Question

Sketch graphs of continuous functions f(x) and g(x) on [0,2] satisfying the following conditions.

Make sure to specify whether the rules result in an underestimate/overestimate and why your chosen rule yields a more accurate approximation.
(a) The Trapezoidal Rule with n=2 is more accurate than the Midpoint Rule.
(b) The right endpoint approximation with n=2 is more accurate than Simpson's Rule.

Answer 1

To meet the conditions in the question, f(x) should be approximately linear for the Trapezoidal Rule to outperform the Midpoint Rule, and g(x) should have a downward concavity, peaking towards x=2 for the right endpoint approximation to surpass Simpson's Rule in accuracy on the interval [0,2].

Step-by-step explanation:

To sketch graphs of continuous functions f(x) and g(x) on the interval [0,2] that meet specific conditions regarding the accuracy of numerical integration methods, we must understand the behavior of each rule. For the Trapezoidal Rule to be more accurate than the Midpoint Rule (condition a), the function likely has a linear or almost linear behavior, as the Trapezoidal Rule tends to be more accurate in such cases because it is akin to a linear approximation.

For the right endpoint approximation to be more accurate than Simpson's Rule (condition b), the function might have an upward concavity decreasing towards x=2, as the right endpoint approximation would capture the larger values towards the end and Simpson's Rule may underestimate the curve if it's concave down. Therefore, a function g(x) that has a concave down curvature with the right endpoint significantly higher than the rest of the function values on [0,2] could fulfill this condition.

Throughout the graphs, it is important to consider the degree of curvature and the behavior of the function at the endpoints to determine if an underestimate or overestimate is produced by each rule.