Unit #3 Rules of Differentiation Homework Packet
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1 Unit #3 Rules of Differentiation Homework Packet In the table below, a function is given. Show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified method. f ( 1. 3 Rewrite f( as a polynomial first. Then apply the power rule to find f '(. f (. 3 Apply the product rule to find f '(. For eercises 3 8, find the derivative of each function. 3. f ( 5. f ( f ( Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 193 Mark Sparks 01
2 Find the slope of the normal line drawn to the graph of each function at the indicated value of. 6. g( sin when = π 7. h( sin (sin cos when = 4 For each of the functions below, find the equation of the tangent line drawn to the graph of g( at the indicated value of. 8. ( ) 4 g when = 4 9. g( cos when = Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 194 Mark Sparks 01
3 Use the table below to complete eercises f( f ( g( g ( If H( f ( g(, what is the equation of the tangent line when = 1? 11. If J( g( sin, what is the value of J '(0)? 1. If ( 4 f ( g( K, what is the slope of the normal line when =? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 195 Mark Sparks 01
4 For eercises 13 and 14, show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified method f ( 3 Rewrite f( as a polynomial first. Then apply the power rule to find f '(. 14. f ( Apply the quotient rule to find f '( Find the equation of the line tangent to the graph of g ( when = Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 196 Mark Sparks 01
5 Find the derivative of each of the following functions. 16. h ( h ( cos g ( ) (1 sin ) f ( ) cos Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 197 Mark Sparks 01
6 Use the table below to complete eercises 0. f( f ( g( g ( f ( 0. If H(, what is the equation of g( the tangent line when = 1? 3 cos 1. If J(, what is the value of f ( J '(0)?. If 4 f ( K(, what is the slope of the normal line when =? 3 g( Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 198 Mark Sparks 01
7 3. If f ( ) csc sin, show that f '( ) cot cos. 4. Find the equation of the line tangent to the graph of f ( ) tan sin when. 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 199 Mark Sparks 01
8 In eercises 5 30, find the derivative of each of the following functions f 6. ( f ( 3 7. h ( g ( 9 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 00 Mark Sparks 01
9 9. f ( p ( 1 For eercises 31 and 3, find the value of the derivative of the function at the given point. 31. ( ) 1 g sin when 3. f ( ) sin cos when 4 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 01 Mark Sparks 01
10 The graph of the function f ( 5 is pictured to the right. Use this function to complete eercises 33. Find the values of f (3) and f '(3). 34. Find the equation of the line tangent to the graph of f( when = 3 and graph this line on the grid with f(. 35. Find the equation of the normal line to the graph of f( when = 3 and graph this line on the grid with f(. 36. At what value(s) of does the graph of h ( 1 1 have a horizontal tangent? Show your work and give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 0 Mark Sparks 01
11 Use the table below to complete eercises f( f ( g( g ( If H( f ( g(, is the graph of H( increasing or decreasing when = 1? Give a reason for your answer. 38. If P( f ( g( 3, what is the value of P '(0)? 39. Find the equation of the normal line to the graph of h( tan(3 when 1. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 03 Mark Sparks 01
12 In eercises 40 49, find the derivative of the function. Epress your answer in simplest factored form. 40. F 3 ( e 41. P( e 4. H ln ( e 43. g( 3e 44. ( ) J lne F( ln(3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 04 Mark Sparks 01
13 46. K ( ln F( e ln T ( 49. e P( Find the equation of the tangent line to the graph of ln y when = 1. 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 05 Mark Sparks 01
14 Use the graph of H(, pictured above, to complete eercises Graphically, identify a value of at which the function is continuous but not differentiable. Give a reason for your answer. 5. Write an equation of H( and show analytically that H( is, in fact, continuous at the value that you identified in eercise 51. Show and eplain your work. 53. Show analytically that H( is, in fact, not differentiable at the value that you identified in eercise 51. Show and eplain your work. 54. Given the graph of H( pictured above, find the equation of the tangent line to the graph of P( H( when = 3. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 06 Mark Sparks 01
15 A continuous function, h(, is described in the table below. Use the information to complete eercises < < < < 3 3 < < 5 5 h( Increasing & Concave Up 1 Decreasing & Concave Up Increasing & Concave Up Sketch a graph of h(. 56. Estimate the values of h' ( ) and h '(1). Do these values support the claim that h( is increasing on the interval 4 < < 0? Give a reason for your answer. 57. Determine the values at which h( is not differentiable. Give a reason for why h( is not differentiable at this/these value(s). 58. On what interval(s) of is h '( 0? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 07 Mark Sparks 01
16 59. At what value(s) of will the graph of f( = e 3 have a tangent line whose slope is 1? 60. The graph of y = 9 is parallel to the normal line to the graph of f( when = 5. What is the value of f '(5)? Justify your answer. 61. Let f be defined by the function 3, 1 f (. a b, 1 a. If the function is continuous at = 1, what is the relationship between a and b? Eplain your reasoning using limits. b. Find the unique values of a and b that will make f both continuous and differentiable at = 1. Show your analysis using limits. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 08 Mark Sparks 01
17 For questions 6 65, refer to the table of values to the right. f f ' g g ' Find the value of [ f ()]'. 63. Find the value of [ g (1)]' 64. Find the equation of the line tangent to the graph of 1 g when = if 3 g '( ). 65. If h( = g(f(), what is the equation of the line normal to the graph of h when =? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 09 Mark Sparks 01
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 119 Mark Sparks 2012
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