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Exercise 7.6.11 Using Simpson's rule on a parabola f(x), even with just two subintervals, gives the exact value of the integral, because the parabolas used to approximate f will be f itself. Remarkably, Simpson's rule also computes the integral of a cubic function f(x) = ax³ + bx² +cx+d exactly. Show this is true by showing that X2 x2 - Xo 3.2 This does require a bit of messy algebra, so you may prefer to use Sage. f(x) dx = -(ƒ(xo)+4ƒ((xo+x₂)/2) + f(x₂)).

Exercise 7.6.11 Using Simpson's rule on a parabola f(x), even with just two subintervals, gives the exact value of the integral, because the parabolas used to approximate f will be f itself. Remarkably, Simpson's rule also computes the integral of a cubic function f(x) = ax³ + bx² +cx+d exactly. Show this is true by showing that X2 x2 - Xo 3.2 This does require a bit of messy algebra, so you may prefer to use Sage. f(x) dx = -(ƒ(xo)+4ƒ((xo+x₂)/2) + f(x₂)).

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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Exercise 7.6.11 Using Simpson's rule on a parabola f(x), even with just two subintervals, gives the exact value of the integral, because the parabolas used to approximate f will be f itself. Remarkably, Simpson's rule also computes the integral of a cubic function f(x) = ax³ + bx² +cx+d exactly. Show this is true by showing that px2 [²²5 X2 - XO 3.2 -(ƒ(xo)+4ƒ((xo+x₂)/2) This does require a bit of messy algebra, so you may prefer to use Sage. f(x) dx = +ƒ(x₂)).
Transcribed Image Text:Exercise 7.6.11 Using Simpson's rule on a parabola f(x), even with just two subintervals, gives the exact value of the integral, because the parabolas used to approximate f will be f itself. Remarkably, Simpson's rule also computes the integral of a cubic function f(x) = ax³ + bx² +cx+d exactly. Show this is true by showing that px2 [²²5 X2 - XO 3.2 -(ƒ(xo)+4ƒ((xo+x₂)/2) This does require a bit of messy algebra, so you may prefer to use Sage. f(x) dx = +ƒ(x₂)).
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