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Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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422 CHAPTER 7 LOGARITHMIC AND EXPONENTIAL FUNCTIOI 51. Suppose the slope of the curve y = f-(x) at (4, 7) is . Find f'(7). 52. Suppose the slope of the curve y = f(x) at (4, 7) is . Find (f-1)'(7). %3D T. Further Explorations 53. Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. If f(x) = x² + 1, then f-'(x) = 1/(x² + 1). b. If f(x) = 1/x, then f-(x) = 1/x. c. When restricted to the largest possible intervals, the function f(x) = x + x has three different inverses. d. When restricted to the largest possible intervals, a tenth-degree %3D polynomial could have ten different inverses. e. If f(x) = 1/x, then (f(x))' = -1/x². %3D 54. Piecewise linear function Consider the function f(x) = \x| – 2|x - . a. Find the largest possible intervals on which f is one-to-one. b. Find explicit formulas for the inverse of f on the intervals in part (a). 55-58. Finding all inverses Find all the inverses associated with the following functions and state their domains. 55. f(x) = (r + 1)3
Transcribed Image Text:422 CHAPTER 7 LOGARITHMIC AND EXPONENTIAL FUNCTIOI 51. Suppose the slope of the curve y = f-(x) at (4, 7) is . Find f'(7). 52. Suppose the slope of the curve y = f(x) at (4, 7) is . Find (f-1)'(7). %3D T. Further Explorations 53. Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. If f(x) = x² + 1, then f-'(x) = 1/(x² + 1). b. If f(x) = 1/x, then f-(x) = 1/x. c. When restricted to the largest possible intervals, the function f(x) = x + x has three different inverses. d. When restricted to the largest possible intervals, a tenth-degree %3D polynomial could have ten different inverses. e. If f(x) = 1/x, then (f(x))' = -1/x². %3D 54. Piecewise linear function Consider the function f(x) = \x| – 2|x - . a. Find the largest possible intervals on which f is one-to-one. b. Find explicit formulas for the inverse of f on the intervals in part (a). 55-58. Finding all inverses Find all the inverses associated with the following functions and state their domains. 55. f(x) = (r + 1)3
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