Chapter 8: Q. 37 (page 692)

In Exercises 41–48 in Section 8.2, you were asked to find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of x 0. Here give Lagrange’s form for the remainder $R_{4} (x)$ .
$\cos x, \frac{π}{2}$

Short Answer

Expert verified

Ans: $R_{4} (x) = - \frac{\sin c}{120} {(x - \frac{π}{2})}^{5}$

Step by step solution

Step 1. Given information.

given,

$\cos x, \frac{π}{2}$

Step 2. Consider the given function,

The derivatives of the function $f (x) = \cos x$ are

$\begin{aligned} f^{'} (x) = \frac{d}{d x} [\cos x] \\ = - \sin x \end{aligned}$

Also,

$\begin{aligned} f^{''} (x) = \frac{d}{d x} [- \sin x] \\ = - \frac{d}{d x} [\sin x] \\ = - \cos x \end{aligned}$

Again

$\begin{aligned} f^{'''} (x) = - \frac{d}{d x} [\cos] \\ = - (- \sin x) \\ = \sin x \end{aligned}$

Also,

$\begin{aligned} f^{''''} (x) = \frac{d}{d x} [\sin x] \\ = \cos x \end{aligned}$

Implies that

$f^{(4)} (x) = \cos x$

Finally,

$\begin{aligned} f^{(5)} (x) = \frac{d}{d x} [\cos x] \\ = - \sin x \end{aligned}$

Step 3. Now,

by the Lagrange's form for the remainder, if f is a function that can be differentiated n+1 times in some open interval / containing the point $x_{0}$ and $R_{n} (x)$ be the nth remainder for $f$ at $x = x_{0}$ . Then there exists at least one c between $x_{0}$ and x such that

$R_{n} (x) = \frac{f^{(n + 1)} (c)}{(n + 1)!} {(x - x_{0})}^{n + 1}$

Since $f^{(5)} (x) = - \sin x and x_{0} = \frac{π}{2} then$

$R_{4} (x) = \frac{f^{5} (c)}{5!} {(x - \frac{π}{2})}^{5}$

That is,

$R_{4} (x) = - \frac{\sin c}{120} {(x - \frac{π}{2})}^{5}$

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In Exercises 41–48 in Section 8.2, you were asked to find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of x 0. Here give Lagrange’s form for the remainder $R_{4} (x)$ .
$\cos x, \frac{π}{2}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the given function,

Step 3. Now,

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In Exercises 41–48 in Section 8.2, you were asked to find the fourth Taylor polynomial P4(x)for the specified function and the given value of x 0. Here give Lagrange’s form for the remainder R4(x). cosx,π2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the given function,

Step 3. Now,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Decision Maths

Pure Maths

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 41–48 in Section 8.2, you were asked to find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of x 0. Here give Lagrange’s form for the remainder $R_{4} (x)$ .
$\cos x, \frac{π}{2}$