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Consider cos 0 d0. What is the best-fit parabola to cos e using the (a) three points 69 = 0, 01 = 7/4, 02= 1/2? (b) What is the area under the curve of this parabola? How does this approx- imate area compare to the exact value of the integral?

Consider cos 0 d0. What is the best-fit parabola to cos e using the (a) three points 69 = 0, 01 = 7/4, 02= 1/2? (b) What is the area under the curve of this parabola? How does this approx- imate area compare to the exact value of the integral?

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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Just how accurate is the quadratic approximation in Simpson's rule? Let's see, using two simple functions where we know the exact value of the integral. (a) Consider cos 0 do. What is the best-fit parabola to cos 0 using the three points 6, = 0, 01 = T/4, 02 = 1/2? (b) What is the area under the curve of this parabola? How does this approx- imate area compare to the exact value of the integral? (c) Repeat parts (a) and (b) for the integral e dr. Use the points xo = 0, a1 = }, and r2 = 1. Note: In practice, one would of course divide the range of integration into more sub- intervals. But this problem shows how for certain integrals, Simpson's rule can be quite good even with a large step size!
Transcribed Image Text:Just how accurate is the quadratic approximation in Simpson's rule? Let's see, using two simple functions where we know the exact value of the integral. (a) Consider cos 0 do. What is the best-fit parabola to cos 0 using the three points 6, = 0, 01 = T/4, 02 = 1/2? (b) What is the area under the curve of this parabola? How does this approx- imate area compare to the exact value of the integral? (c) Repeat parts (a) and (b) for the integral e dr. Use the points xo = 0, a1 = }, and r2 = 1. Note: In practice, one would of course divide the range of integration into more sub- intervals. But this problem shows how for certain integrals, Simpson's rule can be quite good even with a large step size!
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