Approximating e^x to within 0.025 by Taylor polynomial in interval [-1,1]

Question

I want to use $R_{(5)}(x)$ to show that $e^x$ can be approximated by the polynomial $T_{(4)}(x)$ to within $0.025$ for all values of $x$ on the interval $[-1,1]$. I just don't know how to treat the interval.

$$R_{(5)}(x) = e^\xi\frac{x^5}{5!} $$ ( where $\xi$ is between $0$ and $x$)

Using $|x| \le 1$, $|\xi| \le 1$, and $e < 3$

$$R_{(5)}(x) \le \left\lvert e^\xi\frac{x^5}{5!} \right\rvert \\ = 3\frac{1^5}{5!} \\ = 0.025$$

Would this be enough to show what I wanted to show?

Answer 1

Yes that is enough.

The Lagrange Error bound computes the MAXIMUM possible error on the interval, so any real error between the polynomial and the function will be less than or equal to the computed maximum.