What value of x will give the largest error in an exponentiall function using Lagranges remainder?

Question

a) Determine the Taylor polynomial of order $1$ to $F(x)$ around $x=0$.

b) Determine an approximation to $F(1/2)$ that deviates at most $1/8$ from the exact value.

$F(x) = \int_{0}^{x} e^{-t^2}$

This is what I did

a) $p(x) = x$ (I don't need help with a)

b)

I have this:

$E(x) = \frac{f^{n+1} (z) (x-c)^{n+1}}{(n+1)!}$

$f''(x) = -2te^{-t^2}$ since $n+1 = 2$ and in Lagranges formula I put in $c = 0$ and $x = 1/2$

$E(\frac12) = \frac{-2ze^{-z^2} (\frac12)^2}{2}= -ze^{-z^2} (\frac12)^2 = \frac{-ze^{-z^2}}{4}$

$0<t<\frac12$ and the largest error I will get is when $t=\frac12$ so $z=\frac12$

$\frac{-(\frac12)e^{-(\frac12)^2}}{4} = - \frac1{8e^{1/4}}$.

The absolute value of this is greater than $1/8$

But this is how they did it in the answer sheet. They did everything like me except for the last part. Here:

$\frac{te^{-t^2}}{4} < \frac12 \frac{e^{-t^2}}{4} < \frac12 \frac{e^{-0^2}}{4} = \frac18$

> Since $t≥0$, $\frac{te^{-t^2}}{4}≥ 0$ and thus the error is less than $\frac18$.

I don't really understand this last step. Why did they put different values for $t$? First they did t=$\frac12$ and then $t=0$. I need help understading why I am wrong and thier answer.

Answer 1

The reference solution uses the mean-value form for the remainder term in Taylor series. In your case that means that the error term is equal to $-\frac{1}{4} z e^{-z^2} $ for some value $z$ between the point where you evaluate the function (which is $1/2$) and the point around which you approximate (which is $0$).

Even though you don't know the exact value of $z$, you can now make an upper bound for this term in range $0 \le z \le 1/2$. This is exactly what they do:

• $z \le 1/2$ on this range,
• $z \ge 0 \implies -z \le 0 \implies -z^2 \le 0$, and thus $e^{-z^2} \le 1$ on this same range,
• finally, you can multiply those two bounds and and a factor of $1/4$ to get the bound $1/4 \times 1/2 \times 1 = 1/8$ on the error term.

I am not entirely sure what you have attempted to do with the absolute value step in your solution given you have used the negative value for the bound in the very next step. Could you elaborate on your attempt in this part a bit more? (if still needed)