As always, you are expected to rework my examples successfully on your own BEFORE attempting the homework.

1. Answer each in a complete and coherent sentence. What are the global errors of:
   1. the trapezoidal rule,
   2. the rectangle method,
   3. Simpson's–3/8 rule, and
   4. Simpson's–1/3 rule?
   (You should know these.)

2. When using Simpson's–3/8 rule, what must be true of the number of subintervals? Answer in a complete and coherent sentence.

3. Do Problems 3–6 in this problem set.

4. Modify the Maple code for Simpson's–1/3 rule (see this previous assignment) to turn it into Simpson's–3/8 rule.
   Then use it to approximate the sample problem (integral of sin x on the interval [0, π]) using 6 subintervals.
   You should get the answer we obtained in class. (Also see the pseudocode for Simpson's–3/8 rule.)

5. Now use your Simpson's–3/8 code to re-do Problem 4 above using 12 subintervals. You should get the answer we obtained in class.

6. Use the Simpson's–3/8 rule code to approximate the integral of   e^–x2  on the interval [0,3] using:
   1. 9 subintervals        Answer:   0.88619 38607 33682 32028
   2. 18 subintervals      Answer:   0.88620 71620 01857 85809
   3. Use Richardson extrapolation to obtain an improved estimate.
      Answer:   +0.88620 80487 53069 56062
   4. Without knowing the actual errors, determine how much more accurate we expect result (b) to be than result (a). Answer in a complete and coherent sentence.
   5. What is the error estimate of the result in (b)?      Answer:   +0.000000 886751 211702 520374
   6. What is the error estimate of the result in (b) in parts per million?      Answer:    +1.0006 ppm

7. Play around with the code to approximate integrals of other functions on different intervals.
   

8. In class I derived the formula for the trapezoidal rule. In a previous assignment, you were supposed to derive the formula for Simpson's–1/3 rule. Now you will derive the composite formula for Simpson's–3/8 rule:
   1. First construct the Newton-Gregory interpolating polynomial P_[0–3](x) containing points indexed [0,1,2,3]. It approximates function f(x) on Section 1.

   2. Integrate the polynomial on Section 1: [x₀, x₃] and simplify the result.
      You should obtain    I₁ = (3/8) · h · ( f₀ + 3 f₁ + 3 f₂ + f₃ ).
      (You might want to review Problem 3 in this Assignment.)

   3. Use the formula for I₁ on Section 1 as the template to construct the formulas for I₂ on Section 2, I₃ on Section 3, etc.

   4. Add   I₁ + I₂ + I₃ + · · ·   to obtain the composite Simpson's–3/8 rule.

9. Answer each in a complete and coherent sentence. How much do we expect the accuracy to increase if we double the number of subintervals using:
   1.   Simpson's–3/8 rule?
   2.   the trapezoidal rule?
   3.   Simpson's–1/3 rule?
   4.   the rectangle method learned in Calc-2?

   Did you omit a VERY IMPORTANT WORD in your answers above?

10. Answer each in a complete and coherent sentence. How much do we expect the accuracy to increase if we quadruple the number of subintervals using:
    1.   Simpson's–3/8 rule?
    2.   the trapezoidal rule?
    3.   Simpson's–1/3 rule?
    4.   the rectangle method learned in Calc-2?

    Did you forget a VERY IMPORTANT WORD in your answers above?

11. Is the rectangle method for approximating integrals a Newton–Cotes method? Explain why or why not.

Newton-Cotes Methods:  See a summary of some of the Newton–Cotes integration rules.

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