10.1 | Taylor Polynomials


The Taylor polynomial of degree \(n\) centered at \(x=a\) or the Taylor polynomial about \(x =a\) is given by
$$
P_n(x) = f(a) + f'(a) (x-a) + \frac{f^{\prime \prime}(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n
$$
and \(f(x) \approx P_n(x)\) near \(x = a\).

Find the first, second, third, fourth, fifth, and sixth degree Maclaurin polynomial for
$$
f(x) = e^x.
$$


The \(n^{th}\)-degree Maclaurin polynomial is given by:
$$
M_n(x) = \sum_{i = 0}^n \frac{f^{(i)}(0)}{i!}x^i
$$

\(\displaystyle i\) \(\displaystyle f^{(i)}(x)\) \(\displaystyle \frac{f^{(i)}(0)}{i!}x^i\)
0 \(e^x\) \(1\)
1 \(e^x\) \(x\)
2 \(e^x\) \(\frac{1}{2}x^2\)
3 \(e^x\) \(\frac{1}{3!}x^3\)
4 \(e^x\) \(\frac{1}{4!}x^4\)
5 \(e^x\) \(\frac{1}{5!}x^5\)
6 \(e^x\) \(\frac{1}{6!}x^6\)

Using the table on the left, we obtain:
$$
\begin{aligned}
M_1(x) & = 1 + x \\
M_2(x) & = 1 + x + \frac{x^2}{2} \\
M_3(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} \\
M_4(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!}\\
M_5(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!}\\
M_6(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!}\\
\end{aligned}
$$

E 10.1 Exercises