# Trapezoidal Rule: Maximum error in approximation?

1. Mar 12, 2016

### Jess Karakov

1. The problem statement, all variables and given/known data
Suppose that T4 is used to approximate the ∫ from 0 to 3 of f(x)dx, where -2 ≤ f ''(x) ≤ 1 for all x. What is the maximum error in the approximation?

2. Relevant equations
|ET|≤ (K(b-a)^3)/(12n^2)

3. The attempt at a solution
So I know how to find the error of the trapezoidal rule using the above equation, but I do not understand how to find the maximum error in an approximation.
To find the max error I would find the max/mins of f ''(x), right? But I don't know how to do that when f(x) is not given

2. Mar 12, 2016

### Ray Vickson

You can't. All you can do is find an upper bound on the absolute error, so your actual error may be a lot less than your bound. Just find an upper bound on $|f''(x)|$ over $x \in [0,3]$.

Note: people hardly ever find best bounds by finding actual maxima of things like $|f''(x)|$; typically, they are satisfied with decent bounds.

3. Mar 12, 2016

### Jess Karakov

So K would just be 2?

4. Mar 12, 2016

### Ray Vickson

You tell me.

5. Mar 13, 2016

yes....