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3.1 |
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 | 1. We constructed a variety of linear approximations in this section. Approximations can be “good” approximations or “bad” approximations. Explain why it can be said that y = x is a good approximation to y = sinx near x = 0 but y = 1 is not a good approximation to y = cosx near x = 0. (Hint: Look at the graphs of y = sin x and y = x on the same axes, then do the same with y = cos x and y = 1.) |
| | 2. Briefly explain in terms of tangent lines why the approximation in
Example 1.2 gets worse as x gets farther from 8. |
| | 3. A friend is struggling with L'Hôpital's Rule. When asked to work a problem, your friend says, “First, I plug in for x and get 0 over 0. Then I use the quotient rule to take the
derivative. Then I plug x back in. ”Explain to your friend what the mistake is and how to correct it. |
| | 4. Suppose that two runners begin a race from the starting line, with one runner initially going twice as fast as the other. If f (t) and g(t) represent the positions of the runnersat time t  0, explain why we can assume that f (0) = g(0) = 0 and  Explain in terms of the runners' positions why L'Hôpital's Rule holds: that is, 
|
| In exercises 5-12, find the
linear approximation to f (x) at x=x0. Graph the
function and its linear approximation.
|
| 5. | 6. f (x) = (x + 1)1/3, x0 = 0 |
| 7. | 8. f (x) = 2/x, x0 = 1 |
| 9. f (x) = sin 3x, x0 = 0 | 10. f (x) = sin x, x0 =  |
| 11. f (x) = e2x, x0 = 0 | 12. 
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| In exercises 13-16, find the
linear approximation at
x=0 to show that the following commonly used approximations are valid for “small”
x. Compare the approximate and exact values for
x=0.01, x=0.1 and x=1.
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| 13. tan x x | 14.  |
| 15. | 16. ex 1 + x
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| In exercises 17-22, use
linear approximations to estimate the quantity.
|
| 17.sin 1 | 18. sin  |
| 19. | 20.  |
| 21. | 22.ln 2.8 (Hint: ln e = 1.) |
| 23. For exercises 19-21, compute the error (the
absolute value of the difference between the exact value and the linear approximation). |
| 24. Thinking of exercises 19-21 as numbers of the form  , denote the errors as e( x) (where x = 0.04, x = 0.08 and x = 0.16). Based on these three computations, determine a
constant c such that e(x)  c(x)2. |
| 25. Use a computer algebra system (CAS) to determine the
range of x's in exercise 13 for which the approximation is accurate to within 0.01. That is, find x such that |tan x - x| <0.01. |
| 26. Use a CAS to determine the
range of x's in exercise 16 for which the approximation is accurate to within 0.01. That is, find x such that |ex - (1 + x)|<0.01. |
| In exercises 27-30, use
linear interpolation to estimate the desired quantity.
|
| 27. A company estimates that f (x) thousand software games can be sold at the price of $x as given in the table.
Estimate the number of games that can be sold at (a) $24 and (b) $36.
|
| x | 20 | 30 | 40 |
| f (x) | 18 | 14 | 12 |
|
| 28. A vending company estimates that f (x) cans of soft drink can be sold in a day if the temperature is x°F as given in the table.
Estimate the number of cans that can be sold at (a) 72° and (b) 94°.
|
| x | 60 | 80 | 100 |
| f (x) | 84 | 120 | 168 |
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| 29. An animation director enters the position f (t) of a character'shead after t frames of the movie as given in the table.
If the computer software uses interpolation to determine the intermediate positions, determine the position of the head at frame number (a) 208 and (b) 232.
|
| t | 200 | 220 | 240 |
| f (t) | 128 | 142 | 136 |
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| 30. A sensor measures the position f (t) of a particle t microseconds after a collision as given in the table.
Estimate the position of the particle at time (a) t = 8 and (b) t = 12.
| |
| In exercises 31-42, use L'Hôpital's Rule to evaluate the limit.
|
| 31. | 32. |
| 33. | 34. |
| 35. | 36. |
| 37. | 38. |
| 39. | 40. |
| 41. | 42.
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| 43. Compute  and compare your result to that of
Example 1.5. |
| 44. Compute  and compare your result to that of
Example 1.6. |
| 45. Use your results from exercises 43 and 44 to evaluate  and  without doing any calculations. |
| 46. If  , what can be said about  ? Explain why knowing that
 for a 0 does not tell you anything about  . |
| 47. Find all errors in the string
 
Then determine the correct value of the limit. |
| 48. Find all errors in the string
 .
Then determine the correct value of the limit. |
| 49. Starting with  , cancel sin to get  , then cancel x's to get  . This answer is correct. Are either of the steps used valid? Use
linear approximations to argue that the first step is likely to give a correct answer. |
| 50. Evaluate  for nonzero constants n and m. |
| 51. Evaluate  for any
constant c. |
| 52. Evaluate  for any
constant c. |
| 53. In section 1.1, we briefly discussed the position of a baseball thrown with the unusual knuckleball pitch. The left/right position (in feet) of a ball thrown with spin rate  and a particular grip at time t seconds is f () = (2.5/)t - (2.5/42) sin4t. Treating t as a
constant and as the variable (change to x if you like), show that  for any value of t. (Hint: Find a common denominator and use L'Hôpital's Rule.) Conclude that this pitch does not move left or right at all. |
| 54. In this exercise, we look at a knuckleball thrown with a different grip than that of exercise 53. The left or right position (in feet) of a ball thrown with spin rate and this new grip at time t seconds is f () = (2.5/42) - (2.5/42) sin(4t + /2). Treating t as a
constant and as the variable (change to x if you like), find  . Your answer should depend on t. By graphing this
function of t, you can see the path of the pitch (use a domain of 0 t 0.68 ). Describe this pitch. |
| 55. A water wave of length L meters in water of depth d meters has velocity v; satisfying the equation
 .
Treating L as a constant and thinking of v2 as a function f (d), use a
linear approximation to show that f (d) 9.8d for small values of d. That is, for small depths the velocity of the wave is approximately  and is independent of the wavelength L. |
| 56. Planck's law states that the energy density of blackbody radiation of wavelength x is given by
 .
Use the linear approximation in exercise 16 to show that f (x) 8 kT/x4, which is known as the Rayleigh-Jeans law. |
 | 57. In this exercise, we introduce Taylor series (explored in depth in
Chapter 8). Start with the limit  . Briefly explain why this means that for x close to 0, sin x x. Graph y = sinx and y = x to see why this is true. If you look far enough away from x = 0, the graph of y = sin x eventually curves noticeably. We will find polynomials of higher order to match this curving. Show that  . This means that sin x - x 0 or (again) sin x x. Show that  . This says that if x is close to 0, then  or  . Graph these two functions to see how well they match up. To continue, |
| compute  and  for the |
| appropriate approximation f (x). At this point, look at the pattern of terms you have (Hint: 6 = 3! and 120 = 5!). Using this pattern,
approximate sin x with an 11th-degree polynomial and graph the two functions. |
