Question

True or False If \(f\) is a positive, continuous, increasing function on \([a, b],\) then LRAM gives an area estimate that is less than the true area under the curve. Justify your answer.

Short answer

True. If \(f\) is a positive, continuous, increasing function on [a, b], then LRAM gives an area estimate that is less than the true area under the curve. This is because, for an increasing function, the function's value at the left endpoint (used by LRAM) will be less than the actual value within the interval. Hence, the area of the rectangle drawn is less than the actual area under the curve in that interval.

Step-by-step solution

Understand LRAM

The Left Rectangular Approximation Method or LRAM uses the left endpoints of each interval [a, b] to approximate the area under the curve. It creates rectangles with heights determined by the function's value at the chosen left endpoints.

Understand the nature of Function \(f\)

Function \(f\) is given as an increasing function. This means that the function's value at left endpoint of any interval would be less than the function's value at some point within the interval.

Applying LRAM to Function \(f\)

When we apply LRAM to an increasing function like \(f\), the rectangles created for the approximation will have less area than the true area under the curve. This is because the height of the rectangles (given by function's value at left endpoint) is less than the actual height of the curve within the interval.