1. Do Problem 2 in this problem set.

2. Modify the Maple code for the trapezoidal rule (see the previous assignment) to turn it into Simpson's–1/3 rule.
   Then use it to approximate the sample problem (integral of sin x on the interval [0, π]) using 10 subintervals.
   (Also see the pseudocode for Simpson's–1/3 rule.)

3. Now re-run your Simpson's–1/3 rule code using 20 subintervals. You should get the results handed out in class.

4. Use the Simpson's–1/3 rule code to approximate the integral of   e^–x2  on the interval [0,3] using:
   1. 10 subintervals      Answer:   0.88620 65522 46007 52234
   2. 20 subintervals      Answer:   0.88620 72892 43422 90023
   3. Use Richardson extrapolation to obtain an improved estimate.
      Answer:   0.88620 73383 76583 92542
   4. Without knowing the actual errors, determine how much more accurate result (b) should be than result (a).

5. Play around with the code to approximate integrals of other functions f(x) on different intervals [a, b].
   

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

   2. Integrate the polynomial on Section 1: [x₀, x₂] and simplify the result.
      You should obtain    I₁ = (h/3) · ( f₀ + 4 f₁ + f₂ ).

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

   4. Add I₁ + I₂ + I₃ + · · ·   to obtain the composite Simpson's–1/3 rule.

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