Read sections 4.6, 4.7, 4.8, and 4.9. of the text
(Thomas/Finney). I will not be covering 4.9 in class, but you will be
responsible for the material anyway. It's material that's best
understood by "doing".
Make sure you are able to solve all the core problems in
these sections.
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section 4.6: 18, 34, 40, 44, 49, 50. Do these problems without using
the material in sections 4.7 on.
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section 4.7: 4, 10, 19, 20, 30, 44, 46, 48, 50, 52, 68, 78, 80
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section 4.8: 8, 10, 19, 24, 28, 30, 32, 36a
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section 4.9: 8, 22. Do problem 8 using either a calculator or maple,
but _don't_ use the maple programs "trap" and "simp" that I provide in
the worksheet below.
Here's a sample worksheet where I coded up and used the
Trapezoidal rule and Simpson's rule:
sample worksheet for help.
Problems with Maple? Here are the
commands I used.
A hint on making your maple life easier: make seven copies of my maple
worksheet, prob1.mws, ..., prob7.mws This willl save you a huge amount
of typing since you can make minor modifications of prob1.mws and
prob2.mws to do the trapezoidal rule problem and minor modificiations
of prob3.mws, ... prob7.mws to do the integrability problems.
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:
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Find the exact value of the integral by doing the integral. Call
this exact value T.
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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.
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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?
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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)?
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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.
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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?
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Using the above information, how large would you take n if you wanted
E_n from Simpsons rule to be smaller than 10^(-10)?
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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.
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Following the worksheet, present evidence that suggests that 1/x _is_
_not_ integrable on [0,1].
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Following the worksheet, present evidence that suggests that 1/x _is_
integrable on [.1,1].
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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].
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Following the worksheet, present evidence that suggests that 1/x^a _is_
integrable on [.1,1]. (Use the same a as above.)
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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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Math 140