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18. f(x) = x ln (x + 1) - 1; xo = 1.7

18. f(x) = x ln (x + 1) - 1; xo = 1.7

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question 18 and 29. Show all work, thank you!
### Finding Intersection Points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations. **Problems:** 27. \( y = \sin x \) and \( y = \frac{x}{2} \) 28. \( y = e^x \) and \( y = x^3 \) 29. \( y = \frac{1}{x} \) and \( y = 4 - x^2 \) 30. \( y = x^3 \) and \( y = x^2 + 1 \) 31. \( y = 4\sqrt{x} \) and \( y = x^2 + 1 \) 32. \( y = \ln x \) and \( y = x^3 - 2 \) Use these equations as a guide to find the intersection points. Graphing the functions may help estimate where the intersections occur, aiding the use of Newton’s method for more precise solutions.
Transcribed Image Text:### Finding Intersection Points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations. **Problems:** 27. \( y = \sin x \) and \( y = \frac{x}{2} \) 28. \( y = e^x \) and \( y = x^3 \) 29. \( y = \frac{1}{x} \) and \( y = 4 - x^2 \) 30. \( y = x^3 \) and \( y = x^2 + 1 \) 31. \( y = 4\sqrt{x} \) and \( y = x^2 + 1 \) 32. \( y = \ln x \) and \( y = x^3 - 2 \) Use these equations as a guide to find the intersection points. Graphing the functions may help estimate where the intersections occur, aiding the use of Newton’s method for more precise solutions.
**Practice Exercises** **13–19. Finding roots with Newton’s method** For the given function \( f \) and initial approximation \( x_0 \), use Newton’s method to approximate a root of \( f \). Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in *Example 1*. 13. \( f(x) = x^2 - 10; \, x_0 = 3 \) 14. \( f(x) = x^3 + x^2 + 1; \, x_0 = -1.5 \) 15. \( f(x) = \sin x + x - 1; \, x_0 = 0.5 \) 16. \( f(x) = e^x + x - 5; \, x_0 = 1.6 \) 17. \( f(x) = \tan x - 2x; \, x_0 = 1.2 \) 18. \( f(x) = x \ln (x + 1) - 1; \, x_0 = 1.7 \) 19. \( f(x) = \cos^{-1} x - x; \, x_0 = 0.75 \)
Transcribed Image Text:**Practice Exercises** **13–19. Finding roots with Newton’s method** For the given function \( f \) and initial approximation \( x_0 \), use Newton’s method to approximate a root of \( f \). Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in *Example 1*. 13. \( f(x) = x^2 - 10; \, x_0 = 3 \) 14. \( f(x) = x^3 + x^2 + 1; \, x_0 = -1.5 \) 15. \( f(x) = \sin x + x - 1; \, x_0 = 0.5 \) 16. \( f(x) = e^x + x - 5; \, x_0 = 1.6 \) 17. \( f(x) = \tan x - 2x; \, x_0 = 1.2 \) 18. \( f(x) = x \ln (x + 1) - 1; \, x_0 = 1.7 \) 19. \( f(x) = \cos^{-1} x - x; \, x_0 = 0.75 \)
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