Mathematics
hunterslutzky
2016-04-10 13:49:42
Find the taylor polynomial t3(x) for the function f centered at the number a. f(x) = eâ4xsin(2x), a = 0
ANSWERS
eva209
2016-04-10 16:09:54

[latex]e^{-4x}=displaystylesum_{n=0}^inftyfrac{(-4x)^n}{n!}=1+(-4x)+dfrac{(-4x)^2}2+dfrac{(-4x)^3}6+cdots[/latex] [latex]e^{-4x}=1-4x+8x^2-dfrac{32x^3}3+cdots[/latex] [latex]sin2x=displaystylesum_{n=0}^{infty}frac{(-1)^k(2x)^{2k+1}}{(2k+1)!}=(2x)-dfrac{(2x)^3}6+cdots[/latex] [latex]sin2x=2x-dfrac{4x^3}3+cdots[/latex] [latex]e^{-4x}sin2x=left(1-4x+8x^2-dfrac{32x^3}3+cdots ight)left(2x-dfrac{4x^3}3+cdots ight)[/latex] [latex]e^{-4x}sin2x=2x-8x^2+dfrac{44x^3}3+cdots[/latex] [latex]implies T_3(x)=2x-8x^2+dfrac{44x^3}3[/latex]

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