### Section 1.3

Problems 1, 8 Write the programs using matlab.
hint on #8: "help exp"

### Appendix A

Problems 2, 4b, 6

### Section 7.1

Problems 2b, 3b, 7b, 7c, 9

### A problem that generalizes the averaging proof we did in class...

First, assume that x_1 < x_2 < x_3 and that w_1 + w_2 + w_3 = 1, where w_1, w_2, and w_3 >= 0. Assume that the function f is continuous on the interval [x_1,x_3]. Prove that there is some point x_0 in [x_1,x_3] such that f(x_0) = w_1 f(x_1) + w_2 f(x_2) + w_3 f(x_3).
Now, assume that x_1 < x_2 < ... < x_n and that w_1 + w_2 + ... + w_n = 1, where w_1, w_2, ... w_n >= 0. Assume that the function f is continuous on the interval [x_1,x_n]. Prove that there is some point x_0 in [x_1,x_n] such that f(x_0) = w_1 f(x_1) + w_2 f(x_2) + ... + w_n f(x_n).

### Matlab problem

We are going to study how long it takes to solve a linear algebra problem using matlab.

First, do "help slash". Read this. Now define "A = rand(10,10), b = rand(10,1), x = A\b, A*x-b". How do you know whether you've successfully solved A*x = b? We'll dicuss solving the problem this way versus solving it with x = inv(A)*b later.

Now take n = 50, 100, 150, 200, 250, 300, 350, and 400. Create a vector N with those values. For each value of n, create a random matrix A, a random vector b and solve A*x = b. Find how long this took, using "flops", "cputime", and "tic, toc". Use this information to create three vectors T1, T2, and T3 that have the running time. Plot them versus N on separate plots using "subplot(3,1,1)", "subplot(3,1,2)", and "subplot(3,1,3)" ("help subplot"). Do you have a conjecture on how these functions are as a function of n? Can you defend your conjecture by plotting the functions differently? (hint: think logarithm). Can you defend your conjecture by fitting the vectors T1, T2, and T3 with appropriate functions of n?

Repeat the above, but solving it using x = inv(A)*b; (Hint, you can do all of them simultaneously, and just define the extra three vectors as S1, S2, and S3.