Section 7.4

Problems 5, 7, 9, 11-13

NOTE on #5: On problem 5, you can either prove all three Richardson extrapolations have a particular error (in addition to deriving them) or you can program each one and test it. If you choose to program them, the use the test function f(x) = exp(x), find the derivative at x=2, and use h=.1/2, .1/4, .1/8, .1/16, and .1/32.

NOTE on #12: Matlab is in double precision, so read the instructions as "double precision". Use the test function f(x) = exp(x), find the derivative at x=2, and use h=.1/2, .1/4, .1/8, .1/16, ...

Problem 1a: Find and prove the error formula for the approximation you found in #9a

Problem 1b: Program the two approximations you found in #9a and #9b and verify that your programs are correct by doing an example and finding the ratios. Again, use f = exp(x), approximate the derivative near x=2, and use h=.1/2, .1/4, .1/8, .1/16, and .1/32.

Problem 2a: Use Simpsons rule to approximate the integral from 1 to 2 of cos(x^2) (sqrt(x))''. Use the midpoint rule to approximate the second derivative (don't do the derivative by hand!). What errors do you expect? Write a program and verify that the ratios went to what you expected. Divide into 2, 4, 8, ... 256 intervals.

Problem 2b: Find an approximation of the second derivative, f'', that would work well with Simpsons rule. It should be accurate enough, but not overly accurate. Write a program and verify that the ratios went to what you expected. Divide into 2, 4, 8, ... 256 intervals.