# quotient

102 CHAPTER 3 DIFFERENTIATION 3.1 SUMMARY f'(a) = lim f(a+h)-f(a) • The difference quotient: f(a+h)-f(a) h The difference quotient is the slope of the secant line through the points P = (a, f(a)) and Q = (a +h, f(a+h)) on the graph of f. • The derivative f'(a) is defined by the following equivalent limits: f(x) – f(a) = lim -0 h x x – a If the limit exists, we say that f is differentiable at x = a. By definition, the tangent line at P = (a, f(a)) is the line through P with slope f'(a) (assuming that f'(a) exists). Equation of the tangent line in point-slope form: y-f(a) = f'(a)(x-a) • To calculate f'(a) using the limit definition: Step 1. Write out the numerator of the difference quotient. Step 2. Divide by h and simplify. Step 3. Compute the derivative by taking the limit. • For small values of h, we have the estimate f'(a) f(a+h)-f(a) h 3.1 EXERCISES Preliminary Questions 1. Which of the lines in Figure 11 are tangent to the curve? 3. Find a and h such that f(a+h)-f(a) is equal to the slope of h the secant line between (3, f (3)) and (5, / (5)). FIGURE 11 4. Which derivative is approximated by tan( + 0.0001) -1 0.0001 5. What do the following quantities represent in terms of the graph of f(x) = sinx? sin 1.3 – sin 0.9 (a) sin 1.3 – sin 0.9 (b) (c) ‘(0.9) 0.4 2. What are the two ways of writing the difference quotient? Exercises 1. Let f(x) = 5×2. Show that (3+h) = 5h2 + 306 +45. Then show that S(3+h)-f(3) = 5h + 30 h and f'(3) by taking the limit as h 0 2. Let f(x) = 2×2 – 3x – 5. Show that the secant line through (2. f(2)) and (2+h, f (2+h)) has slope 2h +5. Then use this for- mula to compute the slope of: (a) The secant line through (2. f (2)) and (3. f(3)) (b) The tangent line at x = 2 (by taking a limit) In Exercises 3-8, compute f'(a) in two ways, using Eq. (1) and Eq. (2). 3. S(x) = x2 +9x, a = 0 4. S(x) = x2 + 9x, a = 2 5. S(x) = 3×2 + 4x +2, a = -1 6. S(x)=x”, a = 2 7. f(x) = x + 2x, a = 1 8. f(x) = 5; a = 2 In Exercises 9-12, refer to Figure 12. 9. Find the slope of the secant line through (2, 2)) and (2.5, S (2.5)). Is it larger or smaller than f'(2)? Explain. (2+h)-f(2) 10. Estimate for h = -0.5. What does this quantity represent? Is it larger or smaller than /’ (2)? Explain. h 104 CHAPTER 3 3 DIFFERENTIATION 50. GU Let /(x) = cotx. Estimate f'(4) graphically by zooming in on a plot of / near x = 51. Determine the intervals along the x-axis on which the derivative in Figure 16 is positive. (c) Similarly, compute f'(x) to four decimal places for x = 1, 2, 3, 4. (d) Now compute the ratios f'(x)/S’O) for x = 1, 2, 3, 4. Can you guess an approximate formula for f'(x)? у 4.0 + 3.5 3.0 2.5 2.0 1.5 1.0 0.5 2 3 FIGURE 18 Graph of f(x) = 2*. 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 FIGURE 16 h 52. Sketch the graph of f(x) = sin x on [0, 1] and guess the value of f'(4). Then calculate the difference quotient at x = for two small positive and negative values of h. Are these calculations consistent with your guess? In Exercises 53-58, each limit represents a derivative f'(a). Find f(x) and a (5+h)3 – 125 x3 – 125 53. lim 54. lim h0 h x-5 sin ( + h) – 0.5 4 55. lim 56. lim h0 h xx- h 5h – 1 57. lim 58. lim h h0 h 59. Apply the method of Example 7 to f(x) = sin x to determine 1′ () accurately to four decimal places. 63. GO Sketch the graph of f(x) = x5/2 on [0,6). (a) Use the sketch to justify the inequalities for h>0: f(4) -(4-h) ) < h (b) Use (a) to compute f'(4) to four decimal places. (c) Use a graphing utility to plot y = f(x) and the tangent line at x = 4, utilizing your estimate for f'(4). 64. GU Verify that P = (1, ) lies on the graphs of both \$(x) = 1/(1+x?) and L(x) = 1 + (x – 1) for every slope m. Plot y = f(x) and y = L(x) on the same axes for several values of m until you find a value of m for which y = L(x) appears tangent to the graph of /. What is your estimate for f'(1)? 65. GU Use a plot of f(x) = x* to estimate the value c such that f'(c) = 0. Find e to sufficient accuracy so that 25 (c+h)-f(c) h < 0.006 for h= +0.001 60. Apply the method of Example 7 to f(x) = cos x to de- termine f'() accurately to four decimal places. Use a graph off to explain how the method works in this case. 61. For each graph in Figure 17, determine whether f'(1) is larger or smaller than the slope of the secant line between x = 1 and x=1+h for h > 0. Explain. ( y = f(x) 66. GU Plot f(x) = x’ and y = 2x + a on the same set of axes for several values of a until the line becomes tangent to the graph. Then estimate the value c such that f'(c) = 2. In Exercises 67-73, estimate derivatives using the symmetric differ- ence quotient (SDQ), defined as the average of the difference quotients at h and -h: 11(a+h) – f(a) f(a – h) – f(a) h -h f(a+h)-f(a – h) 4 2h The SDQ usually gives a better approximation to the derivative than the difference quotieni. 67. The vapor pressure of water at temperature T (in kelvins) is the atmospheric pressure P at which no net evaporation takes place. Use the following table to estimate P'(T) for T = 303, 313, 323, 333, 343 by computing the SDQ given by Eq. (4) with h = 10. T (K) 293 303 313 323 333 343 353 P (atm) 0.0278 0.0482 0.0808 0.1311 0.2067 0.3173 0.4754 y = f(x) -X 1 1 (A) (B) FIGURE 17 62. Refer to the graph of f(x) = 2* in Figure 18. (a) Explain graphically why, for h > 0, f(-h)-f(0) s(h)-f(0) 0, f(-h)-f(0) s(h)-f(0)
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