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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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BuyFindarrow_forward

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem

Finding Taylor Polynomials In Exercises 15 and 16, use a symbolic differentiation utility to find the Taylor polynomial (centered at 0) of degree (a) 2, (b) 4, (c) 6, and (d) 8.

f ( x ) = e 4 x 2

(a)

To determine

To calculate: The Taylor polynomial centered at 0 of degree 2 for the function f(x)=e4x2 and then find the Taylor approximation using differential utility.

Explanation

Given Information:

The provided function is f(x)=e4x2.

Formula used:

Power series:

ex=n=0xnn!=1+x1!+x22!+x33!++xnn!+=1+x+x22+x36+

Taylor polynomial of degree m for a function f(x) with power series n=0anxn is given by:

Sm(x)=a0+a1x+a2x2+a3x3++amxm

Calculation:

Consider the function f(x)=e4x2.

Substitute 4x2 for x in the power series for ex=n=0xnn! as,

e4x2=1+(4x2)1!+(4x2)22!+(4x2)33!++(4x2)nn!+=14x2+42

(b)

To determine

To calculate: The Taylor polynomial centered at 0 of degree 4 for the function f(x)=e4x2 and then find the Taylor approximation using differential utility.

(c)

To determine

To calculate: The Taylor polynomial centered at 0 of degree 6 for the function f(x)=e4x2. and then find the Taylor approximation using differential utility.

(d)

To determine

To calculate: The Taylor polynomial centered at 0 of degree 8 for the function f(x)=e4x2 and then find the Taylor approximation using differential utility.

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