Q. 56 In Exercises 49鈥�56 find the Ta... [FREE SOLUTION]

Chapter 8: Q. 56 (page 680)

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of $x_{0}$ .
56. $\sin 3 x, \frac{蟺}{6}$

Short Answer

Expert verified

The Taylor series of the function $\sin 3 x$ at $x = \frac{蟺}{6}$ is $p_{n} (x) = {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k + 1} 3^{2 k}}{(2 k)!} {(x - \frac{蟺}{6})}^{2 k}$

Step by step solution

Step 1. Given information.

We have given that $f (x) = \sin 3 x$ and $x_{0} = \frac{蟺}{6}$ .

Step 2. Table of the Taylor series

Any function $f$ with a derivative of order $n$ , the Taylor series at $x = \frac{蟺}{6}$ is given by,

$P_{n} (x) = f (\frac{蟺}{6}) + f^{'} (\frac{蟺}{6}) (x - \frac{蟺}{6}) + \frac{f^{''} (\frac{蟺}{6})}{2!} {(x - \frac{蟺}{6})}^{2} + \frac{f^{'''} (\frac{蟺}{6})}{3!} {(x - \frac{蟺}{6})}^{3} + . . . .$

So, let us find the derivatives of the given function and construct a table of the Taylor series for the function $f (x) = \sin 3 x$ at $\frac{蟺}{6}$ .

Step 3. Required Taylor series

Therefore, the Taylor series of the function $f (x) = \sin 3 x$ at $x = \frac{蟺}{6}$ .

$P_{n} (x) = 1 + 0 路 (x - \frac{蟺}{6}) + \frac{- 3^{2}}{2!} {(x - \frac{蟺}{6})}^{2} + \frac{0}{3!} {(x - \frac{蟺}{6})}^{3} + . . . + \frac{{(- 1)}^{k} 3^{2 k}}{(2 k)!} {(x - \frac{蟺}{6})}^{2 k} + \frac{0}{(2 k + 1)!} {(x - \frac{蟺}{6})}^{2 k + 1} + . . .$

otherwise, it can be written as

$P_{n} (x) = {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k + 1} 3^{2 k}}{(2 k)!} {(x - \frac{蟺}{6})}^{2 k}$

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91影视

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of $x_{0}$ .
56. $\sin 3 x, \frac{蟺}{6}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Table of the Taylor series

Step 3. Required Taylor series

One App. One Place for Learning.

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91影视

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of x0.56.sin3x,蟺6

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Table of the Taylor series

Step 3. Required Taylor series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Discrete Mathematics

Pure Maths

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of $x_{0}$ .
56. $\sin 3 x, \frac{蟺}{6}$