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 | 1. Explain to a non-calculus-speaking friend how to (mechanically) use the power rule. Decide if it is better to give separate explanations for positive and negative exponents; integer and noninteger exponents; other special cases. |
| | 2. In the 1700s, mathematical “proofs” were, by modern standards, a bit fuzzy and lacked rigor. In 1734, the Irish metaphysician Bishop Berkeley wrote The Analyst to an “infidel mathematician” (thought to be Edmund Halley of Halley's comet fame). The accepted proof at the time of the power rule may be described as follows.
If x is incremented to x+h , then xn is incremented to (x + h)n. It follows that   Now, let the increment h vanish, and the
derivative is nxn-1.
Bishop Berkeley objected to this argument.
“But it should seem that the reasoning is not fair or conclusive. For when it is said, let the increments vanish, the former supposition that the increments were something, or that there were increments, is destroyed, and yet a consequence of that supposition is retained. Which... is a false way of reasoning. Certainly when we suppose the increments to vanish, we must
suppose... everything derived from the supposition of their existence to vanish with them. ”
Do you think Berkeley's objection is fair? Is it logically acceptable to assume something exists to draw one deduction, and then assume that the same thing does not exist to avoid having to accept other consequences? For example, can you use the principle of “always tell the truth” to get a friend to tell you something, but then get mad when part of the truth is unflattering to you? Mathematically speaking, how does the limit avoid Berkeley's objection of the increment h both existing and not existing? |
| | 3. The historical episode in exercise 2 is just one part of an ongoing conflict between people who blindly use mathematical techniques without proof and those who insist on a full proof before permitting anyone to use the technique. To which side are you sympathetic? Defend your position in an essay. Try to anticipate and rebut the other side's arguments. |
| | 4. Explain the first two terms in the expansion (x+h)n = xn+nhxn-1+ , where n is a positive integer. Think of multiplying out (x+h)(x+h) (x+h) (x+h) ; how many terms would include xn ? xn-1 ? |
| In exercises 5-24, find the
derivative of each function.
|
| 5. f (x) = x3-2x+1 | 6. f (x) = 2x4-3x2+x |
| 7. f (x) = 3x2-4 | 8. f (x) = x9-3x5+4x2+x |
| 9. g(x) = 4 | 10. g(x) = -4x |
| 11.  | 12.  |
| 13.  | 14.  |
| 15.  | 16.  |
| 17. f (s) = 2s3/2-3s-1/3 | 18. f (t ) = 3t  -2t 1.3 |
| 19.  | 20.  |
| 21. f (x) = x(3x2- ) | 22. f (x) = (x+1)(3x2-4) |
| 23.  | 24. 
|
| In exercises 25-34, compute the indicated
derivative.
|
| 25. f '' (x) for f (x) = x4+3x2-2
|
| 26. 
|
| 27. 
|
| 28. 
|
| 29. 
|
| 30. f '' ' (t ) for f (t ) = t 6-3t 3+2
|
| 31. f (4)(x) for f (x) = x4+3x2-2
|
| 32. f (5)(x) for f (x) = x10-3x4+2x-1
|
| 33. 
|
| 34. 
|
| In exercises 35-38, use the given position
function to find the velocity and acceleration functions.
|
| 35. s(t ) = -16t 2+40t+10 | 36. s(t ) = 12t 3-6t-1 |
| 37.  | 38. 
|
| In exercises 39-42, the given
function represents the height of an object. Compute the velocity and acceleration at time
t = a. Is the object going up or down? Is the speed of the object increasing or decreasing?
|
| 39. h(t ) = -16t 2+40t+5, a = 1
|
| 40. h(t ) = -16t 2+40t+5, a = 2
|
| 41. h(t ) = 10t 2-24t, a = 2
|
| 42. h(t ) = 10t 2-24t, a = 1
|
| In exercises 43-46, find an equation of the tangent line to
y = f (x) at x = a.
|
| 43. f (x) = 4-2x, a = 1 | 44. f (x) = x2-2x+1, a = 2 |
| 45. f (x) = x2-2, a = 0 | 46. f (x) = 3x+4, a = 2
|
| In exercises 47 and 48, determine the value(s) of
x for which the tangent line to y = f (x) is horizontal. Graph the
function and determine the graphical significance of each point.
|
| 47. f (x) = x3-3x+1 | 48. f (x) = x4-2x2+2
|
| In exercises 49 and 50, determine the value(s) of
x for which the slope of the tangent line to
y = f (x) does not exist. Graph the function and determine the graphical significance of each point.
|
| 49. f (x) = x2/3 | 50. f (x) = x1/3
|
| In exercises 51 and 52, one curve represents a
function
f (x) and the other two represent f ' (x) and f ''
(x). Determine which is which.
|
| 51.
|
| 52.
|
| In exercises 53 and 54, find a general formula for the
nth derivative f (n)(x).
|
| 53. f (x) = | 54. 
