Question

Show that Trapezoidal Rule is exact for any polynomial of degree 1, and Simpson's Rule is exact for any polynomial of degree n ≤ 3. Use N=3 and integrate from 0 to 1 for convenience.

Answer 1

Trapezoidal Rule is exact for polynomials of degree 1, Simpson's Rule is exact for polynomials of degree n ≤ 3.

Step-by-step explanation:

The Trapezoidal Rule and Simpson's Rule are numerical methods for approximating definite integrals. The Trapezoidal Rule approximates the integral by dividing the interval into trapezoids and summing their areas. It is exact for any polynomial of degree 1 because the area under a straight line can be exactly calculated using a trapezoid.

Simpson's Rule approximates the integral by dividing the interval into parabolic curves and summing their areas. It is exact for any polynomial of degree n ≤ 3 because the area under a parabolic curve can be exactly calculated using Simpson's Rule.