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 | 1. We have talked about sketching representative graphs, but it is often impossible to draw a graph correctly to scale that shows all of the properties we might be interested in. For example, try to generate a computer or calculator graph that shows all three local extrema of x4 - 25x3 - 2x2 + 80x - 3. When two extrema have y - coordinates of approximately -60 and 50, it takes a very large graph to also show a point with y = -40,000 ! If an accurate graph cannot show all the points of interest, perhaps a freehand sketch like the one shown below is needed.
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| There is no scale shown on the graph because we have distorted different portions of the graph in an attempt to show all of the interesting points. Discuss the relative merits of an “honest” graph with a consistent scale but not showing all the points of interest versus a caricature graph which distorts the scale but does show all the points of interest. |
| | 2. While studying for a test, a friend of yours says that a graph is not allowed to intersect an asymptote. While it is often the case that graphs don't intersect asymptotes, there is definitely not any rule against it. Explain why graphs can intersect a horizontal asymptote any number of times (Hint: Look at the graph of e - xsin x ) but can't pass through a vertical asymptote. |
| | 3. Explain why polynomials never have vertical or horizontal asymptotes. |
| | 4. Explain how the graph of f (x) = cos x - x in example 6.6 relates to the graphs of y = cos x and y = - x. Based on this discussion, explain how to sketch the graph of y = x + sin x. |
| In exercises 5-44, graph the function and completely discuss the graph as in example 6.2.
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| 5. f (x) = x3 - 3x2 + 3x | 6. f (x) = x3 - 9x + 1 |
| 7. f (x) = x4 - 3x2 + 2x | 8. f (x) = x4 + 8x -2 |
| 9. f (x) = x5 - 2x3 + 1 | 10. f (x) = x6 - 10x5 - 7x4 + 80x3 + 12x2 - 192x |
| 11. | 12. |
| 13. f (x) = x + 4/x | 14. f (x) = (x2 -1)/x |
| 15. f (x) = sin x - cos x | 16. f (x) = cos 3x |
| 17. | 18. f (x) = xe - 4x |
| 19. f (x) = xln x | 20. f (x) = xln x2 |
| 21. | 22. |
| 23. | 24. |
| 25. | 26. |
| 27. | 28. |
| 29. | 30. |
| 31. f (x) = x + sin x | 32. f (x) = 2x + sin 2x |
| 33. f (x) = x5 - 5x | 34. |
| 35. | 36. |
| 37. f (x) = x1/5(x + 1) | 38. |
| 39. | 40. |
| 41. f (x) = e - 2/x | 42. |
| 43. | 44. f (x) = x10e - x
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| In exercises 45-50, the “family of functions” contains a parameter
c. The value of c affects the properties of the functions. Determine what differences, if any, there are for
c being zero, positive or negative. Then determine what the graph would look like for very large positive
c's and for very negative c's.
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| 45. f (x) = x4 + cx2 | 46. f (x) = x4 + cx2 + x |
| 47. | 48. |
| 49. f (x) = sin (cx) | 50.
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| 51. In a variety of applications, researchers model a phenomenon whose graph starts at the origin, rises to a single maximum and then drops off to a horizontal asymptote of y = 0. For example, the probability density function of events such as the time from conception to birth of an animal and the amount of time surviving after contracting a fatal disease might have these properties. Show that the family of functions xe - bx has these properties for all positive constants b. What effect does b have on the location of the maximum? In the case of the time since conception, what would b represent? In the case of survival time, what would b represent? |
| 52. The “FM” in FM radio stands for frequency modulation, a method of transmitting information encoded in a radio wave by modulating (or varying) the frequency. A basic example of such a modulated wave is f (x) = sin (x + cos x). Use computer-generated graphs of f (x), f '(x) and f '' (x) to try to locate all relative extrema of f (x) . |
| 53. A rational function is a function of the form  , where p(x) and q(x) are polynomials. Is it true that all rational functions have vertical asymptotes? Is it true that all rational functions have horizontal asymptotes? |
| 54. It can be useful to identify asymptotes other than vertical and horizontal. For example, the parabola x2 is an asymptote of f (x) if  and/or  . Show that x2 is an asymptote of f (x) =  . Graph y = f (x) and zoom out until the graph looks like a parabola. (Note: the effect of zooming out is to emphasize large values of x .) |
| A function f (x) has a slant asymptote y = mx + b (
m 0 ) if  and/or  . In exercises 55-60, find the slant asymptote (use long division to rewrite the function). Then, graph the function and its asymptote on the same axes. |
| 55. | 56. |
| 57. | 58. |
| 59. | 60.
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| In exercises 61-64, find a function whose graph has the given asymptotes.
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| 61. x = 1, x = 2 and y = 3
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| 62. x = -1, x = 1 and y = 0
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| 63. x = -1, x = 1, y = -2 and y = 2
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| 64. x = 1, y = 2 and x = 3
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 | 65. For each function, find a polynomial p(x) such that  .
(a)  (b)  (c) 
Show by zooming out that f (x) and p(x) look similar for large x. The first term of a polynomial is the term with the highest power (e.g., x3 is the first term of x3 - 3x + 1 ). Can you zoom out enough to make the graph of f (x) look like the first term of its polynomial asymptote? State a very quick rule enabling you to look at a rational function and determine the first term of its polynomial asymptote (if one exists). |
| | 66. One of the natural enemies of the balsam fir tree is the spruce budworm, which attacks the leaves of the fir tree in devastating outbreaks. Define N(t) to be the number of worms on a particular tree at time t. A mathematical model of the population dynamics of the worm must include a term to indicate the worm's death rate due to its predators (e.g., birds). The form of this term is often taken to be  for positive constants A and B. Graph the functions   and  for x > 0. Based on these graphs, discuss why  is a plausible model for the death rate by predation. What role do the constants A and B play? The possible stable population levels for the spruce budworms are determined by intersections of the graphs y = r(1 - x/k) and y =  . Here, x = N/A, r is proportional to the birthrate of the budworms and k is determined by the amount of food available to the budworms. Note that y = r(1 - x/k) is a line with y - intercept r and x - intercept k. How many solutions are there to the equation  ? (Hint: The answer depends on the values of r and k .) One current theory is that outbreaks are caused in situations where there are three solutions and the population of budworms jumps from a small population to a large population. |
