ChaoticLlama
Junior Member
- Joined
- Dec 11, 2004
- Messages
- 199
I've just started learning Taylor Polynomials and there is one question that I do not think is right. When I graph the original function and the polynomial, they relly don't look the same at x=0.
My goal is to create a fourth degree Taylor Polynomial for the function y = 1/Sqrt[1+x]
\(\displaystyle \L\
\begin{array}{c}
y = \frac{1}{{(1 + x)^{\frac{1}{2}} }} \\
y' = \frac{{ - 1}}{{2(1 + x)^{\frac{3}{2}} }} \\
y'' = \frac{3}{{4(1 + x)^{\frac{5}{2}} }} \\
y''' = \frac{{ - 15}}{{8(1 + x)^{\frac{7}{2}} }} \\
y^{iv} = \frac{{105}}{{16(1 + x)^{\frac{9}{2}} }} \\
\end{array}\\)
And since we are approximating around x = 0...
\(\displaystyle \L\
\begin{array}{c}
y(0) = 1 \\
y'(0) = \frac{{ - 1}}{2} \\
y''(0) = \frac{3}{4} \\
y'''(0) = \frac{{ - 15}}{8} \\
y^{iv} (0) = \frac{{105}}{{16}} \\
\end{array}\\)
creating the polynomial...
\(\displaystyle \L\begin{array}{l}
P_4 (x) = f(0) + f'(0)x + \frac{{f''(x)}}{{2!}}x^2 + \frac{{f'''(x)}}{{3!}}x^3 + \frac{{f^{iv} (x)}}{{4!}}x^4 \\
P_4 (x) = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{{15}}{{48}}x^3 + \frac{{105}}{{384}}x^4 \\
\end{array}\\)
I hope i've transcribed everything correctly.
Why doesn't this polynomial look anything like the graph?
My goal is to create a fourth degree Taylor Polynomial for the function y = 1/Sqrt[1+x]
\(\displaystyle \L\
\begin{array}{c}
y = \frac{1}{{(1 + x)^{\frac{1}{2}} }} \\
y' = \frac{{ - 1}}{{2(1 + x)^{\frac{3}{2}} }} \\
y'' = \frac{3}{{4(1 + x)^{\frac{5}{2}} }} \\
y''' = \frac{{ - 15}}{{8(1 + x)^{\frac{7}{2}} }} \\
y^{iv} = \frac{{105}}{{16(1 + x)^{\frac{9}{2}} }} \\
\end{array}\\)
And since we are approximating around x = 0...
\(\displaystyle \L\
\begin{array}{c}
y(0) = 1 \\
y'(0) = \frac{{ - 1}}{2} \\
y''(0) = \frac{3}{4} \\
y'''(0) = \frac{{ - 15}}{8} \\
y^{iv} (0) = \frac{{105}}{{16}} \\
\end{array}\\)
creating the polynomial...
\(\displaystyle \L\begin{array}{l}
P_4 (x) = f(0) + f'(0)x + \frac{{f''(x)}}{{2!}}x^2 + \frac{{f'''(x)}}{{3!}}x^3 + \frac{{f^{iv} (x)}}{{4!}}x^4 \\
P_4 (x) = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{{15}}{{48}}x^3 + \frac{{105}}{{384}}x^4 \\
\end{array}\\)
I hope i've transcribed everything correctly.
Why doesn't this polynomial look anything like the graph?