1. Homework, Page 124
1. Let f (x) = 3x2. Show that f (2+h) =3h2 + 12h Then show that and compute f ′(2) by taking the limit as h → 0.
Rogawski Calculus
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2. Homework, Page 124
Compute f ′(a) in two ways, using Equations (1) and (2) 5.
Rogawski Calculus
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3. Homework, Page 124 5. Continued Rogawski Calculus
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4. Homework, Page 124
Refer to the function whose graph is shown. 9. Estimate for h = 0.5 and h = –0.5. Are these numbers larger or smaller than f ′ (2)? Explain. The value of f ′ (2) lies between these two values, as illustrated in Figure 9.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 124
Refer to the function whose graph is shown. 13. Which is larger, f ′ (5.5) or f ′ (6.5)? The value of f ′ (5.5) is larger than the value of f ′ (5.5) because the slope of the curve is steeper at x = 5.5 than it is at x = 6.5
Rogawski Calculus
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6. Homework, Page 124
Use the limit definition to find the derivative of the linear function. 17. g (x) = 9 – t.
Rogawski Calculus
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7. Homework, Page 124
21. Let . Show that Then use this formula to compute f ′(9) (by taking the limit).
Rogawski Calculus
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8. Homework, Page 124
Compute the derivative at x = a using the limit definition and find an equation of the tangent line. 25. f (x) = x3, a = 2
Rogawski Calculus
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9. Homework, Page 124
Compute the derivative at x = a using the limit definition and find an equation of the tangent line. 29. f (x) = x–1, a = 3
Rogawski Calculus
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10. Homework, Page 124
Compute the derivative at x = a using the limit definition and find an equation of the tangent line. 33.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

11. Homework, Page 124
Compute the derivative at x = a using the limit definition and find an equation of the tangent line. 37.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

12. Homework, Page 124
41. What is the equation of the tangent line at x = 3, assuming that f (3) = 5 and f ′(3) = 2?
Rogawski Calculus
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13. Homework, Page 124 45. f ′(1) appears to be about – 0.5
Rogawski Calculus
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14. Homework, Page 124
49. The vapor pressure of water is defined as the atmospheric pressure P at which no net evaporation takes place. See graph and table. (a) Which is larger, P′ (300) or P′ (350)? From the graph, P′ (300) < P′ (350) (b) Estimate P′ (T) for T = 303, 313, 323, 333, 343 using the table and the average of the difference quotients for h = 10 and h = –10.
T (K)
P (atm)
293
0.0278
333
0.2067
303
0.0482
343
0.3173
313
0.0808
353
0.4754
323
0.1311
Rogawski Calculus
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15. Homework, Page 124
49. (b) Estimate P′ (T) for T = 303, 313, 323, 333, 343 using the table and the average of the difference quotients for h = 10 and h = –10.
Rogawski Calculus
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16. Homework, Page 124
Each of the limits represents a derivative f '(a). Find f (x) and a.
Rogawski Calculus
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17. Homework, Page 124
Each of the limits represents a derivative f '(a). Find f (x) and a.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

18. Homework, Page 124
61. Let (a) Plot f (x) over [1, –1]. Then zoom in near x = 0 until the graph appears straight and estimate the slope f '(x). The slope appears to be about –0.1. (b) Use your estimate to find an approximate equation to the tangent line at x = 0. Plot this line and the graph on the same axes.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

19. Homework, Page 124
65. Apply the method of Example 6 to f (x) = sin x to determine accurately to four decimal places. The difference quotient yields the value of accurate to four decimal places with h = ± The value according to our calculators is ……
h > 0
Diff Quot
h < 0
0.01
-0.01
0.001
-0.001
0.0001
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

20. Homework, Page 124
69. Figure 18 gives the average antler weight W of a red deer as a function of age t. Estimate the slope of the tangent line to the graph at t = 4. For which values of t is the slope of the tangent line equal to zero? For which is it negative? The slope of the tangent appears to equal zero at t = 10 and t = The slope is negative on [10, 11.5].
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

21. Chapter 3: Differentiation Section 3.2: The Derivative as a Function
Jon Rogawski Calculus, ET First Edition
Chapter 3: Differentiation
Section 3.2: The Derivative as a Function
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

22. The Derivative of a Function

23. The Derivative as a Function
If y = f (x), we may also write y′ or y' (x) for f '(x).
The domain of f '(x) consists of all values of x in the domain of f (x) for which the limit exists.
Function f (x) is said to be differentiable on (a, b) if f '(x) exists for all x in (a, b).
If f '(x) exists for all values of x in the domain of f (x), then we say f (x) is differentiable.

24. Example: Finding the Derivative of a Function

25. Example: Finding the Derivative of a Function

26. Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

27. Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company