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1+ Vx+ x 31. dx X,

1+ Vx+ x 31. dx X,

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Hi I need help answering only 31 and 45 thank you.

47. f"(x) = 7x and f' (1) = -1, f(1) = 10 55. Given the graph of f below, sketch the graph of the an- tiderivative F of f that passes through the origin. What do the graphs of the other antiderivatives of f look like? y 48. f"(x) = 5e* and f' (0) = 3, f(0) = 5 4 49. f"(0) = sin 0 and f' (T) = 2, f(T) = 4 50. f"(x) = 0 and f'(1) = 3, f(1) = 1 56. Given the graph of f below, sketch the graph of the an- tiderivative F of f that passes through the origin. What do the graphs of the other antiderivatives of f look like? 51. f'(x) = and f(1) = 2 y 52. f'(x) = and f(4) = 0 2 53. An object is moving so that its velocity at time t is given by v(t) = 3/t. If the object was at the origin at time t = 0, find it's position s(t) at time t. Review 54. A nickel dropped from the top of the North Dakota State Capital Building has acceleration a(t) = -32 ft/sec? (ig- noring air resistance), initial velocity v(0) = 0, and initial height s(0) = 241.67 ft. How long will it take the nickel to hit the ground? 57. Use information gained from the first and second deriva- 1 tives to sketch f(x) = ex + 1' 58. Given y = x'e* cos x, find dy.
Transcribed Image Text:47. f"(x) = 7x and f' (1) = -1, f(1) = 10 55. Given the graph of f below, sketch the graph of the an- tiderivative F of f that passes through the origin. What do the graphs of the other antiderivatives of f look like? y 48. f"(x) = 5e* and f' (0) = 3, f(0) = 5 4 49. f"(0) = sin 0 and f' (T) = 2, f(T) = 4 50. f"(x) = 0 and f'(1) = 3, f(1) = 1 56. Given the graph of f below, sketch the graph of the an- tiderivative F of f that passes through the origin. What do the graphs of the other antiderivatives of f look like? 51. f'(x) = and f(1) = 2 y 52. f'(x) = and f(4) = 0 2 53. An object is moving so that its velocity at time t is given by v(t) = 3/t. If the object was at the origin at time t = 0, find it's position s(t) at time t. Review 54. A nickel dropped from the top of the North Dakota State Capital Building has acceleration a(t) = -32 ft/sec? (ig- noring air resistance), initial velocity v(0) = 0, and initial height s(0) = 241.67 ft. How long will it take the nickel to hit the ground? 57. Use information gained from the first and second deriva- 1 tives to sketch f(x) = ex + 1' 58. Given y = x'e* cos x, find dy.
Terms and Concepts 25. dx 1. Define the term "antiderivative" in your own words. x – 7x dx 26. 2. Is it more accurate to refer to "the" antiderivative of f(x) or "an" antiderivative of f(x)? 5 - 3 27. xp X +X 3. Use your own words to define the indefinite integral of f(x). 2 4. Fill in the blanks: "Inverse operations do the 28. u° – 2u° – u +: 7 du things in the order." 5. What is an "initial value problem"? 29. + 4)(2u + 1) du 6. The derivative of a position function is a func- 30. + 3t + 2) dt tion. 7. The antiderivative of an acceleration function is a 1+ Vx+ x 31. dx function. | sin' x+ cos? x dx Problems 32. In Exercises 8-40, evaluate the given indefinite integral. 33. 2 + tan? 0 do 8. 3x dx 34. sec t(sec t + tan t) dt 1 - sin? t dt sin? t 9. 35. xp |(10 – 2) dx sin 2x dx sin x 10. 36. 4 + 6u du 37. 11. dt Vu sin 0 + sin 0 tan? 0 do 12. dt 38. sec2 39. 2 +t dt 13. dt | V + V* dx 40. 14. dx 41. This problem investigates why Theorem 32 states that 15. sec 0 9 de dx = In |x| + C. (a) What is the domain of y = In x? 16. sin 0 de (b) Find (In x). 17. (sec x tan x + cSc x cot x) dx (c) What is the domain of y = In(-x)? (d) Find (In(-x)). 18. dt 2 (e) You should find that 1/x has two types of antideriva- tives, depending on whether x > 0 or x < 0. In |(2+ + 3)° dt 20. ( + 3)(? – 21) dt 19. one expression, give a formula for dx that takes these different domains into account, and explain your answer. In Exercises 42-52, find f(x) described by the given initial value problem. 21. dx 22. e* dx 42. f'(x) = sin x and f(0) = 2 43. f'(x) = 5e* and f(0) = 10 44. f'(x) = 4x³ – 3x and f(-1) = 9 45. f'(x) = sec² x and f(/4) = 5 23. dx 4x° – 7 dx 24. 46. f"(x) = 5 and f'(0) = 7, f(0) = 3 227
Transcribed Image Text:Terms and Concepts 25. dx 1. Define the term "antiderivative" in your own words. x – 7x dx 26. 2. Is it more accurate to refer to "the" antiderivative of f(x) or "an" antiderivative of f(x)? 5 - 3 27. xp X +X 3. Use your own words to define the indefinite integral of f(x). 2 4. Fill in the blanks: "Inverse operations do the 28. u° – 2u° – u +: 7 du things in the order." 5. What is an "initial value problem"? 29. + 4)(2u + 1) du 6. The derivative of a position function is a func- 30. + 3t + 2) dt tion. 7. The antiderivative of an acceleration function is a 1+ Vx+ x 31. dx function. | sin' x+ cos? x dx Problems 32. In Exercises 8-40, evaluate the given indefinite integral. 33. 2 + tan? 0 do 8. 3x dx 34. sec t(sec t + tan t) dt 1 - sin? t dt sin? t 9. 35. xp |(10 – 2) dx sin 2x dx sin x 10. 36. 4 + 6u du 37. 11. dt Vu sin 0 + sin 0 tan? 0 do 12. dt 38. sec2 39. 2 +t dt 13. dt | V + V* dx 40. 14. dx 41. This problem investigates why Theorem 32 states that 15. sec 0 9 de dx = In |x| + C. (a) What is the domain of y = In x? 16. sin 0 de (b) Find (In x). 17. (sec x tan x + cSc x cot x) dx (c) What is the domain of y = In(-x)? (d) Find (In(-x)). 18. dt 2 (e) You should find that 1/x has two types of antideriva- tives, depending on whether x > 0 or x < 0. In |(2+ + 3)° dt 20. ( + 3)(? – 21) dt 19. one expression, give a formula for dx that takes these different domains into account, and explain your answer. In Exercises 42-52, find f(x) described by the given initial value problem. 21. dx 22. e* dx 42. f'(x) = sin x and f(0) = 2 43. f'(x) = 5e* and f(0) = 10 44. f'(x) = 4x³ – 3x and f(-1) = 9 45. f'(x) = sec² x and f(/4) = 5 23. dx 4x° – 7 dx 24. 46. f"(x) = 5 and f'(0) = 7, f(0) = 3 227
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