4.5. Comparison of Rules

Try these first

1. 1.

   What is the order of accuracy for the Trapezoidal Rule?

   Hint

   Think about how the approximation improves when you increase the number of intervals.

2. 2.

   Can the Trapezoidal Rule be used with any number of intervals?

   Hint

   Recall the conditions for the different rules.

3. 3.

   What is the order of accuracy for Simpson's 1/3 Rule?

   • O(h)
   • O(h²)
   • O(h⁴)

   Hint

   Consider how the order relates to precision in integration.

4. 4.

   True or False: The Trapezoidal Rule requires the number of subintervals to be even.

   • True
   • False

   Hint

   Recall the specific rules associated with each method.

5. 5.

   Given a smooth curve y = x³ over the interval [0, 1], derive both the Trapezoidal Rule and Simpson's 1/3 Rule estimates for n = 4.

   Hint

   Make sure to calculate the average of the endpoints for Trapezoidal.

6. 6.

   If a function is oscillating and you want better accuracy using Simpson’s 3/8 Rule, what should you consider about your number of intervals?

   Hint

   Look at how you set your data points in practice.