### Section 7.2

Problems 1c, 5bc, 6bc, 7, 10, 12, 15-17

### Understanding the proof I did in class for the error formula for the trapezoidal rule... part 1

Halfway through the proof of the error formula for the trapezoidal rule, we reached

int_{x_i}^{x_{i+1}} f''(c_x) (x-x_i) (x-x_{i+1})/2 dx.

That is, the integral from x_i to x_{i+1} of the function f''(c_x) (x-x_i) (x-x_{i+1})/2. The delicate thing is... c_x depends on x, so you can't just pull it out of the integral, leaving something that you know how to integrate. At this point, I mumbled the words "integral mean value theorem" (Theorem A.3) and then wrote

- f''(c_i) h^3/12.

Fill in the missing steps and explain why they're valid.

### Understanding the proof I did in class for the error formula for the trapezoidal rule... part 2

In class, I proved the error formula for the trapezoidal rule by first proving a lemma that had to do with approximating functions by linear functions and then using this to prove the error formula for the trapezoidal rule.

Repeat this but for the left-hand rule. You need to figure out the analogue of the lemma and prove it. And then prove the analogue of the theorem, figuring out the sign, the power of h, the constant, whether or not there's a (b-a), and what sorts of derivatives of f are required.

### Matlab problem

Use the error formula for the left-hand rule that you found in the above and propose a "corrected left-hand rule". Code this up by modifying left_hand.m and demonstrate that you've created a scheme that's more accurate than the original left-hand rule. (To demonstrate this, show that the errors decrease more quickly for the corrected scheme.)