Homework, Page 124 1. Let f (x) = 3x2. Show that f (2+h) =3h2 + 12h + 12. Then show that and compute f ′(2) by taking the limit as h → 0. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 124 Compute f ′(a) in two ways, using Equations (1) and (2) 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 124 5. Continued Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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

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 Copyright © 2008 W. H. Freeman and Company

Homework, Page 124 Use the limit definition to find the derivative of the linear function. 17. g (x) = 9 – t. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 124 21. Let . Show that Then use this formula to compute f ′(9) (by taking the limit). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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 Copyright © 2008 W. H. Freeman and Company

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 Copyright © 2008 W. H. Freeman and Company

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

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

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 Copyright © 2008 W. H. Freeman and Company

Homework, Page 124 45. f ′(1) appears to be about – 0.5 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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 Copyright © 2008 W. H. Freeman and Company

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 Copyright © 2008 W. H. Freeman and Company

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

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

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

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 = ± 0.00001. The value according to our calculators is 0.7071067812…… h > 0 Diff Quot h < 0 0.01 0.70356 -0.01 0.71063 0.001 0.70675 -0.001 0.70746 0.0001 0.70707 -0.0001 0.70714 0.00001 0.70710 -0.00001 0.70711 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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 = 11.5. The slope is negative on [10, 11.5]. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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

The Derivative of a Function

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.

Example: Finding the Derivative of a Function

Example: Finding the Derivative of a Function

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

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