Exam on Hawkes

FALL 2020 – Calculus Final Exam Review (Answer Key) 1. Consider the following one-sided limit. → 18− ( 2 − 324 − 18 ) Step 1. Approximate the limit by filling in the table. Round to the nearest thousandth. Step 2. Determine the value of the one-sided limit. Answer: ____________________ 2. Use the graph of = ( ) to find the limits: Step 1. Find → −3 − ( ). Answer: _______________ Step 2. Find → 1 − ( ). Answer: ____________________ Step 3. Find → −1 − ( ). Answer: ____________________ Step 4. Find → −1 + ( ). Answer: ____________________ 3. Use the graph to find the indicated limits. Step 1. Find → −1 − ( ). Answer: ____________________ Step 2. Find → −1 + ( ). Answer: ____________________ Step 3. Find → −1 ( ). Answer: ____________________ 4. Find the limit algebraically by factoring the expression first. → 2 ( 4 2 − 3 − 10 − 2 ) Answer: ____________________ 5. Consider the graph of ( ). What is the average rate of change of ( ) from 1 = 4 to 2 = 7? I need help writing my essay – research paper write your answer as an integer or simplified fraction. Answer: ____________________ 6. ( ) is the velocity in meters per hour a snowmobile is traveling at hours. Which of the following does the slope ′ ( ) represent? A) Direction the snowmobile is traveling at x hours. B) Average rate of change in the velocity of the snowmobile in meters per hour at x hours. C) Distance in meters the snowmobile has traveled in x hours. D) Rate of change of the velocity of the snowmobile at x hours. 7. Consider the function ( ) = 4 3 − 2 2 − + 8. Step 1. Interpret the meaning of (1) = 9. A) The average rate of change from = 0 to = 1 is 9. B) The slope of the tangent line at = 1 is 9. C) The slope of the secant line from (−1) to (1) is 9. D) The value of the function evaluated at = 1 is 9. Step 2. Interpret the meaning of ′ (1) = 7. A) The slope of the curve at = 1 is 7. B) The average rate of change from = −1 to = 1 is 7. C) The height of the function at = 1 is 7. D) The average rate of change from = 0 to = 1 is 7. 8. Find the derivative for the following function. = − 3 7 Answer: ____________________ 9. Find the derivative for ( ) = 5 2 + 2 3 Answer: ____________________ 10. Find the derivative for ( ) = 5 3 + 3 − 2 5 Answer: ____________________ 11. Use algebraic techniques to rewrite ( ) = (3 + 5)(3 + 4) as a sum or difference; then find ′ ( ). Answer: ____________________ 12. Use algebraic techniques to rewrite ( ) = −3 6 + 4 3 2 as a sum or difference; then find ′ ( ). Answer: ____________________ 13. For the function ( ) = −7 3 − 8 2 − 3 , Step 1. Find the slope of the tangent line at = −5. Answer: ____________________ Step 2. Find the equation of the tangent line at = −5. Answer: ____________________ 14. A rock is falling. It is ( ) = −16 2 + 410 feet off the ground after seconds. Step 1. Find the instantaneous rate of change of the rock’s position at = 2 seconds. Answer: ____________________ Step 2. When will the rock be 394 feet off the ground? Answer: ____________________ 15. For the function ( ) = 2 3 + 3 2 − 7 , find the slope of the tangent line at = −3. Answer: =_______________ 16. Find the limit. → −9 − (5 2 + 6) Answer: ____________________ 17. Find the limit. → −2 − ( 18 − −19 + 2 ) Answer: ____________________ 18. Find the limit. → +∞ ( 15 11 2 + 10) Answer: ____________________ 19. Find the limit. → −∞ ( −5 2 − 3 2 2 + 9 ) Answer: ____________________ 20. Find the limit. → −∞ ( 3 − 27 2 + 3 + 9 ) Answer: ____________________ 21. Find the limit. → −3 + (−18 + 10 + 3 ) Answer: ____________________ 22. Find the limit. → −9 (√−10 + 14 + 16) Answer: ____________________ 23. Find the limit. → +∞ ( −3 2 + 16 + 4 5 3 + 4 2 + 2 + 5 ) Answer: ____________________ 24. Consider the function ( ) = { −4 < 2 7 − 18 ≥ 2 . Step 1. Find → 2 − ( ). Answer: ____________________ Step 2. Find → 2 + ( ). Answer: ____________________ Step 3. Find → 2 ( ). Answer: ____________________ 25. Use the graph of = ( ) to answer the question regarding the function. Step 1. Find → 1 − ( ). A) ____________ B) Does Not Exist Step 2. Find → 1 + ( ). Answer: ____________________ Step 3. Find (1). Answer: ____________________ Step 4. Is ( ) continuous at = 1? A) Yes B) No 26. Use the graph of = ( ) to answer the question regarding the function. Step 1. Find → −2 − ( ). A) ____________ B) Does Not Exist Step 2. Find → −2 + ( ). Answer: ____________________ Step 3. Find (−2). Answer: ____________________ Step 4. Is ( ) continuous at = −2? A) Yes B) No 27. Consider the following function: ( ) = { 4 2 − 9 − 3 < −4 5 2 − 6 + 2 ≥ −4 Step 1. At what -value is the function discontinuous? Answer: ____________________ Step 2. What type of discontinuity is at the discontinuous point? A) Non-Removable Discontinuity B) Removable Discontinuity C) Jump Discontinuity 28. Use the Product Rule or Quotient Rule to find the derivative. ( ) = (− 3 − 7)(−2 −1 + 6) Answer: ′ ( ) =_______________ 29. Use the Product Rule or Quotient Rule to find the derivative. ( ) = 8 3 + 16 2 − 16 − 32 2 + 4 Answer: ′ ( ) =_______________ 30. Given (−6) = −2, ′ (−6) = −18, (−6) = −12, and ′ (−6) = 7, find the value of ℎ ′ (−6) based on the function below. ℎ( ) = ( ) ( ) Answer: ℎ ′ (−6) =_______________ 31. Use the Product Rule or Quotient Rule to find the derivative. ( ) = (3 3 + 9)(3 4 − 1) Answer: ′ ( ) =_______________ 32. Use the Product Rule or Quotient Rule to find the derivative. ( ) = 3 6 − 2 4 3 − 5 Answer: ′ ( ) =_______________ 33. Find the derivative for the given function. Ace my homework – Write your answer using positive and negative exponents instead of fractions and use fractional exponents instead of radicals. ℎ( ) = (7 4 + 9) 3 Answer: ____________________ 34. Find the derivative for the given function. Ace my homework – Write your answer using positive and negative exponents instead of fractions and use fractional exponents instead of radicals. ( ) = ( 4 + 4 −9 2 + 10) 3 Answer: ____________________ 35. Consider the function. ( ) = − 2 + 6 − 6 Step 1. Find all values of that correspond to horizontal tangent lines. Select “None” if the function does not have any values of that correspond to horizontal tangent lines. Answer: ____________________ Step 2. Determine the open intervals on which the function is increasing and on which the function is decreasing. Enter ø to indicate the interval is empty. Answer: ____________________ 36. Consider the function. ( ) = 3 3 − 18 2 + 36 − 26 Step 1. Find all values of that correspond to horizontal tangent lines. Select “None” if the function does not have any values of that correspond to horizontal tangent lines. Answer: ____________________ Step 2. Determine the open intervals on which the function is increasing and on which the function is decreasing. Enter ø to indicate the interval is empty. Answer: ____________________ 37. Consider the function. ( ) = + 7 − 1 Step 1. Find all values of that correspond to horizontal tangent lines. Select “None” if the function does not have any values of that correspond to horizontal tangent lines. Answer: ____________________ Step 2. Determine the open intervals on which the function is increasing and on which the function is decreasing. Enter ø to indicate the interval is empty. Answer: ____________________ 38. Consider the function: ( ) = 3(−3 2 + 48) 2 + 4 Step 1. Find the critical values of the function. Separate multiple answers with commas. Answer: ____________________ Step 2. Use the First Derivative Test to find any local extrema. Enter any local extrema as an ordered pair. Answer: ____________________ 39. Consider