4. Interpolation & Numerical Methods

Try these first

1. 1.

   Explain the Trapezoidal Rule in your own words.

   Hint

   Think about how you would use rectangles versus trapezoids.

2. 2.

   What do you need to ensure for Simpson’s Rule to work?

   Hint

   Consider the formula that defines Simpson's 1/3 Rule.

3. 3.

   What is the main purpose of numerical integration?

   • To find exact solutions
   • To approximate definite integrals
   • To graph functions

   Hint

   Think about why we would use numerical methods.

4. 4.

   True or False: Simpson's 1/3 Rule can be applied to odd numbers of intervals.

   • True
   • False

   Hint

   Think about the conditions for the formula.

5. 5.

   Using both the Trapezoidal Rule and Simpson's 1/3 Rule, approximate the integral of f(x)=ln(x) from 1 to 2 using four intervals. Compare results.

   Hint

   Calculate each step carefully and ensure to divide the range correctly.

6. 6.

   Create a Python program to implement and compare the results of the Gauss-Kronrod quadrature method with Simpson's 3/8 Rule for a complex function like sin(x).

   Hint

   Focus on structuring your function correctly and handling intervals effectively.