Increasing and Decreasing Functions

Increasing and Decreasing Functions
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3.4

1. Suppose that f (0) = 2 and f (x) is an increasing function. To sketch the graph of y = f (x), you could start by plotting the point (0, 2). Filling in the graph to the left, would you move your pencil up or down? How does this fit with the definition of increasing?

2. Suppose you travel east on an east-west interstate highway. You reach your destination, stay a while and then return home. Explain the First Derivative Test in terms of your velocities (positive and negative) on this trip.

3. Suppose that you have a differentiable function f (x) with two critical numbers. Your computer has shown you a graph that looks like the one below.

Discuss the possibility that this is a representative graph: that is, is it possible that there are any important points not shown in this window?

4. Suppose that the function in exercise 3 has three critical numbers. Explain why the graph is not a representative graph. Explain how you would change the graphing window to show the rest of the graph.

In exercises 5-14, find (by hand) the intervals where the function is increasing and decreasing. Verify your answers by graphing both f (x) and f '(x).

5. y = x³ - 3x + 2 6. y = x³ + 2x² + 1

7. y = x⁴ - 8x² + 1 8. y = x³ - 3x² - 9x + 1

9. y = (x + 1)^2/3 10. y = (x -1)^1/3

11.y = sin 3x 12.y = sin²x

13. y = e^{x² - 1} 14.y = ln(x² - 1)

In exercises 15-34, find the x-coordinates of all extrema and sketch a graph.

15. y = x³ + 2x² - x -1 16. y = x³ + 4x -2

17. y = x⁴ + 2x² - x + 2 18. y = x⁵ + 2x⁴ - x² + 1

19. 20.

21.y = xe^{- 2x} 22.y = x²e^{- x}

23.y = ln x² 24. y = e^{- x²}

25. 26.

27. 28.

29.y = sin x + cos x 30.y = cos x - x

31. 32. y = 2x^1/2 - 4x^{- 1/2}

33. y = x^2/3 - 2x^{- 1/3} 34. y = x^4/3 + 4x^1/3

In exercises 35-42, find the x-coordinates of all extrema and sketch graphs showing global and local behavior of the function.

35. y = x³ - 13x² - 10x + 1

36. y = x³ + 15x² - 70x + 2

37. y = x⁴ - 15x³ - 2x² + 40x -2

38. y = x⁴ - 16x³ -0.1x² + 0.5x -1

39. y = x⁵ - 200x³ + 605x -2

40. y = x⁴ -0.5x³ -0.02x² + 0.02x + 1

41. y = (x² + x + 0.45)e^{- 2x}

42. y = x⁵ ln 8x²

In exercises 43-46, sketch a graph of a function with the given properties.

43. f (0) = 1, f (2) = 5, f ' (x) < 0 for x < 0 and x > 2, f ' (x) > 0 for 0 < x < 2 .

44. f ( -1) = 1, f (2) = 5, f ' (x) < 0 for x < -1 and x > 2, f ' (x) > 0 for -1 < x < 2, f ' ( -1) = 0, f ' (2) does not exist.

45. f (3) = 0, f ' (x) < 0 for x < 0 and x > 3, f '(x) > 0 for 0 < x < 3, f ' (3) = 0, f (0) and f ' (0) do not exist.

46. f (1) = 0, , f ' (x) < 0 for x < 1, f ' (x) > 0 for x > 1, f ' (1) = 0 .

47. Suppose an object has position function s(t), velocity function v(t) = s' (t) and acceleration function a(t) = v'(t). If a(t) > 0, then the velocity v(t) is increasing. Sketch two possible velocity functions, one with velocity getting less negative and one with velocity getting more positive. In both cases, sketch possible position graphs. The position graph should be curved (nonlinear). Does it look more like part of an upward-curving parabola or a downward-curving parabola? We look more closely at curving in section 3.5.

48. Repeat exercise 41 for the case where a(t) < 0 .

49. Prove Theorem 4.2 (The First Derivative Test).

50. Give a graphical argument that if f (a) = g(a) and f ' (x)> g' (x) for all x > a, then f (x) > g(x) for all x > a. Use the Mean Value Theorem to prove it.

In exercises 51-54, use the result of exercise 50 to verify the inequality.

51. for x>1 52.x>sin x for x>0

53. e^x > x + 1 for x > 0 54.x - 1>ln x for x>1

55. Give an example showing that the following statement is false. If f (0) = 4 and f (x) is a decreasing function, then the equation f (x) = 0 has exactly one solution.

56. Determine if the following statement is true or false: If f (0) = 4 and f (x) is an increasing function, then the equation f (x) = 0 has no solutions.

57. Suppose that the total sales of a product after t months is given by thousand dollars. Compute and interpret s' (t) .

58. In exercise 57, show that s' (t) > 0 for all t>0. Explain in business terms why it is impossible to have s' (t) < 0 .

59. In this exercise, you will play the role of professor and construct a tricky graphing exercise. The first goal is to find a function with local extrema so close together that they're difficult to see. For instance, suppose you want local extrema at x = - 0.1 and x = 0.1. Explain why you could start with f '(x) = (x -0.1)(x + 0.1) = x² -0.01. Look for a function whose derivative is as given. Graph your function to see if the extrema are “hidden. ” Next, construct a polynomial of degree 4 with two extrema very near x = 1 and another near x = 0.

60. In this exercise, we look at the ability of fireflies to synchronize their flashes. (To see a remarkable demonstration of this ability, see David Attenborough's video series Trials of Life.) Let the function f (t) represent an individual firefly's rhythm, so that the firefly flashes whenever f (t) equals an integer. Let e(t) represent the rhythm of a neighboring firefly, where again e(t) = n, for some integer n, whenever the neighbor flashes. One model of the interaction between fireflies is f ' (t) = + A sin[e(t) - f (t)] for constants and A. If the fireflies are synchronized (e(t)f (t)), then f ' (t) = , so the fireflies flash every 1/ time units. Assume that the difference between e(t) and f (t) is less than. Show that if f (t)<e(t), then f '(t) >. Explain why this means that the individual firefly is speeding up its flash to match its neighbor. Similarly, discuss what happens if f (t)>e(t).

61. The HIV virus attacks specialized T cells that trigger the human immune system response to a foreign substance. If T(t) is the population of uninfected T cells at time t (days) and V(t) is the population of infectious HIV in the bloodstream, a model that has been used to study AIDS is given by the following differential equation that describes the rate at which the population of T cells changes.

If there is no HIV present [that is, V(t) = 0] and T(t) = 1000, show that T' (t) = 0. Explain why this means that the T-cell count will remain constant at 1000 (cells per cubic mm). Now, suppose that V(t) = 100. Show that if T(t) is small enough, then T' (t) > 0 and the T-cell population will increase. On the other hand, if T(t) is large enough, then T' (t) < 0 and the T-cell population will decrease. For what value of T(t) is T' (t) = 0 ? Even though this population would remain stable, explain why this would be bad news for the infected human.