the function: ( ) = 3 − 3 2 − 24 + 4 Step 1. Find the critical values of the function. Separate multiple answers with commas. Answer: ____________________ Step 2. Use the First Derivative Test to find any local extrema. Enter any local extrema as an ordered pair. Answer: ____________________ 40. Consider the function ( ) = 4 3 − 108 on the interval [−4, 4]. Find the absolute extrema for the function on the given interval. Express your answer as an ordered pair ( , ( )). Answer: Absolute Maximum: _______________ Absolute Minimum: _______________ 41. Consider the function ( ) = 4 3 − 12 2 − 288 on the interval [−7, 7]. Find the absolute extrema for the function on the given interval. Express your answer as an ordered pair ( , ( )). Answer: Absolute Maximum: _______________ Absolute Minimum: _______________ 42. Consider the function: ( ) = 7 3 − 8 2 + 7 Step 1. Find ′′( ). Answer: ____________________ Step 2. Evaluate ′′(−5), ′′(8), and ′′(6), if they exist. If they do not exist, select “Does Not Exist”. A) ″ (−5) =____________ B) ″ (−5) Does Not Exist A) ″ (8) =____________ B) ″ (8) Does Not Exist A) ″ (6) =____________ B) ″ (6) Does Not Exist 43. Consider the function: ( ) = 3 2 − 6√ 4 + 8 Step 1. Find ′′( ). Answer: ____________________ Step 2. Evaluate ′′(2), ′′(8), and ′′(3), if they exist. If they do not exist, select “Does Not Exist”. A) ″ (2) =____________ B) ″ (2) Does Not Exist A) ″ (8) =____________ B) ″ (8) Does Not Exist A) ″ (3) =____________ B) ″ (3) Does Not Exist 44. Consider the function: ( ) = 5 3 − 2 + 6 − 6 Find ′′( ). Answer: ____________________ 45. Consider the function: ( ) = −4 3 − 36 2 + 2 + 9 Step 1. Determine the intervals on which the function is concave upwards or concave downwards. A) Concave Up: ____________ B) Never Concave Up A) Concave Down: ____________ B) Never Concave Down Step 2. Locate any points of inflection. Enter your answer as ( , )-pairs. A) Points of Inflection: ____________ B) None 46. Consider the function: ( ) = √5 + 1 Step 1. Find ′′( ). Answer: ____________________ Step 2. Evaluate ′′(3), ′′(7), and ′′(6), if they exist. If they do not exist, select “Does Not Exist”. A) ″ (3) =____________ B) ″ (3) Does Not Exist A) ″ (7) =____________ B) ″ (7) Does Not Exist A) ″ (6) =____________ B) ″ (6) Does Not Exist 47. Use the Second Derivative Test to find all local extrema, if the test applies. Otherwise, use the First Derivative Test. ( ) = 6 3 + 81 2 + 360 A) Local Maxima: ____________ B) No Local Maxima A) Local Minima: ____________ B) No Local Minima 48. Consider the function: ( ) = −4 3 + 6 2 + 240 − 12 Step 1. Find the second derivative of the given function. Answer: ____________________ Step 2. Use the Second Derivative Test to locate any local maximum or minimum points in the graph of the given function. A) Local Maxima: ____________ B) No Local Maxima A) Local Minima: ____________ B) No Local Minima 49. Use the Second Derivative Test to find all local extrema, if the test applies. Otherwise, use the First Derivative Test. ( ) = 2 3 + 6 2 − 48 + 18 A) Local Maxima: ____________ B) No Local Maxima A) Local Minima: ____________ B) No Local Minima 50. Consider the function: ( ) = (2 2 + 11) 2 Step 1. Find the second derivative of the given function. Answer: ____________________ Step 2. Use the Second Derivative Test to locate any local maximum or minimum points in the graph of the given function. A) Local Maxima: ____________ B) No Local Maxima A) Local Minima: ____________ B) No Local Minima 51. Use the Second Derivative Test to find all local extrema, if the test applies. Otherwise, use the First Derivative Test. ( ) = 4 + 16 3 − 7 A) Local maxima: No local maxima; Local minima: No local minima B) Local maxima: No local maxima; Local minima: (−12,0) C) Local maxima: (−12, −6919); Local minima: No local minima D) Local maxima: No local maxima; Local minima: (−12, −6919) 52. Find the following indefinite integral. ∫(−4 2 − 6) Answer: ____________________ 53. Find the following indefinite integral. ∫( −2 + 5 − 4 5 + 8 ) Answer: ____________________ 54. Find the following indefinite integral. ∫(−2 − 8 −3 + 3 7 ) Answer: ____________________ 55. Perform the indicated multiplication and then integrate. ∫ 3 (2 − 9) Answer: ____________________ 56. Evaluate the definite integral below. 