Let s(x)=lnx for 1≤x≤5. The interval is divided into four subintervals of equal length Δx=1. Three different rectangular approximations of ∫15s(x)dx are considered:
• one uses the left endpoint of each subinterval,
• one uses the right endpoint of each subinterval, and
• one uses the point halfway between the endpoints of each subinterval.
(a) Compute the rectangular approximation that uses the left endpoints.
(b) Compute the rectangular approximation that uses the right endpoints.
(c) Compute the rectangular approximation that uses the points x=1.5,2.5,3.5,4.5.
(d) Using the first and second derivatives of s, identify which of the three approximations from parts (a)–(c) is closest to the actual value of ∫15s(x)dx. Provide a complete justification based on the behavior of s on [1,5].