A 53-54 Use the definition of a derivative to find f'(x) and f"(x). Then graph f.f, and f" on a common screen and check to see if ify each your answers are reasonable. – 3x 54. f(x) = x 53. f(x) = 3x + 2x + 1 55. If f(x) = 2x² – x', find f'(x), f"(x), f"(x), and f(ª(x). Graph f. f', f", and f"on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives? 56. (a) The graph of a position function of a car is shown, where s is measured in feet andt in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at t = 10 seconds? fy each 100 10 20 (b) Use the acceleration curve from part (a) to estimate the jerk at t = 10 seconds. What are the units for jerk? he 57. Let f(x) = Vx. (a) If a # 0, use Equation 2.7.5 to find f'(a). (b) Show that f'(0) does not exist. (c) Show that y = x has a vertical tangent line at (0, 0). (Recall the shape of the graph of f. See Figure 1.2.13.) and your 2/3 58. (a) If g(x) = x2, show that g'(0) does not exist. (b) If a + 0, find g'(a). (c) Show that y = x2/3 has a vertical tangent line at (0, 0). (d) Illustrate part (c) by graphing y = x2/3. 59. Show that the function f(x) = |x – 6 is not differentiable at 6. Find a formula for f' and sketch its graph. 60. Where is the greatest integer function f(x) = [ x ] not differentiable? Find a formula for f' and sketch its graph. 61. (a) Sketch the graph of the function f(x) = x|x|. (b) For what values of x is f differentiable? (c) Find a formula for f!. 62. (a) Sketch the graph of the function g(x) = x + |x. (b) For what values of x is g differentiable? (c) Find a formula for g'. 63. Recall that a function f is called even if f(-x) = f(x) for all x in its domain and odd if f(-x) = -f(x) for al such x. Prove each of the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function. arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit occurs so widely, it is given a special name and notation. 4 Definition The derivative of a function f at a number a, denoted by f'(a), is gnsi o lo sgole orli of f prime of a."obiznos l f(a + h) – f(a) lim (v),f if this limit exists. ab noitsvoado 1ogqu ort moil - a and h approaches 0 if and only if x approaches a. Therefore an equivalent way of stating the definition of the derivative, as If we write xr = a + h, then we have h Sbnuon we saw in finding tangent lines, is GjoCIA omit leonog B Vlioolov UGLJ f'(a) = lim (v)f – (x)f 5. EXAMPLE 4 Find the derivative of the function f(x) = x² - 8x + 9 at the number a. %3D

A 53-54 Use the definition of a derivative to find f'(x) and f"(x). Then graph f.f, and f" on a common screen and check to see if ify each your answers are reasonable. – 3x 54. f(x) = x 53. f(x) = 3x + 2x + 1 55. If f(x) = 2x² – x', find f'(x), f"(x), f"(x), and f(ª(x). Graph f. f', f", and f"on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives? 56. (a) The graph of a position function of a car is shown, where s is measured in feet andt in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at t = 10 seconds? fy each 100 10 20 (b) Use the acceleration curve from part (a) to estimate the jerk at t = 10 seconds. What are the units for jerk? he 57. Let f(x) = Vx. (a) If a # 0, use Equation 2.7.5 to find f'(a). (b) Show that f'(0) does not exist. (c) Show that y = x has a vertical tangent line at (0, 0). (Recall the shape of the graph of f. See Figure 1.2.13.) and your 2/3 58. (a) If g(x) = x2, show that g'(0) does not exist. (b) If a + 0, find g'(a). (c) Show that y = x2/3 has a vertical tangent line at (0, 0). (d) Illustrate part (c) by graphing y = x2/3. 59. Show that the function f(x) = |x – 6 is not differentiable at 6. Find a formula for f' and sketch its graph. 60. Where is the greatest integer function f(x) = [ x ] not differentiable? Find a formula for f' and sketch its graph. 61. (a) Sketch the graph of the function f(x) = x|x|. (b) For what values of x is f differentiable? (c) Find a formula for f!. 62. (a) Sketch the graph of the function g(x) = x + |x. (b) For what values of x is g differentiable? (c) Find a formula for g'. 63. Recall that a function f is called even if f(-x) = f(x) for all x in its domain and odd if f(-x) = -f(x) for al such x. Prove each of the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function. arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit occurs so widely, it is given a special name and notation. 4 Definition The derivative of a function f at a number a, denoted by f'(a), is gnsi o lo sgole orli of f prime of a."obiznos l f(a + h) – f(a) lim (v),f if this limit exists. ab noitsvoado 1ogqu ort moil - a and h approaches 0 if and only if x approaches a. Therefore an equivalent way of stating the definition of the derivative, as If we write xr = a + h, then we have h Sbnuon we saw in finding tangent lines, is GjoCIA omit leonog B Vlioolov UGLJ f'(a) = lim (v)f – (x)f 5. EXAMPLE 4 Find the derivative of the function f(x) = x² - 8x + 9 at the number a. %3D

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter4: Exponential And Logarithmic Functions

Section: Chapter Questions

Problem 3CC: If xis large, which function grows faster, f(x)=2x or g(x)=x2?

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