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 \(x = a\).
$$
P_n(x) = f(a) + f'(a) (x-a) + \frac{f^{\prime \prime}(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n
$$
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:
$$
\begin{aligned}
M_1(x) & = 1 + x \\
M_2(x) & = 1 + x + \frac{x^2}{2} \\
M_3(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} \\
M_4(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!}\\
M_5(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!}\\
M_6(x) & = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!}\\
\end{aligned}
$$