7 8 y 9 44. Find dy/d by implicit differentiation. 9 9y y 6 4. Find dy/d by implicit differentiation. 7 7 y y 16 46. Find dy/d by implicit differentiation. sin 9cos 7y 47. Find dy/d by implicit differentiation and evaluate it at the given point. 4y 6,, 4 48. Find an equation of the tangent line to the graph of the function given below at the given point. ( y 10) ( 6), 4.80,.00. (The coefficients below are given to two decimal places.) 49. Find an equation of the tangent line to the graph of the function given below at the given point.,, 9 10y y 4 0 (The coefficients below are given to two decimal places.) Page 8
0. A point is moving along the graph of the function y 6 6 such that d/dt = centimeters per second. Find dy/dt for the given values of. (a) (b) 4 1. A point is moving along the graph of the function y 1 9 7 such that d/dt = 4 centimeters per second. Find dy/dt when 4.. A point is moving along the graph of the function y sin such that d/dt = 8 centimeters per second. Find dy/dt when. 11. Area The radius, r, of a circle is decreasing at a rate of 4 centimeters per minute. Find the rate of change of area, A, when the radius is 6 4. Determine whether Rolle's Theorem can be applied to the function f ( ) ( + )( + )( + 1), 1. If Rolle's Theorem can be on the closed interval applied, find all numbers c in the open interval (, 1) such that f( c) 0. Page 9
. Determine whether Rolle's Theorem can be applied to the function f ( ) ( )( ),. If Rolle's Theorem can be applied, on the closed interval find all numbers c in the open interval (,) such that f( c) 0. 6. Determine whether the Mean Value Theorem can be applied to the function f ( ) on the closed interval [4,10]. If the Mean Value Theorem can be applied, find all f(10) f(4) numbers c in the open interval (4,10) such that f() c. 10 4 7. Determine whether the Mean Value Theorem can be applied to the function f ( ) on the closed interval [0,8]. If the Mean Value Theorem can be applied, find all numbers f(8) f(0) c in the open interval (0,8) such that f() c. 8 0 8. The function line. s( t) t 1t 6 t + 6 describes the motion of a particle moving along a (a) Find the velocity function of the particle at any time t; (b) Identify the time intervals when the particle is moving in a positive direction; (c) Identify the time intervals when the particle is moving in a negative direction; and (d) Identify the times when the particle changes its direction. 9. The function s( t) 4t t describes the motion of a particle moving along a line. (a) Find the velocity function of the particle at any time t; (b) Identify the time intervals when the particle is moving in a positive direction; (c) Identify the time intervals when the particle is moving in a negative direction; and (d) Identify the times when the particle changes its direction. 60. The function s( t) t 1t describes the motion of a particle moving along a line. (a) Find the velocity function of the particle at any time t; (b) Identify the time intervals when the particle is moving in a positive direction; (c) Identify the time intervals when the particle is moving in a negative direction; and (d) Identify the times when the particle changes its direction. Page 10
61. Find the points of inflection and discuss the concavity of the function f ( ) sin + cos on the interval (0, ). 6. Find the points of inflection and discuss the concavity of the function f ( ) 7 cos on the interval [0, ]. 6. Find the limit. 4 8 lim 8 7 64. Find the limit. 6 + 7 lim 7 6. Find two positive numbers with product of 4 and sum that is minimum. 66. Find the length and width of a rectangle that has perimeter 40 meters and a maimum area. 67. Determine the dimensions of a rectangular solid (with a square base) with maimum volume if its surface area is 100 meters. Page 11
68. Find the dimensions of the rectangle of maimum area bounded by the -ais and y-ais and the graph of y. 69. A rectangular page is to contain 169 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used. Page 1
Answer Key 1. 7.. 9, 1, 689 4.,, 1. 1/ 6. 1/ 7. 1 8. 7, + + 9., + 14 10. Does not eist. 11. 0 1. Continuous everywhere 1. Continuous everywhere 14. 8 (Removable), 4 (Not removable) 1. 16. 17. f ( ) 4 8 18. Both A and D 19. f ( ) 1 +16 0. 4 f( ) +10 1. f( ) 4. 7 f( ) 7 7. 478 f () 7 4. 18 f (4). f () 964 6. h'() 4 7 7 8 1 7 8 7. 0 and 4 8. 9. There are no points at which the graph has a horizontal tangent. 0. 4 Page 1
1. 6. 7 t f '( t) 6t t. 4 R'( t) t sin t t cos t. 4 g '( v) v cos v v sin v 4. 4 g '( s) s sin s 4s cos s. 10 R'( ) 10 6. 14tt P'( t) t 7. ( )cos sin g'( ) 8. 8 f ( ) 9. 10 H''( s) s 40. Q"( t) t sec t(0 t sec t 10t tan t t tan t) 41. dy d y 4. dy 1y d y 1 4. 1 7 dy 1 d 8y 44. 4 dy 9 9y d y 9 4. 4 6 dy 7 y 6 7 d 7y 46. dy cos d 6sin 7y 47. dy 9 d 4 4 48. y 0.1 +. 49. y 1.+ 0.67 0. dy 7 dt dy dt 144 Page 14
1. dy 88 dt 801. dy 4cos dt 11. da 48 dt 4. Both A and B. 7 Rolle's Theorem applies; 6. MVT applies; 7 7. 8 MVT applies; 8. (a) v( t) t 4t 6; (b) 0, 6, ; (c),6 ; (d) t = and t = 6. 9. (a) v( t) 4 t ; (b) 0,1 ; (c) 1, ; (d) t = 1. 60. (a) v( t) t 1; (b) 6, ; (c) 0,6 ; (d) t = 6. 61. None of the above 6. Concave downward on, ; concave upward on 0,,,. Inflection points at and 6. 1 64. 0 6., 66. 10, 10 67. 6 Square base side ; height 6 68. Length 1; width 0. 69. 1, 1 Page 1