homework 10

## Homework 10, Due Friday 11/17 (for everyone)

#### Do the following problems in maple and hand in your print-outs. These problems will be graded, so make sure to do it!

You've done problem 8 of section 4.9 as the book instructed. Now do the following with the same integral:

• Find the exact value of the integral by doing the integral. Call this exact value T.

• Use my maple routine to use the trapezoidal rule with 4, 8, 16, 32, and 64 subintervals to find the approximations T_4, T_8, T_16, T_32, and T_64 of the integral. How are they behaving? Check that T_4 agrees with what you found by hand in problem #8.

• Find the errors E_4 = T - T_4, E_8 = T - T_8, E_16 = T - T_16, E_32 = T - T_32, and E_64 = T - T_64. How are they behaving? Give the ratios E_4/E_8, E_8/E_16, E_16/E_32, and E_32/E_64. How are they behaving?

• Using the above information, how large would you take n if you wanted E_n from the Trapezoidal rule to be smaller than 10^(-10)?

• Use my maple routine to use Simpsons rule with 4, 8, 16, 32, and 64 subintervals to find the approximations S_4, S_8, S_16, S_32, and S_64 of the integral. How are they behaving? Check that S_4 agrees with what you found by hand in problem #8.

• Find the errors E_4 = T - S_4, E_8 = T - S_8, E_16 = T - S_16, E_32 = T - S_32, and E_64 = T - S_64. How are they behaving? Give the ratios E_4/E_8, E_8/E_16, E_16/E_32, and E_32/E_64. How are they behaving?

• Using the above information, how large would you take n if you wanted E_n from Simpsons rule to be smaller than 10^(-10)?

• Which rule is better for finding the answer with an error smaller than 10^(-10)? Simpsons rule or Trapezoidal rule? Why?

#### Now we switch directions and look at integrability. In class, I claimed that if you try to integrate 1/x over [0,1] then the Riemann sums go to infinity. I also claimed this about the Riemann sums for 1/x^(1.001) and 1/x^2. In the worksheet, I test the case of f(x) = 1/sqrt(x). It blows up at x = 0, but it's actually integrable on [0,2]. It's integrable on [0,b] for any b. Warning! Read the problems below carefully! Make sure that you notice whether you're reading "is" or "is not" and be careful about what interval the integration is over.

• Following the worksheet, present evidence that suggests that 1/x _is_ _not_ integrable on [0,1].

• Following the worksheet, present evidence that suggests that 1/x _is_ integrable on [.1,1].

• Following the worksheet, present evidence that suggests that 1/x^a _is_ _not_ integrable on [0,1]. (You get to choose your favorite exponent a > 0.) Go over your notes from class and prove that 1/x^a is not integrable on [0,1].

• Following the worksheet, present evidence that suggests that 1/x^a _is_ integrable on [.1,1]. (Use the same a as above.)

• Following the worksheet, present evidence that suggests that 1/x^a _is_ integrable on [0,1]. (You get to choose your favorite exponent a > 0. Don't use a = 1/2 since that's the one I did myself!)

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