Always rework my examples BEFORE attempting the homework.

1. What are the global errors of the trapezoidal rule and Simpson's–1/3 rule? (You should know these.)

2. When using Simpson's–1/3 rule, what must be true of the number of subintervals?

3. 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 ).
      (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 formula for I₂ on Section 2, I₃ on Section 3, ... , I_n/2 on Section n/2.

   4. Add   I₁ + I₂ + I₃ + · · · + I_n/2   to obtain the composite Simpson's–1/3 rule.

4. Do Problem 2 in this problem set.

5. 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 6 subintervals.
   (Also see the pseudocode for Simpson's–1/3 rule.)

6. Now re-run your Simpson's–1/3 rule code using 12 subintervals. You should get the result obtained in the lecture example.
   
7. 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, estimate how much more accurate result (b) should be than result (a).
   5. What is the error estimate of the result in (b)?     Answer:   4.913316102519277e-08
   6. What is the error estimate of the result in (b) in parts per million?     Answer:   0.05544 ppm

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

9. Answer each in a complete and coherent sentence:
   How much would the error decrease if we triple the number of subintervals:
   1. – using the trapezoidal rule?
   2. – using Simpson's–1/3 rule?
   3. – using the rectangle method learned in Calc-2?

See this example of Simpson's–1/3 rule.