Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This is a homework problem that I am having trouble with.

Given $f: \mathbb{R} \to \mathbb{R}$ is twice differentiable ans $f''(x) \geq 0$ on the interval $x \in [a,b]$. Prove that: $$ M_k(f) \leq \int_a^b f(x) \mathrm{d}x \leq T_k(f) $$

Where $M_k(f)$ is the composite Midpoint rule and $T_k(f)$ is the composite trapezoid rule.

I can see this conceptually. $f$ is convex so the whole function lies under a secant line through $a$ and $b$ which indicates $\int_a^b f(x) \mathrm{d}x \leq T_k(f)$, and I think this also indicates the left part. But I have a hard time proving this.

Any tips to get me in the right direction?

share|improve this question
    
Have you tried proving for the case of a single interval? –  J. M. Oct 16 '11 at 17:31
    
Yes. I tried a Taylor expansion on a single interval which shows that the area I(f) consists of M(f) + a second order derivitive term + error. Which indicates the left part if the error is zero. Im not sure what to do with the error term tho. –  Zagga Oct 16 '11 at 17:36
    
Has the Euler-Maclaurin formula ever been mentioned in your class, at least in passing? –  J. M. Oct 16 '11 at 17:43
    
No it has not been mentioned. –  Zagga Oct 16 '11 at 17:45

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.