# 9.7 | Taylor Polynomials and Approximations

Definitions of $$n$$th Taylor Polynomial and $$n$$th Maclaurin Polynomial
If $$f$$ has $$n$$ derivatives at $$c$$, then the polynomial
$$P_n(x) = \sum_{i=0}^n \frac{f^{(i)}(c)}{i!}(x-c)^i$$
is called the $$n$$th Taylor polynomial for $$f$$ at $$c$$.
If $$c=0$$, then
$$P_n(x) = \sum_{i=0}^n \frac{f^{(i)}(0)}{i!}x^i$$
is also called the $$n$$th Maclaurin polynomial for $$f$$
Taylor’s Theorem If $$f$$ has derivatives up to order $$n+1$$ in an interval $$I$$ containing $$c$$, then for each $$x$$ in $$I$$, there exists a number $$z$$ between $$x$$ and $$c$$ such that
\begin{aligned} f(x) & = f(c) + f'(c)(x-c) + \frac{f^{\prime \prime}(c)}{2}(x-c)^2 + \cdots + \frac{f^{(n)}(c)}{n!}(x-c)^n + R_n(x) \\ & = P_n(x) + R_n(x) \end{aligned}
where
$$R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!}(x-c)^{n+1}$$

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:
$$M_n(x) = \sum_{k = 0}^n \frac{f^{(k)}(0)}{k!}x^k$$

$$\displaystyle k$$ $$\displaystyle f^{(k)}(x)$$ $$\displaystyle \frac{f^{(k)}(0)}{k!}x^k$$
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}