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3.3 |
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 | 1. Using one or more graphs, explain why the Extreme Value Theorem is true. Is the conclusion true if we drop the hypothesis that f (x) is a continuous function? Is the conclusion true if we drop the hypothesis that the interval is closed? |
| | 2. Using one or more graphs, argue that Fermat's Theorem is true. Discuss how Fermat's Theorem is used. Restate the theorem in your own words to make its use more clear. |
| | 3. Suppose that f (t) represents your elevation after t hours on a mountain hike. If you stop to rest, explain why f '(t) = 0. Discuss the circumstances under which you would be at a
local maximum, local minimum or neither. |
| | 4. Mathematically, an if/then statement is usually strictly one-directional. When we say “If A then B” it is generally not the case that “If B then A” is also true: when both are true, we say “A if and only if B” which is abbreviated to “A iff B. ” Unfortunately, common English usage is not always this precise. This occasionally causes a problem interpreting a mathematical theorem. To get this straight, consider the statement, “If you wrote a best-selling book, then you made a lot of money. ” Is this true? How does this differ from its converse, “If you made a lot of money, then you wrote a best-selling book. ” Is the converse always true? Sometimes true? Apply this logic to both the Extreme Value Theorem and Fermat's Theorem: state the converse and decide if it is sometimes true or always true. |
| In exercises 5-32, find all critical numbers and determine whether each represents a
local maximum, local minimum or neither.
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| 5. f (x) = x2 + 5x -1 | 6. f (x) = - x2 + 4x + 2 |
| 7. f (x) = x3 - 3x + 1 | 8. f (x) = x3 + 3x + 1 |
| 9. f (x) = x3 - 3x2 + 3x | 10. f (x) = x3 - 3x2 + 6x |
| 11. f (x) = x4 - 3x3 + 2 | 12. f (x) = x4 + 6x2 -2 |
| 13. f (x) = x3/4 - 4x1/4 | 14. f (x) = (x2/5 - 3x1/5)2 |
| 15. f (x) = x3 - 2x2 - 4x | 16. f (x) = x5 - 20x2 + 1 |
| 17. f (x) = sin xcos x, [0,2 ] | 18. f (x) = sin x + cos x,[0,2 ] |
| 19. | 20. |
| 21. | 22. |
| 23. f (x) =  (ex + e - x) | 24. f (x) = (ex - e - x) |
| 25. f (x) = x4/3 + 4x1/3 + 4x - 2/3 | 26. f (x) = x7/3 - 28x1/3 |
| 27. | 28. |
| 29. | 30. f (x) = xe - x |
| 31. f (x) = sin x2, [0, ] | 32. f (x) = sin 2x, [0,2 ]
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| In exercises 33-42, find the
absolute extrema of the given function on the indicated interval.
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| 33. f (x) = x3 - 3x + 1, [0,2] | 34. f (x) = x3 - 3x + 1, [ -3,2] |
| 35. f (x) = x4 - 8x2 + 2, [ -3,1] | 36. f (x) = x4 - 8x2 + 2, [ -1,3] |
| 37. f (x) = x2/3, [ -4, - 2] | 38. f (x) = x2/3, [ -1,3] |
| 39. f (x) = sin x + cos x, [0,2 ] | 40. f (x) = sin x + cos x, [ /2, ] |
| 41. f (x) = xsin x + 3, [0,2 ] | 42. f (x) = x2 + ex, [ -2,2]
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| In exercises 43-46, numerically estimate the
absolute extrema of the given function on the indicated interval.
