Last time: Taylor polynomials

(Monday, November 1, 2021)

Given a function $f$ , its Taylor polynomial of order $n$ centered at $x=0$ is the polynomial

P_{n}(x)=C_{0}+C_{1}x+\cdots+C_{n}x^{n}

such that

We found

C_{n}=\frac{f^{(n)}(0)}{n!}

P_{n}(x)=f(0)+f^{\prime}(0)x+\frac{f^{\prime\prime}(0)}{2!}x^{2}+\frac{f^{% \prime\prime\prime}(0)}{3!}x^{3}+\cdots+\frac{f^{(n)}(0)}{n!}x^{n}.

Similarly, the Taylor polynomial of order $n$ centered at $x=a$ is

P_{n}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime\prime}(a)}{2!}(x-a)^{2}+\frac% {f^{\prime\prime\prime}(a)}{3!}(x-a)^{3}+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^{n}.

Warm-up.

What is the $n$ th Taylor polynomial of $e^{x}$ about $x=0$ ?