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I 8. y" + siny = 0; y (0) = 1, y' (0) = 0

I 8. y" + siny = 0; y (0) = 1, y' (0) = 0

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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8.1 EXERCISES In Problems 1-8, determine the first three nonzero terms in the Taylor polynomial approximations for the given initial value problem. 1. y' = x² + y²; 2. y' = y²; y (0) = 2 3. y'= siny+ e*; 4. y'= sin(x+y); 5. x" + tx = 0; 2 y(0) = 1 6. y" + y = 0; 7. y" (0) + y(0)³ = sin0; y(0) = 0, I 8. y" + siny = 0; 3 x(0) = 1, y (0) = 0, 80 y(0) = 0 y(0) = 0 y'(0) = 0 y' (0) = 0 9. (a) Construct the Taylor polynomial p3(x) of degree 3 for the function f(x) = ln x around x = 1. (b) Using the error formula (6), show that y (0) = 1, (0.5)4 In (1.5) - P3 (1.5) | = 0.015625. 4 (c) Compare the estimate in part (b) with the actual error by calculating In (1.5) - P3 (1.5). (d) Sketch the graphs of ln x and p(x) (on the same $ 4 x' (0) = 0 y' (0) = 1 888 % 5 ^ 6 F6 7 * (d) S sa 8 11. Argue equati- (a, b) then th 12. Argue equatic val (a- orders, 13. Duffing with pe y" Let k = nonzerc to the sc 14. Soft v tion giv changes force ky (r = 0) restoring
Transcribed Image Text:8.1 EXERCISES In Problems 1-8, determine the first three nonzero terms in the Taylor polynomial approximations for the given initial value problem. 1. y' = x² + y²; 2. y' = y²; y (0) = 2 3. y'= siny+ e*; 4. y'= sin(x+y); 5. x" + tx = 0; 2 y(0) = 1 6. y" + y = 0; 7. y" (0) + y(0)³ = sin0; y(0) = 0, I 8. y" + siny = 0; 3 x(0) = 1, y (0) = 0, 80 y(0) = 0 y(0) = 0 y'(0) = 0 y' (0) = 0 9. (a) Construct the Taylor polynomial p3(x) of degree 3 for the function f(x) = ln x around x = 1. (b) Using the error formula (6), show that y (0) = 1, (0.5)4 In (1.5) - P3 (1.5) | = 0.015625. 4 (c) Compare the estimate in part (b) with the actual error by calculating In (1.5) - P3 (1.5). (d) Sketch the graphs of ln x and p(x) (on the same $ 4 x' (0) = 0 y' (0) = 1 888 % 5 ^ 6 F6 7 * (d) S sa 8 11. Argue equati- (a, b) then th 12. Argue equatic val (a- orders, 13. Duffing with pe y" Let k = nonzerc to the sc 14. Soft v tion giv changes force ky (r = 0) restoring
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