Explain why or why not Determine whether the following statements are true and g

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Consider the linear function f(x) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3, 6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals. b. A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a, b]. c. For an increasing or decreasing nonconstant function on an interval [a, b] and a given value of n, the value of the midpoint Riemann sum always lies between the values of the left and right Riemann sums.

Explanation / Answer

a) FALSE

    As approximation by using midpoint rule always gives the approximated area only but not exact area

b) FALSE

As the left Riemann sum approximation may or may not overestimate the area.

c) True

As midpoint rule estimation always lies between the right and left riemann sums.

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