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Approximate f by a Taylor polynomial with degree n at the number a. Step 1 The Taylor polynomial with degree n = 3 is T3(x) = f(a) + f '(a)(x − a) + f ''(a) 2! (x − a)2 + f '''(a) 3! (x − a)3. The function f(x) = e2x2 has derivatives f '(x) = $4x e2x2, f ''(x) = $16x2+4 e2x2, and f '''(x) = $48x+64x3 e2x2. Step 3 Therefore, T3(x) = . Submit Skip (you cannot come back)

Respuesta :

Answer:

If you center the series at x=1

[tex]T_3(x) = e^2 + 4e^2 (x-1)+10(x-1)^2 + \frac{56}{3} e^2(x-1)^3 + R(x)[/tex]

Where [tex]R(x)[/tex] is the error.

Step-by-step explanation:

From the information given we know that

[tex]f(x) = e^{2x^2}[/tex]

[tex]f'(x) = 4x e^{2x^2}[/tex]   (This comes from the chain rule )

[tex]f^{(2)}(x) = 4e^{2x^2} (4x^2+1)[/tex]   (This comes from the chain rule and the product rule)

[tex]f^{(3)}(x) = 16xe^{2x^2}(4x^2 + 3)[/tex]  (This comes from the chain rule and the product rule)

If you center the series at x=1  then

[tex]T_3(x) = e^2 + 4e^2 (x-1)+10(x-1)^2 + \frac{56}{3} e^2(x-1)^3 + R(x)[/tex]

Where [tex]R(x)[/tex] is the error.

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