number 5 1. Determine a Taylor Series about x = 0 for the following integral.cos(x)1dx2. Write down T2 (x), T4 (x) and T6 (x) for the Taylor Series of f (x) = sin(x) about x =2Graph all three of the Taylor polynomials and f (x) on the same graph for the interval [-n, 27].3. Write down T2 (x), T3 (x) and T4 (x) for the Taylor Series of f (x) = In(1 – x) about x = -2. Graph all three of the Taylor polynomials and f (x) on the same graph for the interval [-4,0].14. Write down T¡ (x), T3 (x) and T; (x) for the Taylor Series of f (x) :about x = 4. Graph all three of the Taylor polynomials and f (x) on the same graph for the interval [1,5].(6–x)"5. Write down T2 (x), T4 (x) and T6 (x) for the Taylor Series of f (x) = v2+x about æ = 1. Graph all three of the Taylor polynomials and f (x) on the same graph for the interval -2, 4|.

Name: number 5 1. Determine a Taylor Series about x = 0 for the following in
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Description: number 5 1. Determine a Taylor Series about x = 0 for the following integral.cos(x)1dx2. Write down T2 (x), T4 (x) and T6 (x) for the Taylor Series of f (x) = sin(x) about x =2Graph all three of the Taylor polynomials and f (x) on the same graph for

Given fx=2+x Taylor series at a is: fa+f'a1!x-a+f''a2!x-a2+.... Therefore,…

Answered: 5. Write down T2 (x), T4 (x) and T, (x)…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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number 5

1. Determine a Taylor Series about x = 0 for the following integral.
cos(x)
1
dx
2. Write down T2 (x), T4 (x) and T6 (x) for the Taylor Series of f (x) = sin(x) about x =
2
Graph all three of the Taylor polynomials and f (x) on the same graph for the interval [-n, 27].
3. Write down T2 (x), T3 (x) and T4 (x) for the Taylor Series of f (x) = In(1 – x) about x = -2. Graph all three of the Taylor polynomials and f (x) on the same graph for the interval [-4,0].
1
4. Write down T¡ (x), T3 (x) and T; (x) for the Taylor Series of f (x) :
about x = 4. Graph all three of the Taylor polynomials and f (x) on the same graph for the interval [1,5].
(6–x)"
5. Write down T2 (x), T4 (x) and T6 (x) for the Taylor Series of f (x) = v2+x about æ = 1. Graph all three of the Taylor polynomials and f (x) on the same graph for the interval -2, 4|.

Expert Solution

Step 1

Given $f (x) = \sqrt{2 + x}$

Taylor series at a is:

$f (a) + \frac{f^{'} (a)}{1!} (x - a) \frac{+ f^{''} (a)}{2!} {(x - a)}^{2} + . . . .$

Therefore,

$\begin{array}{rcl} T_{2} {(x)}_{} & = & \sqrt{3} + \frac{1}{2} 3^{- 0.5} (x - 1) - \frac{1}{8} 3^{- 1.5} {(x - 1)}^{2} \\ T_{4} (x) & = & \sqrt{3} + \frac{1}{2} 3^{- 0.5} (x - 1) - \frac{1}{8} 3^{- 1.5} {(x - 1)}^{2} + \frac{3}{72} 3^{- 2.5} {(x - 1)}^{3} - \frac{15}{16 \times 24} 3^{- 2.5} {(x - 1)}^{4} \\ T_{5} (x) & = & \sqrt{3} + \frac{1}{2} 3^{- 0.5} (x - 1) - \frac{1}{8} 3^{- 1.5} {(x - 1)}^{2} + \frac{3}{72} 3^{- 2.5} {(x - 1)}^{3} - \frac{15}{16 \times 24} 3^{- 2.5} {(x - 1)}^{4} + \frac{105}{32 \times 24 \times 120} 3^{- 3.5} {(x - 1)}^{5} \\ - \frac{945}{74 \times 24 \times 120 \times 720} 3^{- 4.5} {(x - 1)}^{6} \end{array}$

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