3 ∫ 1 −5 3 Enter your answer in exact form or rounded to two decimal places. Answer: ____________________ 57. Evaluate the definite integral below. 4 ∫ 3 ( −6 2 − 6) Enter your answer in exact form or rounded to two decimal places. Answer: ____________________ 58. Evaluate the definite integral below. 5 ∫ 1 (2 + 4 ) Enter your answer in exact form or rounded to two decimal places. Answer: ____________________ 59. Evaluate the following definite integral. Ace my homework – Write the exact answer. Do not round. ∫ ( 2 + 5 − 12) 4 2 Answer: _______________ 60. Find the following indefinite integral. ∫( − 5 6 ) Answer: ____________________ 61. Find the following indefinite integral. ∫( 1 2 1 8 + 7) Answer: ____________________ 62. Find the following indefinite integral. ∫(12 + 3 ) Answer: ____________________ 63. Find the following indefinite integral. ∫(5 9 − 6 + 1 5 ) Answer: ____________________ 64. Simplify the indicated quotient and then integrate. ∫ −5 6 + 5 − 8 6 Answer: ____________________ 65. Consider the function: ( ) = −2 3 − 5 2 − 9 + 4 Step 1. Find ′′( ). Answer: ____________________ Step 2. Evaluate ′′(4), ′′(−9), and ′′(0), if they exist. If they do not exist, select “Does Not Exist”. A) ″ (4) =____________ B) ″ (4) Does Not Exist A) ″ (−9) =____________ B) ″ (−9) Does Not Exist A) ″ (0) =____________ B) ″ (0) Does Not Exist 66. Consider the function: ( ) = 2 + 5 −6 − 8 Step 1. Find ′′( ). Answer: ____________________ Step 2. Evaluate ′′(10), ′′(7), and ′′(−3), if they exist. If they do not exist, select “Does Not Exist”. A) ″ (10) =____________ B) ″ (10) Does Not Exist A) ″ (7) =____________ B) ″ (7) Does Not Exist A) ″ (−3) =____________ B) ″ (−3) Does Not Exist 67. Consider the function: ( ) = 5 2 − √ + 6 Find ′′( ). Answer: ____________________ 68. Consider the function: ( ) = 9 2 + 8 − 10 Step 1. Determine the intervals on which the function is concave upwards or concave downwards. A) Concave Up: ____________ B) Never Concave Up A) Concave Down: ____________ B) Never Concave Down Step 2. Locate any points of inflection. Enter your answer as ( , )-pairs. A) Points of Inflection: ____________ B) None 69. Consider the function: ( ) = 3 + 72√ − 5 Step 1. Determine the intervals on which the function is concave upwards or concave downwards. A) Concave Up: ____________ B) Never Concave Up A) Concave Down: ____________ B) Never Concave Down Step 2. Locate any points of inflection. Enter your answer as ( , )-pairs. A) Points of Inflection: ____________ B) None 1. Step 1.Correct Answer: 35, 35.9, 35.99, 35.999 Step 2.Correct Answer: 36 2. Step 1.Correct Answer: → −3 − ( ) = −1 Step 2.Correct Answer: → 1 − ( ) = −1 Step 3.Correct Answer: → −1 − ( ) = 8 Step 4.Correct Answer: → −1 + ( ) = −2 3. Step 1.Correct Answer: −4 Step 2.Correct Answer: 1 Step 3.Correct Answer: Does Not Exist 4. Correct Answer: 13 5. Correct Answer: 2 3 6. Correct Answer: Rate of change of the velocity of the snowmobile at hours. 7. Step 1.Correct Answer: The value of the function evaluated at = 1 is 9. Step 2.Correct Answer: The slope of the curve at = 1 is 7. 