|
| 55. Find a second-degree
polynomial (of the form ax2+bx+c ) such that f (0) = -2, f ' (0) = 2 and f '' (0) = 3. |
| 56. Find a second-degree
polynomial (of the form ax2+bx+c ) such that f (0) = 0, f ' (0) = 5 and f '' (0) = 1. |
| 57. Show that the formula  rule. Explain the more general formula  graphically. |
| 58. Show that the formula  is a special case of the power rule. Explain the more general formula  graphically. |
| 59. A public official solemnly proclaims, “We have achieved a reduction in the rate at which the national debt is increasing. ” If d(t ) represents the national debt, which
derivative of d(t ) is being reduced? What can you conclude about the size of d(t ) itself? |
| 60. A rod made of an inhomogeneous material extends from x = 0 to x = 4. The mass of the portion of the rod from x = 0 to x = t is given by m(t ) = 3t 2 kg. Compute m' (t ) and explain why it represents the density of the rod. |
| 61. For most land animals, the relationship between leg width w and body length b follows an equation of the form w = c  b3/2 for some
constant c. Show that if b is large enough, w' (b) > 1. Conclude that for larger animals, leg width (necessary for support) increases faster than body length. Why does this put a limitation on the size of land animals? |
| 62. Suppose the
function v(d) represents the average speed in m/s of the world record running time for d meters. For example, if the fastest 200 - meter time ever is 19.32s, then v(200) = 200/19.32  10.35. Compare the
function f (d) = 26.7d-0.177 to the values of v(d) , which you will have to research and compute, for distances ranging from d = 400 to d = 2000. Explain what v' (d) would represent. |
| In exercises 63-70, find a
function with the given derivative.
|
| 63. f ' (x) = 4x3 | 64. f ' (x) = 5x4 |
| 65. f ' (x) = x4 | 66. f ' (x) = x5 |
| 67. f ' (x) = 3x-1 | 68. f ' (x) = 4-x2 |
| 69. f ' (x) = | 70. 
|
 | 71. A plane is cruising at an altitude of 2 miles at a
distance of 10 miles from an airport. Choosing the airport to be at the point (0, 0), the plane starts at (10, 2). The plane starts its descent and lands at the airport. Sketch a picture of a reasonable flight path y = f (x) , where y represents altitude and x gives the ground
distance from the airport. (Think about it as you draw!) Explain what the derivative f ' (x) represents. (Hint: It's not velocity.) Explain why it is important and/or necessary to have f (0) = 0 , f (10) = 2 , f ' (0) = 0 and f ' (10) = 0. The simplest
polynomial that can meet these requirements is a cubic polynomial f (x) = ax3+bx2+cx+d (note: four requirements, four constants). Find values of the constants a, b, c and d to fit the flight path. (Hint: Start by setting f (0) = 0 and then
set f ' (0) = 0. You may want to use your CAS to solve the equations.) Graph the resulting
function; does it look right? Suppose that airline regulations prohibit a derivative of  or larger. Why might such a regulation exist? Show that the flight path you found is illegal. Argue that in fact all flight paths meeting the four requirements are illegal. Therefore, the descent needs to start further away than 10 miles. Find a flight path with descent starting 20 miles that meets all requirements. |
| | 72. We discuss a graphical interpretation of the second
derivative in Chapter 3. You can discover the most important aspects of that here. For f (x) = x4-2x2-1 , solve the equations f ' (x) = 0 and f '' (x) = 0. What do the solutions of the equation f ' (x) = 0 represent graphically? The solutions of the equation f '' (x) = 0 are a little harder to interpret. Looking at the graph of f (x) near x = 0 , would you say that the graph is curving up or curving down? Compute f '' (0). Looking at the graph near x = 2 and x = -2 , is the graph curving up or down? Compute f '' (2) and f '' (-2). Where does the graph change from curving up to curving down and vice versa? Hypothesize a relationship between f '' (x) and the curving of the graph of y = f (x). Test your hypothesis on a variety of functions. (Try y = x4-4x3.) |
| | 73. In the enjoyable book Surely You're Joking Mr.
Feynman, physicist Richard Feynman tells the story of a contest he had pitting his brain against the technology of the day (an abacus). The contest was to compute the cube root of 1729.03. Feynman came up with 12.002 before the abacus expert gave up. Feynman admits to some luck in the choice ofthe number 1729.03: he knew that a
cubic foot contains 1728cubic inches. Explain why this told Feynman that the answer is slightly greater than 12. How did he get three digits of accuracy? “I had learned in calculus that for small fractions, the cube root's excess is one-third of the number's excess. The excess, 1.03, is only one part in nearly 2000. So all I had to do is find the fraction 1/1728 , divide by 3 and multiply by 12. ” To see what he did, find an equation of the tangent line to y = x1/3 at x = 1728 and find the y - coordinate of the tangent line at x = 1729.03. |
| | 74. Suppose that you want to find solutions of the equation x3-4x2+2 = 0. Show graphically that there is a solution between x = 0 and x = 1. We will
approximate this solution in stages. First, find an equation of the tangent line to y = x3-4x2+2 at x = 1. Then, determine where this tangent line crosses the x - axis. Show graphically that the x - intercept is considerably closer to the solution than is x = 1. Now, repeat the process: for the new x - value, find the equation of the tangent line, determine where it crosses the x - axis and show that this is closer still to the desired solution. This process of using tangent lines to produce continually improved approximations is referred to as Newton's method. We discuss this in some detail in
Section 3.2. |
