Unit # Understanding the Derivative Homework Packet f ( h) f ( Find lim for each of the functions below. Then, find the equation of the tangent line to h 0 h the graph of f( at the given value of. 1. f (. f (. Find the equation of the line tangent to the graph 4. Find the equation of the line tangent to the graph of f ( at = 1. of f ( at = 6. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 119
For problems 5 9, use the function f (. 5. Find f '( by finding lim h 0 f ( h) h f (. 6. Find the slope of the tangent line drawn to the graph of f( at =. 7. Find the slope of the tangent line drawn to the graph of f( at = 1. 8. Find the equation of the tangent line drawn to the graph of f( at = 1. 9. Find f ( f ( a) lim, where a = 1. a a Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 10
10. The line defined by the equation y ( ) is tangent to the graph of g( at =. What is g( g( ) the value of lim? Show your work and eplain your reasoning. Use the graph of f( pictured to the right to perform the actions in eercises 11 15. Give written eplanations for your choices. 11. Label a point, A, on the graph of y = f( where the derivative is negative. 1. Label a point, B, on the graph of y = f( where the value of the function is negative. 1. Label a point, C, on the graph of y = f( where the derivative is greatest in value. 14. Label a point, D, on the graph of y = f( where the derivative is zero. 15. Label two different points, E and F, on the graph of y = f( where the values of the derivative are opposites. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 11
16. Match the points on the graph of g( with the value of g '( in the table. Value of g '( 1 0 ½ 1 Point on g( 17. The function to the right is such that h(4) = 5 and h '(4) = 1.5. Find the coordinates of A, B, and C. For eercises 18 0, use the function 1 f (. 1 18. Find f '(. 19. Find the equation of the tangent line drawn to the graph of f( at = 0. 0. Find the equation of the normal line drawn to the graph of f( at = 0. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 1
1. Given below are graphs of four functions f(, g(, h(, and p(. Below those graphs are graphs of their derivatives. Label the graphs below as f '(, g '(, h '(, and p '(. The table below represents values on the graph of a cubic polynomial function, h(. Use the table to complete eercises 4. 1 0 1 4 h( 4 0 8 6 0 4 18. Two of the zeros of h( are listed in the table. Between which two values of does the Intermediate Value Theorem guarantee that a third value of eists such that h( = 0? Eplain your reasoning.. Estimate the value of h '(1.5). Based on this 4. Estimate the value of h '( 1.75). Based on value, describe the behavior of h( at = 1.5. this value, describe the behavior of h( at Justify your reasoning. = 1.75. Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 1
For eercises 5 6, find the derivative of each function. Leave your answers with no negative or rational eponents and as single rational functions, when applicable. 5. f ( 5 6. h( 7. h( 7 8. g( 8 5 9. f ( ) cos 0. h( 1. g ( ) sin. p( Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 14
. g ( ( )( 1) 4. h( 5. f ( 6. h( 6 cos 7. For what value(s) of will the slope of the tangent line to the graph of h( 4 be? Find the equation of the line tangent to h( at this/these values. Show your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 15
8. Find the equation of the line tangent to the graph of g( when = 1. 4 9. The line defined by the equation 1 ( y ) is the line tangent to the graph of a function f( when = a. What is the value of f '( a)? Show your work and eplain your reasoning. 40. The line defined by the equation y ( ) is the line tangent to the graph of a function f( at the point (, ). What is the equation of the normal line when =. Eplain your reasoning. 4 41. Determine the value(s) of at which the function f ( 8 has a horizontal tangent. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 16
4. Determine the value(s) of θ at which the function f ( ) cos has a horizontal tangent on the interval [0, π). 4. For what value(s) of k is the line y = 4 9 tangent to the graph of f( = k? For eercises 44 46, determine on what intervals the given function is increasing or decreasing. Also, identify the coordinates of any relative etrema of the function. Show your work and justify your reasoning. 44. f( = + 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 17
45. g( = 6 + 15 46. h( = ( + ) ( 1) 47. Pictured to the right is the graph of f '(. On what interval(s) is the graph of f( increasing or decreasing? Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 18
48. Pictured to the right is the graph of f '(. At what value(s) of does the graph of f( have a relative maimum/minimum? Justify your reasoning. 49. If g '( ( ) ( 1), determine on what intervals the graph of g( is increasing or decreasing and identify the value(s) of at which g( has a relative maimum or minimum. Justify your reasoning and show your work. For eercises 50 5, use the graph of t function, h(, pictured to the right. Use the graph to identify the following. Provide written justification. 50. On what interval(s) is h '( < 0? 51. On what interval(s) is h '( > 0? 5. At what value(s) of does h' ( change from positive to negative? From negative to positive? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 19
1 Consider the quadratic function f ( 4. 5. Sketch an accurate graph of the function. 54. Find f '( and use it to find the absolute maimum of the graph of f(. 55. Estimate the value of f '(0) and eplain what this value represents in terms of the graph of f(. 56. Find the equation of the tangent line to the graph of f( at = 0. Draw a graph of this line. 57. Sketch a graph of the normal line to the tangent line at = 0. What is the equation of this line? 58. Use the equation of the tangent line to approimate f(0.1). Then, find f(0.1) using the equation of f(. Is the approimation an under or over approimation of the actual value of f(0.1)? Based on the graph of f(, why do you suppose this is true? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 10
59. For what function does sin( h) sin lim give the derivative? Find the limit. h 0 h 60. Find 5 5 ( h) lim h 0 h h. 61. Find lim. h 0 h 6. If f (, what is the slope of the normal line to the graph of f( when = 4? 6. If = 5(y + 1) is the equation of the normal line to the graph of f( when = a, find the value of f '( a). Show your work and eplain your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 11
64. On the interval [0, π), find the coordinates of the relative minimum(s) of f ( ) sin. The derivative of a function f( is f '( ( ( 5). Use this derivative for eercises 65 and 66. 65. At what value(s) of does the graph of f( have a relative maimum? Justify your answer. 66. Use the equation of the tangent line to approimate the value of f(.1) if f() =. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 1