Derivative of Exponential and Logarithmic Functions

Derivative of Exponential and Logarithmic Functions
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2.6

1. The graph of f (x) = e^x curves upward in the interval from x = -1 to x = 1. Interpreting f ' (x) = e^x as the slopes of tangent lines and noting that the larger x is, the larger e^x is, explain why the graph curves upward. For larger values of x, the graph of f (x) = e^x appears to shoot straight up with no curve. Using the tangent line, determine if this is correct or just an optical illusion.

2. The graph of f (x) = ln x appears to get flatter as x gets larger. Interpret the derivative as the slopes of tangent lines to determine if this is correct or just an optical illusion.

3. Graphically compare and contrast the functions x², x³, x⁴ and e^x for x > 0. Sketch the graphs for large x (and very large y's) and compare the relative growth rates of the functions. In general, how does the exponential function compare to polynomials?

4. Graphically compare and contrast the functions x^1/2, x^1/3, x^1/4 and ln x for x > 1. Sketch the graphs for large x and compare the relative growth rates of the functions. In general, how does ln x compare to ?

In exercises 5-28, find the derivative of the function.

5. f (x) = 4e^x-x 6. f (x) = sin x-3e^x

7. f (x) = xe^x 8. f (x) = e^xcos x

9. f (x) = x+2^x 10. f (x) = x4^x

11. f (x) = 2e^x+1 12. f (x) = (1/e)^x

13. f (x) = (1/3)^x 14. f (x) = 4^-x

15. f (x) = 4^-x+1 16. f (x) = (1/2)^1-x

17. 18.

19. f (x) = ln 2x 20. f (x) = ln 8x

21. f (x) = ln x³ 22. f (x) = x³ln x

23. f (x) = e^2x = e^xe^x 24. f (x) = e^3x

25. f (x) = x²e^-x 26. f (x) = e^xln x

27. 28.

In exercises 29-36, find an equation of the tangent line to y = f (x) at x = 1

29. f (x) = 3e^x 30. f (x) = 2e^x-1

31. f (x) = 3^x 32. f (x) = 2^x

33. f (x) = xe^x 34.

35. f (x) = x²ln x 36. f (x) = 2ln x³

In exercises 37-40, the value of an investment is given by (t ). Find the instantaneous percentage rate of change.

37. v(t ) = 100 3^t 38. v(t ) = 100 4^t

39. v(t ) = 100 e^t 40. v(t ) = 100 e^-t

41. A bacterial population starts at 200 and triples every day. Find a formula for the population after t days and find the percentage rate of change in population.

42. A bacterial population starts at 500 and doubles every four days. Find a formula for the population after t days and find the percentage rate of change in population.

43. An investment of A dollars receiving r percent (per year) interest compounded continuously will be worth f (t ) = Ae^rt dollars after t years. APY can be defined as [f (1)-A]/A , the relative increase of worth in one year. Find the APY for the following interest rates:
(a) 5% (b) 10% (c) 20% (d) 100 ln 2% (e) 100%

44. Determine the interest rate needed to obtain an APY of (a) 100% and (b) 10%.

45. The motion of a spring is described by f (t ) = e^-tcos t. Compute the velocity at time t. Graph the velocity function. When is the velocity zero? What is the position of the spring when the velocity is zero?

46. The motion of a spring is given by f (t ) = e^-tsin t. Compute the velocity at time t. Graph the velocity function. When is the velocity zero? What is the position of the spring when the velocity is zero?

47. In exercise 45, graphically estimate the value of t > 0 at which the maximum velocity is reached and estimate the position of the spring when this occurs.

48. In exercise 46, graphically estimate the value of t > 0 at which the maximum velocity is reached and estimate the position of the spring when this occurs.

In exercises 49-52, use a CAS or graphing calculator.

49. Find the derivative of f (x) = e^{ln x²} on your CAS. Compare its answer to 2x. Explain how to get this answer and your CAS' answer, if it differs.

50. Find the derivative of on your CAS. The correct answer is that it does not exist. Explain how to get this answer and your CAS' answer, if it differs.

51. Find the derivative of on your CAS. Compare its answer to Explain how to get this answer and your CAS' answer, if it differs.

52. Find the derivative of on your CAS. Compare its answer to 4-2/x. Explain how to get this answer and your CAS' answer, if it differs.

53. Numerically estimate the limit in (6.1) for a = 3 and compare your answer to ln 3.

54. Numerically estimate the limit in (6.1) for and compare your answer to

55. Numerically estimate the derivative of ln x at a = 4 and compare your answer to

56. Numerically estimate the derivative of ln x at a = 5 and compare your answer to

57. The Padé approximation of e^x is the function of the form for which the values of f (0), f ' (0) and f '' (0) match the corresponding values of e^x. Show that these values all equal 1 and find the values of a, b and c that make f (0) = 1, f ' (0) = 1 and f '' (0) = 1. Compare the graphs of f (x) and e^x.

58. In a World Almanac or Internet resource, look up the population of the United States by decade for as many years as are available. If there are not columns indicating growth by decade, both numerically and by percentage, compute these yourself (a spreadsheet is helpful here). The United States has had periods of both linear and exponential growth. Explain why linear growth corresponds to a constant numerical increase; during which decades has the numerical growth been (approximately) constant? Explain why exponential growth corresponds to a constant percentage growth. During which decades has the percentage growth been (approximately) constant?

59. In Chapter 0, we defined e as the limit In this section, e is given as the value of the base a such that There are a variety of other interesting properties of this important number. We discover one here. Graph the functions x² ^{and 2}^x. For x > 0 , how many times do they intersect? Graph the functions x³ ^{and 3}^x. For x > 0 , how many times do they intersect? Try the pair x^2.5 ^and^x and the pair x⁴ ^{and 4}^x. What would be a reasonable conjecture for the number of intersections (x > 0) of the functions x^a and a^x? Explain why x = a is always a solution. Under what circumstances is there another solution less than a? greater than a? By trial and error, verify that e is the value of a at which the “other” solution changes.

60. Using the properties of exponential and logarithmic functions, we can discover derivative rules that preview the chain rule of the next section. Since e^2x = (e^x)², show that Since e^3x = e^xe^2x, show that 3e^3x. Use mathematical induction to prove that ne^nx for any positive integer n. As we will see, the formula holds for any real number n. Using ln x^a = aln x, derive a formula for the derivative of ln x^a. Using ln ax = ln a+ln x, find the derivative of ln ax.