8. Correct Answer: ′ = − 3 7 − 4 7 9. Correct Answer: ′ ( ) = 10 + 6 2 10. Correct Answer: ′ ( ) = 15 2 − 10 4 11. Correct Answer: ′ ( ) = 18 + 27 12. Correct Answer: ′ ( ) = −12 3 + 4 13. Step 1.Correct Answer: The slope of the tangent line at = −5 is −448. Step 2.Correct Answer: = −448 − 1550 14. Step 1.Correct Answer: −64 Step 2.Correct Answer: 1 15. Correct Answer: = 29 16. Correct Answer: 411 17. Correct Answer: 1 2 18. Correct Answer: 0 19. Correct Answer: −5 2 20. Correct Answer: −∞ 21. Correct Answer: +∞ 22. Correct Answer: √104 + 16 23. Correct Answer: 0 24. Step 1.Correct Answer: −4 Step 2.Correct Answer: −4 Step 3.Correct Answer: −4 25. Step 1.Correct Answer: 3 Step 2.Correct Answer: 2 Step 3.Correct Answer: 3 Step 4.Correct Answer: No 26. Step 1.Correct Answer: −3 Step 2.Correct Answer: −3 Step 3.Correct Answer: 1 Step 4.Correct Answer: No 27. Step 1.Correct Answer: −4 Step 2.Correct Answer: Jump Discontinuity 28. Correct Answer: ′ ( ) = −18 2 + 4 − 14 −2 29. Correct Answer: ′ ( ) = 8 30. Correct Answer: ℎ ′ (−6) = 115 72 31. Correct Answer: ′ ( ) = 63 6 + 108 3 − 9 2 32. Correct Answer: ′ ( ) = 36 8 − 90 5 + 24 2 (4 3 − 5) 2 33. Correct Answer: 3(7 4 + 9) 2 (28 3 ) 34. Correct Answer: 3 ( 4 + 4 −9 2 + 10) 2 ( (−9 2 + 10)(4 3) −( 4 + 4)(−18 ) (−9 2 + 10)2 ) 35. Step 1.Correct Answer: 3 Step 2.Correct Answer: Increasing: (−∞, 3), Decreasing: (3, ∞) 36. Step 1.Correct Answer: 2 Step 2.Correct Answer: Increasing: (−∞, ∞), Decreasing: ø 37. Step 1.Correct Answer: None Step 2.Correct Answer: Increasing: ø, Decreasing: (−∞, 1), (1, ∞) 38. Step 1.Correct Answer: = −4, 0, 4 Step 2.Correct Answer: Local Maxima: (0, 6916), Local Minima: (−4, 4), (4, 4) 39. Step 1.Correct Answer: = −2, 4 Step 2.Correct Answer: Local Maxima: (−2, 32), Local Minima: (4, −76) 40. Correct Answer: Absolute Maximum: (−3, 216) Absolute Minimum: (3, −216) 41. Correct Answer: Absolute Maximum: (−4, 704) Absolute Minimum: (6, −1296) 42. Step 1.Correct Answer: ′′( ) = 42 − 16 Step 2.Correct Answer: ′′(−5) = −226, ′′(8) = 320, ′′(6) = 236 43. Step 1.Correct Answer: ′′( ) = 6 + 9 8 −7 4 Step 2.Correct Answer: ′′(2) = 6 + 9 √2 4 32 , ′′(8) = 6 + 9 √8 4 512 , ′′(3) = 6 + √3 4 8 44. Correct Answer: ′′( ) = 30 − 2 45. Step 1.Correct Answer: Concave Up: (−∞, −3), Concave Down: (−3, ∞) Step 2.Correct Answer: Points of Inflection: (−3, −213) 46. Step 1.Correct Answer: ′′( ) = − 25 4 (5 + 1) −3 2 Step 2.Correct Answer: ′′(3) = −25 256, ′′(7) = −25 864, ′′(6) = − 25√31 3844 47. Correct Answer: Local Maxima: (−5, −525), Local Minima: (−4, −528) 48. Step 1.Correct Answer: ′′( ) = −24 + 12 Step 2.Correct Answer: Local Maxima: (5, 838), Local Minima: (−4, −620) 49. Correct Answer: Local Maxima: (−4, 178), Local Minima: (2, −38) 50. Step 1.Correct Answer: ′′( ) = 48 2 + 88 Step 2.Correct Answer: Local Maxima: No Local Maxima, Local Minima: (0, 121) 51. Correct Answer: Local maxima: No local maxima; Local minima: (−12, −6919) 52. Correct Answer: − 4 3 3 − 6 + 53. Correct Answer: −2 ln( ⃒ ⃒ ) + 25 1 5 + 8 + 54. Correct Answer: −2 + 4 −2 + 3 7 + 55. Correct Answer: 2 5 5 − 9 4 4 + 56. Correct Answer: −20 9 57. Correct Answer: −13 2 58. Correct Answer: 8 + 4 5 − 4 ≈ 590.78 59. Correct Answer: 74 3 60. Correct Answer: − 5 6 + 61. Correct Answer: 4 9 9 8 + 7 + 62. Correct Answer: 12 + 3 2 2 + 63. Correct Answer: 1 2 10 − 6 ln( ⃒ ⃒ ) − 1 4 −4 + 64. Correct Answer: −5 + ln( ⃒ ⃒ ) + 8 5 −5 + 65. Step 1.Correct Answer: ′′( ) = −12 − 10 Step 2.Correct Answer: ′′(4) = −58, ′′(−9) = 98, ′′(0) = −10 66. Step 1.Correct Answer: ′′( ) = 168(−6 − 8) −3 Step 2.Correct Answer: ′′(10) = −21 39304, ′′(7) = −21 15625, ′′(−3) = 21 125 67. Correct Answer: ′′( ) = 10 + 1 4 −3 2 68. Step 1.Correct Answer: Concave Up: (−∞, ∞), Concave Down: None Step 2.Correct Answer: Points of Inflection: None 69. Step 1.Correct Answer: Concave Up: (√9 5 , ∞), Concave Down: (0, √9 5 ) Step 2.Correct Answer: Points of Inflection: (√9 5 , −5 + 75√3 5 )

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