Problem 18 - Calculus Early Transcendentals

Question

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. $$f(x)=(1+x)^{-1 / 2}$$

Step-by-Step Solution

Answer

#Answer# The nth-order Taylor polynomials for the function \(f(x) = (1+x)^{-1/2}\) centered at \(0\) are: - For n = 0: \(P_0(x) = 1\) - For n = 1: \(P_1(x) = 1 -\frac{1}{2}x\) - For n = 2: \(P_2(x) = 1 -\frac{1}{2}x + \frac{1}{8}x^2\)

Step 1: Find the derivatives of the function

First, let's find the derivatives of the function for n = 0, 1, and 2. The function is \(f(x) = (1+x)^{-1/2}\) For n = 0: \(f^0(x) = f(x) = (1+x)^{-1/2}\) For n = 1: \(f^1(x) = \frac{-1/2}{(1+x)^{3/2}}\) For n = 2: \(f^2(x) = \frac{3}{4(1+x)^{5/2}}\)

Step 2: Evaluate the derivatives at x = 0

Let's evaluate the derivatives at x = 0: \(f^0(0) = (1+0)^{-1/2} = 1\) \(f^1(0) = \frac{-1/2}{(1+0)^{3/2}} = -\frac{1}{2}\) \(f^2(0) = \frac{3}{4(1+0)^{5/2}} = \frac{3}{4}\)

Step 3: Find the Taylor polynomials

Using the results from Step 2, we can find the Taylor polynomials for n = 0, 1, and 2: For n = 0: \(P_0(x) = f^0(0) = 1\) For n = 1: \(P_1(x) = f^0(0) + f^1(0)\cdot x = 1 -\frac{1}{2}x\) For n = 2: \(P_2(x) = f^0(0) + f^1(0)\cdot x + \frac{f^2(0)}{2!}\cdot x^2 = 1 -\frac{1}{2}x + \frac{1}{8}x^2\)

Step 4: Graph the Taylor polynomials and the original function

Using visualization software or graphing tools, you can now plot the function \(f(x) = (1+x)^{-1/2}\), and the Taylor polynomials: - \(P_0(x) = 1\) - \(P_1(x) = 1 -\frac{1}{2}x\) - \(P_2(x) = 1 -\frac{1}{2}x + \frac{1}{8}x^2\) Make sure to label each curve to distinguish the function from the Taylor polynomials.