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| 43. f (x) = x4 - 3x2 + 2x + 1 on (a) [ -1,1] and (b) [ -3,2]
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| 44. f (x) = x6 - 3x4 - 2x + 1 on (a) [ -1,1] and (b) [ -2,2]
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| 45. f (x) = x2 - 3xcos x on (a) [ -2,1] and (b) [ -5,0]
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| 46. f (x) = xecos on (a) [ -2,2] and (b) [2,5]
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| 47. Repeat exercises 33-38, except instead of finding
extrema on the closed interval, find the extrema on the open interval, if they exist. |
| 48. Briefly outline a procedure for finding
extrema on an open interval (a, b), a procedure for the half-open interval (a, b] and a procedure for the half-open interval [a, b). |
| 49. Sketch a graph of a
function f (x) such that the absolute maximum of f (x) on the interval [ - 2, 2] equals 3 and the absolute minimum does not exist. |
| 50. Sketch a graph of a
continuous function f (x) such that the absolute maximum of f (x) on the interval ( - 2, 2) does not exist and the absolute minimum equals 2. |
| 51. Sketch a graph of a
continuous function f (x) such that the absolute maximum of f (x) on the interval ( - 2, 2) equals 4 and the absolute minimum equals 2. |
| 52. Sketch a graph of a
function f (x) such that the absolute maximum of f (x) on the interval [ - 2, 2] does not exist and the absolute minimum does not exist. |
| 53. Sketch a graph of  for x > 0 and determine where the graph is steepest (that is, find where the
slope is a maximum). |
| 54. Sketch a graph of  and determine where the graph is steepest. (Note: This is an important problem in probability theory.) |
| 55. Sketch a graph showing that y = f (x) = x2 + 1 and y = g(x) = ln x do not intersect. Find x to minimize f (x) - g(x). At this value of x, show that the tangent lines to y = f (x) and y = g(x) are
parallel. |
| 56. Explain graphically why it makes sense that the tangent lines in exercise 55 are parallel, given that at this point the vertical
distance between the graphs is smallest. |
| 57. Give an example showing that the following statement is false (not always true): between any two local minima of f (x) there is a
local maximum. |
| 58. Is the statement in exercise 57 true if f (x) is continuous? |
| 59. A section of roller coaster is in the shape of y = x5 - 4x3 - x + 10, where x is between - 2 and 2. Find all relative
extrema and explain what portions of the roller coaster they represent. Find the location of the steepest part of the roller coaster. |
| 60. Suppose a large computer file is sent over the Internet. If the probability that it reaches its destination without any errors is x, then the probability that an error is made is 1 - x. The field of information theory studies such situations. An important quantity is entropy (a measure of unpredictability), defined by H = - xln x - (1 - x)ln (1 - x), for 0<x<1. Find the value of x that maximizes this quantity. Explain why this value makes sense as the probability that maximizes entropy. |
| 61. Researchers in a number of fields (including population biology, economics and the study of animal tumors) make use of the Gompertz growth curve,  As t  , show that W(t) a and W' (t) 0. Find the maximum growth rate. |
 | 62. In this exercise, we will explore the family of functions f (x) = x3 + cx + 1, where c is a
constant. How many and what types of relative extrema are there? (Your answer will depend on the value of c.) Assuming that this family is indicative of all
cubic functions, list all types of cubic functions. Without looking at specific examples, try to list all types of fourth-order polynomials, sketching a graph of each. |
| | 63. Explore the graphs of e - x, xe - x, x2e - x and x3e - x. Find all
local extrema and determine the behavior as x. You can think of the graph of xne - x as showing theresults of a tug-of-war: xn as x but e - x 0 as x. Describe the graph of xne - x in terms of this tug-of-war. |
| | 64.
Johannes Kepler (1571-1630) is best known as an astronomer, especially for his three laws of planetary motion. However, his discoveries were primarily due to his brilliance as a mathematician. While serving in Austrian Emperor Matthew I's court, Kepler observed the ability of Austrian vintners to quickly and mysteriously compute the capacities of a variety of wine casks. Each cask (barrel) had a hole in the middle of its side (see Figure a). The vintner would insert a rod in the hole until it hit the far corner and then announce the volume. Kepler first analyzed the problem for a cylindrical barrel (see Figure b). The volume of a
cylinder is V = r2h. In Figure b, r = y and h = 2x so V = 2 y2x. Call the rod measurement z. By the Pythagorean Theorem, x2 + (2y)2 = 2. Kepler's mystery was how to compute V given only z. The key observation made by Kepler was that Austrian wine casks were made with the same height-to-diameter ratio (for us, x/y). Let t = x/y and show that z2/y2 = t2 + 4. Use this to replace y2 in the volume formula. Then replace x with  . Show that  . In this formula, t is a
constant so the vintner could measure z and quickly estimate the volume. We haven't told you yet what t equals. Kepler assumed that the vintners would have made a smart choice for this ratio. Find the value of t that maximizes the volume for a given z. This is, in fact, the ratio used in the construction of Austrian wine casks!
Figure a
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Figure b
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