First, construct the sixth degree Taylor polynomial P_6(x) for function f(x) = sin(x^2) about x_0 = 0 The use int_{0}^{1} P_6 (x) dx to approximate the integral int_{0}^{1} sin(x^2)dx. Use 4-digit rounding arithmetic in all calculations. What is the approximate value?

Jason Farmer 2021-02-06 Answered
First, construct the sixth degree Taylor polynomial P6(x) for function f(x)=sin(x2) about x0=0 The use 01P6(x)dx to approximate the integral  01sin(x2)dx. Use 4-digit rounding arithmetic in all calculations. What is the approximate value?
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Clara Reese
Answered 2021-02-07 Author has 120 answers

Taylor series f(x)=sin(x2) is =f(x0)+(xx0)1!f(x0)+(xx0)22!f(x0)± Note that x0=0 So, the polynomial is
f(0)+xf(0)+x22f"(0)
+x36f(0)+x424f(0)
+x5120f(0)
+x6720f6(0)+ (up to sixth degree) Now f(x)=sin(x2)
f(x)=cos(x2)2x
fx)=2[cosx2+xsin(x2)(2x)]
=2[cosx22x2sinx2]
f(x)=2(sin(x2)2x)
4[x2cosx22x+sinx22x]
=4xsin(x2)8x3cosx28xsinx2
=12xsinx28x3cosx2
f4(x)=ddx(f(x))
=12(sinx2+xcosx2 2x)8(cosx23x2x3sinx22x)

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Jeffrey Jordon
Answered 2021-12-14 Author has 2313 answers

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