Chapter 8: Q. 41 (page 680)

$P_{4} (x)$ In Exercises 41–48 find the fourth Taylor polynomial for the specified function and the given value of $x_{0}$ .
$\cos x, \frac{π}{2}$

Short Answer

Expert verified

Ans: The fourth Taylor polynomial for the specified function is $= - (x - \frac{π}{2}) + \frac{1}{6} {(x - \frac{π}{2})}^{3}$ .

Step by step solution

Step 1. Given information:

$f (x) = \cos x$

Step 2. The fourth Taylor polynomial:

Since for any function $f$ with a derivative of order 4 at $x = \frac{π}{2}$ , the fourth Taylor polynomial for $x = \frac{π}{2}$ is given by

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

Therefore, first, find the value of the function along with $f^{'} (x), f^{''} (x), f^{'''} (x)$ and $f^{''''} (x)$ at $x = \frac{π}{2}$

Step 3. Finding the fourth Taylor polynomial through derivative of order 4:

$T h u s, t h e v a l u e o f t h e t h e f u n c t i o n s t x = \frac{π}{2} i s f (\frac{π}{2}) = \cos (\frac{π}{2}) = 0 T h e d e r i v a t i v e s o f t h e f u n c t i o n f (x) = \cos (x) a r e f' (x) = \frac{d}{d x} [\cos x] = - \sin x S o, a t x = \frac{π}{2} f' (\frac{π}{2}) = - \sin (\frac{π}{2}) = - 1 A l s o, f'' (x) = \frac{d}{d x} [- \sin x] = - \cos x S o, a t x = \frac{π}{2} f' (\frac{π}{2}) = - \cos (\frac{π}{2}) = 0 A g a i n, f'''' (x) = \frac{d}{d x} [\sin x] = \cos x S o, a t x = \frac{π}{2} f' (\frac{π}{2}) = \cos (\frac{π}{2}) = 0$

Step 4. Substituting the derivative of order 4 in the fourth Taylor polynomial :

Therefore, the fourth Taylor polynomial for the function $f (x) = \cos x$ is

P_{4} (x) = 0 + - 1 \cdot (x - \frac{π}{2}) + \frac{0}{2!} {(x - \frac{π}{2})}^{2} + \frac{1}{3!} {(x - \frac{π}{2})}^{3} + \frac{0}{4!} {(x - \frac{π}{2})}^{4} = - (x - \frac{π}{2}) + \frac{1}{3!} {(x - \frac{π}{2})}^{3} = - (x - \frac{π}{2}) + \frac{1}{6} {(x - \frac{π}{2})}^{3}

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$P_{4} (x)$ In Exercises 41–48 find the fourth Taylor polynomial for the specified function and the given value of $x_{0}$ .
$\cos x, \frac{π}{2}$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. The fourth Taylor polynomial:

Step 3. Finding the fourth Taylor polynomial through derivative of order 4:

Step 4. Substituting the derivative of order 4 in the fourth Taylor polynomial :

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P4(x)In Exercises 41–48 find the fourth Taylor polynomial for the specified function and the given value of x0.cosx,π2

Short Answer

Step by step solution

Step 1. Given information:

Step 2. The fourth Taylor polynomial:

Step 3. Finding the fourth Taylor polynomial through derivative of order 4:

Step 4. Substituting the derivative of order 4 in the fourth Taylor polynomial :

One App. One Place for Learning.

Most popular questions from this chapter

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$P_{4} (x)$ In Exercises 41–48 find the fourth Taylor polynomial for the specified function and the given value of $x_{0}$ .
$\cos x, \frac{π}{2}$