Chapter 8: Q. 26 (page 680)

Find the fourth Maclaurin polynomial $P_{4} (x)$ for the specified function:
$\sin (3 x)$ .

Short Answer

Expert verified

The fourth Maclaurin polynomial is,

$P_{4} (x) = 3 x - \frac{9}{2} x^{3}$ .

Step by step solution

Step 1. Given Information.

The function is,

$\sin (3 x)$ .

Step 2. Describing the polynomial.

Let $f (x) = \sin (3 x)$ .

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

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

Step 3. Finding the fourth Maclaurin polynomial.

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

$f (0) = \sin (3.0) = \sin (0) = 0$

The derivatives of the function $f (x) = \sin (3 x)$ are,

role="math" localid="1649474290888" $f' (x) = \frac{d (\sin (3 x))}{d x} = 3 \cos (3 x) f' (0) = 3 \cos (3.0) = 3 \cos (0) = 3$

Also,

$f'' (x) = \frac{d (3 \cos (3 x))}{d x} = 3 \frac{d \cos (3 x)}{d x} = - 9 \sin (3 x) f'' (0) = - 9 \sin (3.0) = - 9 \sin (0) = 0$

Also,

$f''' (x) = \frac{d (- 9 \sin (3 x))}{d x} = - 9 \frac{d (\sin (3 x))}{d x} = - 27 \cos (3 x) f''' (0) = - 27 \cos (3.0)) = - 27 \cos (0) = - 27$

Also,

$f'''' (x) = \frac{d (- 27 \cos (3 x))}{d x} = - 27 \frac{d (\cos (3 x))}{d x} = 81 \sin (3 x) f'''' (0) = 81 \sin (3.0) = 81 \sin (0) = 0$

The fourth Maclaurin polynomial is,

$P_{4} (x) = 0 + 3 x + \frac{0}{2!} x^{2} + \frac{(- 27)}{3!} x^{3} + \frac{0}{4!} x^{4} = 3 x - \frac{9}{2} x^{3}$

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Find the fourth Maclaurin polynomial $P_{4} (x)$ for the specified function:
$\sin (3 x)$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Describing the polynomial.

Step 3. Finding the fourth Maclaurin polynomial.

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Find the fourth Maclaurin polynomial P4(x)for the specified function:sin3x.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Describing the polynomial.

Step 3. Finding the fourth Maclaurin polynomial.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Statistics

Mechanics Maths

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find the fourth Maclaurin polynomial $P_{4} (x)$ for the specified function:
$\sin (3 x)$ .