 |
|
 | 1. Notice that for both Rolle's Theorem and the Mean Value Theorem, we have assumed that the
function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Recall that if a
function is differentiable at x = a, then it is also continuous at x = a. Therefore, if we had assumed that the
function was differentiable on the closed interval, we would not have had to mention continuity. We do not do so because the assumption of being differentiable at the endpoints is not necessary. The “ethics” of the statement of a theorem is to only include assumptions that are absolutely necessary. Discuss the virtues of this tradition. Is this common practice in our social dealings, such as financial obligations or personal gossip? |
| | 2. One of the results in this section is that if f ' (x) = g' (x), then g(x) = f (x)+c for some
constant c. Explain this result graphically. |
| | 3. Explain the result of
Corollary 9.1 in terms of position and velocity functions. That is, if two objects have the
same velocity functions, what can you say about the relative positions of the two objects? |
| | 4. As we mentioned, you can derive Rolle's Theorem from the Mean Value Theorem simply by setting f (a) = f (b). Given this, it may seem odd that Rolle's Theorem rates its own name and portion of the book. To explain why we do this, discuss ways in which Rolle's Theorem is easier to understand than the Mean Value Theorem. |
| In exercises 5-8, explain why it is not valid to use the Mean Value Theorem. When the hypotheses are not true, the theorem does not tell you anything about the truth of the conclusion. In three of the four cases, show that there is no value of c that makes the conclusion of the theorem true. In the fourth case, find the value of c.
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| 5.  | 6.  |
| 7. f (x) = tan x, [0,  ] | 8. f (x) = x1/3, [-1, 1]
|
| In exercises 9-14, check the hypotheses of Rolle's Theorem and the Mean Value Theorem and find a value of c that makes the appropriate conclusion true. Illustrate the conclusion with a graph.
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| 9. f (x) = x2+1, [-2, 2] | 10. f (x) = x2+1, [0, 2] |
| 11. f (x) = x3+x2, [0, 1] | 12. f (x) = x3+x2, [-1, 1] |
| 13. f (x) = sin x, [0, /2] | 14. f (x) = sin x, [-, 0]
|
| 15. If f ' (x) > 0 for all x, prove that f (x) is an increasing
function: that is, if a < b, then f (a) < f (b).
|
| 16. If f ' (x) < 0 for all x, prove that f (x) is an decreasing
function: that is, if a < b, then f (a) > f (b).
|
| In exercises 17-24, determine if the
function is increasing, decreasing or neither.
|
| 17. f (x) = x3+5x+1 | 18. f (x) = x5+3x3-1 |
| 19. f (x) = -x3-3x+1 | 20. f (x) = x4+2x2+1 |
| 21. f (x) = ex | 22. f (x) = e-x |
| 23. f (x) = ln x | 24. f (x) = ln x2
|
| 25. Prove that x3+5x+1 = 0 has exactly one solution. |
| 26. Prove that x3+4x-3 = 0 has exactly one solution. |
| 27. Prove that x4+3x2-2 = 0 has exactly two solutions. |
| 28. Prove that x4+6x2-1 = 0 has exactly two solutions. |
| 29. Prove that x3+ax+b = 0 has exactly one solution for a > 0.
|
| 30. Prove that x4+ax2-b = 0 ( a > 0, b > 0 ) has exactly two solutions. |
| 31. Prove that x5+ax3+bx+c = 0 has exactly one solution for a > 0, b > 0.
|
| 32. Prove that a third-degree (cubic)
polynomial has at most three zeros (you may use the quadratic formula). |
| 33. Prove that a fourth-degree
polynomial has at most four zeros. |
| 34. Prove that an nth
degree polynomial has at most n zeros. |
| In exercises 35-42, find all functions
g(x) such that g'(x) = f (x).
|
| 35. f (x) = x2 | 36. f (x) = x3 |
| 37. f (x) = x4 | 38. f (x) = 9x4 |
| 39. f (x) = 1/x2 | 40.  |
| 41. f (x) = sin x | 42. f (x) = cos x
|
| 43. Assume that f (x) is a differentiable
function such that f (0) = f ' (0) = 0 and f '' (0) > 0. Argue that there exists a positive
constant a > 0 such that f (x) > 0 for all x in the interval (0, a). Can anything be concluded about f (x) for negative x's? |
| 44. Show that for any real numbers u and  | u-v |. (Hint: Use the Mean Value Theorem.) |
| 45. For  show that f (x) is continuous on the interval (0, 2), differentiable on the interval (0, 2) and has f (0) = f (2). Show that there does not exist a value of c such that f ' (c) = 0. Which hypothesis of Rolle's Theorem is not satisfied? |
| 46. Assume that f (x) is a differentiable
function such that f (0) = f ' (0) = 0. Show by example that it is not necessarily true that f (x) = 0 for all x. Find the flaw in the following bogus “proof. ” Using the Mean Value Theorem with a = x and b = 0, we have  Since f (0) = 0 and f ' (c) = 0, we have  so f (x) = 0.
|
 | 47. In
Section 2.1, we gave an example of computing the velocity of a moving car. The point of the story was that computing
instantaneous velocity requires us to compute a limit. However, we left an interesting question unanswered. If you have an average velocity of 60 mph over 1 hour and the speed limit is 65 mph, you are unable to prove that you never exceeded the speed limit. What is the longest time interval over which you can average 60 mph and still guarantee no speeding? We can use the Mean Value Theorem to answer the question after clearing up a couple of preliminary questions. First, argue that we need to know the maximum
acceleration of a car, and the maximum positive acceleration may differ from the maximum negative
acceleration. Based on your experience, what is the fastest your car could accelerate (speed up)? What is the fastest your car could decelerate (slow down)? Back up your
estimates with some real data (e.g., my car goes from 0 to 60 in 15 seconds). Call the larger number A (use units of mph per second). Next, argue that if
acceleration (the derivative of velocity) is constant, then the velocity function is
linear. Therefore, if the velocity varies from 55 mph to 65 mph at constant acceleration, the average velocity will be 60 mph. Now, apply the Mean Value Theorem to the velocity
function v(t ) on a time interval [0, T], where the velocity changes from 55 mph to 65 mph at
constant acceleration A:  becomes  and so T = 10/A. How long is the guarantee good for? |
| | 48. Suppose that a pollutant is dumped into a lake at the rate of p' (t ) = t 2-t+4 tons per month. The amount of pollutant dumped into the lake in the first two months is A = p(2)-p(0). Using c = 1 (the midpoint of the interval), estimate A by applying the Mean Value Theorem to p(t ) on the interval [0, 2]. To get a better estimate, apply the Mean Value Theorem to the intervals [0, 1/2], [1/2, 1], [1, 3/2] and [3/2, 2]. (Hint: A = p(1/2)-p(0)+ p(1)-p(1/2)+p(3/2)- p(1) + p(2)-p(3/2). ) If you have access to a CAS, get better estimates by dividing the interval [0, 2] into more and more pieces and try to conjecture the limit of the estimates. You will be well on your way to understanding
Chapter 4 on integration. |
