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Chapter 9: Problem 16

a. Find the nth-order Taylor polynomials of the given function centered at \(0,\) for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=e^{-x}$$

Short Answer

Expert verified
Answer: The 0th, 1st, and 2nd Taylor polynomials of \(f(x) = e^{-x}\) are as follows: 1. \(P_0(x) = 1\) 2. \(P_1(x) = 1 - x\) 3. \(P_2(x) = 1 - x + \frac{1}{2}x^2\) These polynomials approximate the original function \(f(x) = e^{-x}\) by agreeing on the value of the function and its derivatives at x = 0. As the order n of the Taylor polynomial increases, the approximation becomes better. This can be observed by graphing the original function and the three Taylor polynomials. The higher the order, the more closely the polynomial resembles the function over a wider range of x values.

Step by step solution

01

Find the derivatives of the function

To find the Taylor polynomial, we need to evaluate the function and its derivatives at the point \(c = 0\). Let's find the first few derivatives of \(f(x) = e^{-x}\): 1. \(f(x) = e^{-x}\) 2. \(f'(x) = -e^{-x}\) 3. \(f''(x) = e^{-x}\)
02

Evaluate the functions at the point c

Now, we'll evaluate the function and each derivative at \(c = 0\): 1. \(f(0) = e^{-0} = 1\) 2. \(f'(0) = -e^{-0} = -1\) 3. \(f''(0) = e^{-0} = 1\)
03

Compute the Taylor polynomials

Using the evaluated values, we can compute the nth-order Taylor polynomials centered at \(0\), for \(n = 0,1,\) and 2: 1. \(n=0: P_0(x) = f(0) = 1\) 2. \(n=1: P_1(x) = f(0) + f'(0) * x = 1 - x\) 3. \(n=2: P_2(x) = f(0) + f'(0) *x + \frac{f''(0)}{2!} * x^2 = 1 - x + \frac{1}{2}x^2\)
04

Graph the Taylor polynomials and the original function

Now you need to plot the original function \(f(x) = e^{-x}\) and the Taylor polynomials \(P_0(x)\), \(P_1(x)\), and \(P_2(x)\) on the same graph. This can be done using graphing software or graphing calculator. Make sure to label each curve accordingly. You will notice that the Taylor polynomials become better approximations to the original function as the order n increases.

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Most popular questions from this chapter

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{2 k}}{4^{k}}$$

Here is an alternative way to evaluate higher derivatives of a function \(f\) that may save time. Suppose you can find the Taylor series for \(f\) centered at the point a without evaluating derivatives (for example, from a known series). Explain why \(f^{(k)}(a)=k !\) multiplied by the coefficient of \((x-a)^{k} .\) Use this idea to evaluate \(f^{(3)}(0)\) and \(f^{(4)}(0)\) for the following functions. Use known series and do not evaluate derivatives. $$f(x)=e^{\cos x}$$

Explain why or why not ,Determine whether the following statements are true and give an explanation or counterexample. a. The interval of convergence of the power series \(\Sigma c_{k}(x-3)^{k}\) could be (-2,8) b. The series \(\sum(-2 x)^{k}\) converges on the interval \(-\frac{1}{2}

Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) $$\sum_{k=0}^{\infty} e^{-k x}$$

Prove that if \(f(x)=\sum_{k=0}^{\infty} c_{k} x^{k}\) converges with radius of convergence \(R,\) then \(f(x)=\sum_{k=0}^{\infty} c_{k} x^{k}\) converges with radius of convergence \(R,\) then
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