Midpoint Rule and the Error of Approximation

[AmySaunders]

(a) Use the Midpoint Rule and the given data to estimate the value of the integral I. (Give the answer to two decimal places.)

x	f(x)	x	f(x)
0.0	7.1	2.0	6.5
0.4	8.3	2.4	6.6
0.8	7.4	2.8	8.3
1.2	6.8	3.2	7.5
1.6	7.1

(b) If it is known that -5 ≤ f ''(x) ≤ 3 for all x, estimate the error involved in the approximation in part (a). (Give the answer to four decimal places.

I was able to find the Midpoint Approximation fairly easily: 23.92. My question is how to find the error of approximation. I know the equation I should use is (K(b-a)^3)/24n^2

I know that K is the second derivative of whatever the function is, but I don't have a function. How do I find K?

 

[Ishuda]

Go back to a definition:
f'(x) ~ $\displaystyle \frac{f(x-\frac{\delta x}{2}) - f(x+\frac{\delta x}{2})}{\delta x}$
For example
f'(1.4) ~ $\displaystyle \frac{f(1.6) - f(1.2)}{0.4}$

Now the second derivative is just the derivative of the first derivative
f''(x) ~ $\displaystyle \frac{f'(x-\frac{\delta x}{2}) - f'(x+\frac{\delta x}{2})}{\delta x}$

EDIT: You could also use the bounds on the second derivative to put bounds on the estimate (K is maximum magnitude of the second derivative).