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Chapter 8: Q. 22 (page 680)

Find the fourth Maclaurin polynomial P4(x)for the specified function:

ex.

Short Answer

Expert verified

The fourth Maclaurin polynomial is,

P4(x)=1+x+x22+x36+x424.

Step by step solution

01

Step 1. Given Information.

The function is,

ex.

02

Step 2. Describing the polynomial.

Let f(x)=ex.

Since for any function fwith a derivative of order 4at x=0, the fourth Maclaurin polynomial is,

role="math" localid="1649440080995" P4(x)=f(0)+f'(0)x+f''(0)2!x2+f'''(0)3!x3+f''''(0)4x4.

03

Step 3. Finding the fourth Maclaurin polynomial.

The value of the function at x=0is,

f(0)=e0=1

All the derivatives f(x)=exis f(x). Hence, f(0)=1.

Thus the fourth Maclaurin polynomial is,

P4(x)=1+1.x+12!x2+13!x3+14!x4=1+x+x22+x36+x424

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Most popular questions from this chapter

What is a power series in x-x0?

What is meant by the interval of convergence for a power series in x-x0? How is the interval of convergence determined? If a power series in x-x0 has a nontrivial interval of convergence, what types of intervals are possible.

What is a difference between a Maclaurin polynomial and the Maclaurin series for a function f ?

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of x0. Note: These are the same functions and values as in Exercises 41鈥�48.

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If a function f has a Maclaurin series, what are the possibilities for the interval of convergence for that series?
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