### Homework assigned 10/4, due 10/10

### 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.)