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|
 | 1. It is often said that a graph is
concave up if it “holds water. ” This is certainly true for parabolas like y = x2, but is it true for graphs like y = 1/x2 ? It is always helpful to put a difficult concept into everyday language, but the danger is in oversimplification. Do you think that “holds water” is helpful or can it be confusing? Give your own description of
concave up, using everyday language. (Hint: One popular image involves smiles and frowns.) |
| | 2. Find a reference book with the population of the United States since 1800. From 1800 to 1900, the numerical increase by decade increased. Argue that this means that the population curve is
concave up. From 1960 to 1990, the numerical increase by decade has been approximately
constant. Argue that this means that the population curve is near a point of zero concavity. Why does this not necessarily mean that we are at an inflection point? Argue that we should hope, in order to avoid overpopulation, that it is indeed an
inflection point. |
| | 3. The goal of investing in the stock market is to buy low and sell high. But, how can you tell whether a price has peaked or not? Once a stock price goes down, you can see that it was at a peak but then it's too late to do anything about it! Concavity can help. Suppose a stock price is increasing and the price curve is
concave up. Why would you suspect that it will continue to rise? Is this a good time to buy? Now, suppose the price is increasing but the curve is
concave down. Why should you be preparing to sell? Finally, suppose the price is decreasing. If the curve is
concave up, should you buy or sell? What if the curve is concave down? |
| | 4. Suppose that f (t) is the amount of money in your bank account at time t. Explain in terms of spending and saving what would cause f (t) to be decreasing and
concave down; increasing and concave up; decreasing and concave up. |
| In exercises 5-8, estimate the intervals where the
function is concave up and concave down. (Hint: Estimate where the slope is increasing and decreasing.)
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| 5.
| 6.
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7.
| 8.
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| In exercises 9-40, find the intervals of increase and decrease, all
local extrema, the intervals of concavity, all inflection points and sketch a graph.
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| 9. f (x) = x3 - 3x2 + 4 | 10. f (x) = x3 + 3x2 - 6x |
| 11. f (x) = x4 - 2x2 + 1 | 12. f (x) = x4 + 4x -2 |
| 13. f (x) = x + 1/x | 14. f (x) = x2 - 16/x |
| 15. f (x) = x3 - 6x + 1 | 16. f (x) = x3 + 3x -1 |
| 17. f (x) = x4 + 4x3 -1 | 18. f (x) = x4 + 4x2 + 1 |
| 19. f (x) = xe - x | 20. |
| 21. | 22. |
| 23. f (x) = (x2 + 1)2/3 | 24. f (x) = x
ln x |
| 25. f (x) = x2/(x2 -9) | 26. f (x) = x/(x + 2) |
| 27. f (x) = sin x + cos x | 28. f (x) = x + cos x |
| 29. f (x) = e - xsin x | 30. f (x) = e - 2xcos x |
| 31. f (x) = x3/4 - 4x1/4 | 32. f (x) = x2/3 - 4x1/3 |
| 33. | 34. |
| 35. f (x) = x4 - 26x3 + x | 36. f (x) = 2x4 - 11x3 + 17x2 |
| 37. | 38.
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| |
| 39. f (x) = x4 - 16x3 + 42x2 -39.6x + 14
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| 40. f (x) = x4 + 32x3 -0.02x2 -0.8x
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| In exercises 41-46, sketch a graph with the given properties. |
| 41. f (0) = 2, f ' (x) > 0 for all x, f ' (0) = 1, f '' (x) > 0 for x > 0, f '' (x) < 0, for x < 0, f '' (0) = 0
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| 42. f (0) = 1, f ' (x) 0 for all x, f ' (0) = 0, f '' (x) > 0 for x > 0, f '' (x) < 0 for x < 0, f '' (0) = 0
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| 43. f (0) = 0, f ' (x) > 0 for x < -1 and -1 < x < 1, f '(x) < 0 for x > 1, f ''(x) > 0 for x < -1, 0 < x < 1 and x > 1, f '' (x) < 0 for -1 < x < 0
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| 44. f (0) = 2, f ' (x) > 0 for all x, f ' (0) = 1, f '' (x) > 0 for x < 0, f ''(x) < 0 for x > 0
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| 45. f (0) = 0, f ( -1) = -1, f (1) = 1, f ' (x) > 0 for x < -1 and 0 < x < 1, f ' (x) < 0 for -1 < x < 0 and x > 1, f '' (x) < 0 for x < 0 and x > 0
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| 46. f (1) = 0, f ' (x) < 0 for x < 1, f ' (x) > 0 for x > 1, f '' (x) < 0 for x < 1 and x > 1
|
| In exercises 47 and 48, estimate the intervals of increase and decrease, the locations of
local extrema, intervals of concavity and locations of inflection points.
