Midpoint rule and error estimates.

[WallE]

Hello everyone. I am having trouble with this math problem:
(b) Use the error estimate to find the smallest value of n that can be chosen in order to guarantee that the midpoint rule M_n approximates the integral ∫₀¹(e^{x^2})dx to within 10^-5.

The formula |E_m| ≤ (K(b-a)³)/24n² applies for |f "(x)| ≤ K.

My issue is that I don't know how to get a constant for the value K, so I don't know what to do next. Here are the steps I went through...

Part (a) had me solve ∫₀¹(e^{x^2})dx with the midpoint rule for n=6. I ended up getting ~ 1.456409832. This is not very important for my issue, but I thought I'd mention it anyway.

Now I here is my work for part (b)...
f(x) = e^{x^2}, so f '(x) = 2xe^{x^2}, and f "(x) = 2(x2xe^{x^2}+e^{x^2}) = 4x²e^{x^2}+2e^{x^2}
Now, plugging that in, I get | 4x²e^{x^2}+2e^{x^2} | ≤ K.

But how do I find K? In my book, I only get examples where f "(x) ends up being a constant value, so then | f "(x) | ends up being equal to K. What do I do when f "(x) contains variables?


Can someone help me understand this? Thanks.

 

[tkhunny]

f(x) is strictly increasing. Its maximum value over the interval [0,1] is at x = 1
f'(x) is strictly increasing. Its maximum value over the interval [0,1] is at x = 1
f"(x) is strictly increasing. Its maximum value over the interval [0,1] is at x = 1