Chapter 8: Q. 21 (page 680)

Find the fourth Maclaurin polynomial $P_{4} (x)$ for the specified function:
$\cos (x)$ .

Short Answer

Expert verified

The fourth Maclaurin polynomial is,

$P_{4} (x) = 1 - \frac{1}{2} x^{2} + \frac{1}{24} x^{4}$ .

Step by step solution

Step 1. Given Information.

The function is,

$\cos (x)$ .

Step 2. Finding the fourth Maclaurin polynomial.

Let $f (x) = \cos (x)$ .

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

$P_{4} (x) = f (0) + f' (0) x + \frac{f'' (0)}{2!} x^{2} + \frac{f''' (0)}{3!} x^{3} + \frac{f'''' (0)}{4!} x^{4}$ .

Step 3. Finding the derivatives.

The value of the function at $x = 0$ is,

$f (0) = \cos (0) = 1$

Finding the derivatives of the function $f (x) = \cos (x)$ ,

$f' (x) = \frac{d (\cos (x))}{d x} = - \sin (x) f' (0) = - \sin (0) = 0$

Also,

$f'' (x) = \frac{d (- \sin (x))}{d x} = - \frac{d (\sin (x))}{d x} = - \cos (x) f'' (0) = - \cos (0) = - 1$

Also,

role="math" localid="1649439432066" $f''' (x) = \frac{d (- \cos (x))}{d x} = - \frac{d (\cos (x))}{d x} = - (- \sin (x)) = \sin (x) f''' (0) = \sin (0) = 0$

Also,

$f'''' (x) = \frac{d (\sin (x))}{d x} = \cos (x) f'''' (0) = \cos (0) = 1$

Thus, the fourth Maclaurin polynomial is,

$P_{4} (x) = 1 + 0 . x + \frac{(- 1)}{2!} x^{2} + \frac{0}{3!} x^{3} + \frac{1}{4!} x^{4} = 1 - \frac{1}{2} x^{2} + \frac{1}{24} x^{4}$

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Find the fourth Maclaurin polynomial $P_{4} (x)$ for the specified function:
$\cos (x)$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the fourth Maclaurin polynomial.

Step 3. Finding the derivatives.

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the fourth Maclaurin polynomial.

Step 3. Finding the derivatives.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Applied Mathematics

Probability and Statistics

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Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find the fourth Maclaurin polynomial $P_{4} (x)$ for the specified function:
$\cos (x)$ .