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| 47.
| 48.
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| 49. Repeat exercise 47 if the given graph is of f ' (x) instead of f (x) . |
| 50. Repeat exercise 48 if the given graph is of f ' (x) instead of f (x) . |
| 51. Suppose that w(t) is the depth of water in a city's water reservoir. Which would be better news at time t = 0, w'' (0) = 0.05 or w'' (0) = -0.05 or would you need to know the value of w' (0) to determine which is better? |
| 52. Suppose that T(t) is a sick person's temperature at time t. Which would be better news at time t, T'' (0) = 2 or T'' (0) = -2 or would you need to know the value of T' (0) and T(0) to determine which is better? |
| 53. Suppose that a company that spends $ x thousand on advertising sells $ s(x) of merchandise, where s(x) = - 3x3 + 270x2 - 3600x + 18,000. Find the value of x that maximizes the rate of change of sales. (Hint: Read the question carefully!) |
| 54. For the sales
function in exercise 53, find the inflection point and explain why in advertising terms this is the “point of diminishing returns. ” |
| 55. Suppose that it costs a company C(x) = 0.01x2 + 40x + 3600 dollars to manufacture x units of a product. For this cost function, the average cost function is  . Find the value of x that minimizes the average cost. |
| 56. In exercise 55, the cost
function can be related to the efficiency of the production process. Explain why a cost
function that is concave down indicates better efficiency than a cost function that is
concave up. |
| 57. Show that there is an
inflection point at (0,0) for any function of the form f (x) = x4 + cx3, where c is a nonzero
constant. What role(s) does c play in the graph of y = f (x) ? |
| 58. The following examples show that there is not a perfect match between inflection points and places where f '' (x) = 0. First, for f (x) = x6, show that f '' (0) = 0, but there is no
inflection point at x = 0. Then, for g(x) = x| x|, show that there is an
inflection point at x = 0, but that g'' (0) does not exist. |
| 59. Give an example of a
function showing that the following statement is false. If the graph of y = f (x) is
concave down for all x, the equation f (x) = 0 has at least one solution. |
| 60. Determine if the following statement is true or false. If f (0) = 1, f '' (x) exists for all x and the graph of y = f (x) is
concave down for all x, the equation f (x) = 0 has at least one solution. |
| 61. One basic principle of physics is that light follows the path of minimum time. Assuming that the speed of light in the earth's atmosphere decreases as altitude decreases, argue that the path that light follows is
concave down. Explain why this means that the setting sun appears higher in the sky than it really is.
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| 62. Prove
Theorem 5.2 (the Second Derivative Test). (Hint: Think about what the definition of f '' (c) says when f '' (c) > 0 or f '' (c) < 0.)
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 | 63. The
linear approximation that we defined in Section 3.1 is the line having the same location and the same
slope as the function being approximated. Since two points determine a line, two requirements (point, slope) are all that a
linear function can satisfy. However, a quadratic function can satisfy three requirements since three points determine a parabola (and there are three constants in a general
quadratic function ax2 + bx + c ). Suppose we want to define a quadratic approximation to f (x) at x = a. Building on the
linear approximation, the general form is g(x) = f (a) + f ' (a) (x - a) + c(x - a)2 for some
constant c to be determined. In this way, show that g(a) = f (a) and g' (a) = f ' (a). That is, g(x) has the right position and
slope at x = a. The third requirement is that g(x) have the right concavity at x = a, so that g'' (a) = f '' (a). Find the
constant c that makes this true. Then, find such a
quadratic approximation for each of the functions sin x, cos x and ex at x = 0. In each case, graph the original
function, linear approximation and quadratic approximation and describe how close the approximations are to the original functions. |
| | 64. In this exercise, we explore a basic problem in genetics. Suppose that a species reproduces according to the following probabilities: p0 is the probability of having no children, p1 is the probability of having one offspring, p2 is the probability of having two offspring, . . ., pn is the probability of having n offspring and n is the largest number of offspring possible. Explain why for each i, we have 0 pi 1 and p0 + p1 + p2 +  + pn = 1. We define the
function F(x) = p0 + p1x + p2x2 + + pnxn. The smallest non-negative solution of the equation F(x) = x for 0 x 1 represents the probability that the species becomes extinct. Show graphically that if p0 > 0 and F' (1) > 1, then there is a solution of F(x) = x with 0 < x < 1. Thus, there is a positive probability of survival. However, if p0 > 0 and F' (1) < 1, show that there are no solutions of F(x) = x with 0 < x < 1. (Hint: First show that F is increasing and
concave up.) |
