As we saw in class, left, right, and mid-point estimates only approximate the area between a positive function f(x) and the x-axis on an interval [a,b]. Consider a function f(x), such that f'(x) > 0 on [a,b]. If f''(x) > 0 on [a,b], would a midpoint estimate over- or over-estimate the area between the function and the x-axis on [a,b]? What if f''(x) < 0 on [a,b]? What if f''(x)=0 on [a,b]? Use a drawing to explain your answer. Hint: Just one rectangle should help you see what is happening.
As we saw in class, left, right, and mid-point estimates only approximate the area between a positive function f(x) and the x-axis on an interval [a,b]. Consider a function f(x), such that f'(x) > 0 on [a,b]. If f''(x) > 0 on [a,b], would a midpoint estimate over- or over-estimate the area between the function and the x-axis on [a,b]? What if f''(x) < 0 on [a,b]? What if f''(x)=0 on [a,b]? Use a drawing to explain your answer. Hint: Just one rectangle should help you see what is happening.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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As we saw in class, left, right, and mid-point estimates only approximate the area between a positive function f(x) and the x-axis on an interval [a,b]. Consider a function f(x), such that f'(x) > 0 on [a,b]. If f''(x) > 0 on [a,b], would a midpoint estimate over- or over-estimate the area between the function and the x-axis on [a,b]? What if f''(x) < 0 on [a,b]? What if f''(x)=0 on [a,b]? Use a drawing to explain your answer. Hint: Just one rectangle should help you see what is happening.
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