Problem 102 - Applied Calculus

Question

Suppose that you have a positive, increasing, concave down function and you approximate the area under it by a Riemann sum with midpoint rectangles. Will the Riemann sum overestimate or underestimate the actual area? [Hint: Make a sketch.

Step-by-Step Solution

Answer

The Riemann sum overestimates the actual area.

Step 1: Understand the Characteristics

The given function is positive, increasing, and concave down. This means that the slope of the function is positive (the function is going up) and the second derivative is negative (the slope is decreasing).

Step 2: Sketch the Function

To illustrate, sketch the function on a coordinate plane. A curve that is increasing and concave down may look like the upper part of a dome or a curve that flattens as it rises.

Step 3: Concept of Midpoint Riemann Sum

In a midpoint Riemann sum, each rectangle's height is determined by the function value at the midpoint of its base. Rectangles are drawn under the curve using these heights to approximate the area.

Step 4: Analyze One Rectangle

Draw a single midpoint rectangle under the curve. Since the function is concave down, you'll notice that the rectangle extends above the curve at each midpoint. This is because the function is decreasing in steepness.

Step 5: Determine Sum's Overestimation or Underestimation

Because the function is concave down, each midpoint rectangle crosses above the curve, thus covering more area than exists just below the curve. This results in each rectangle contributing to an overestimation of the total area under the curve.

Key Concepts

Concave Down Function

A concave down function is one where the curve bends downward like the upper part of a dome. This means that the slope, or steepness, of the function is decreasing as you move along the curve. Another key characteristic is that its second derivative is negative, indicating the rate of change of the slope is negative as well.

Such a function might initially rise quickly, but as you progress along the curve, it starts to rise more slowly. Imagine a skier racing downhill. At the top of a hill, they speed up quickly, but as they approach the flat end, their speed decreases. This behavior is what a concave down curve displays. The slope is still positive, as the curve goes up, but it's growing at a slowing pace.

Recognizing a concave down function is important in calculus, especially when using Riemann sums, as it affects how we approximate areas under curves.

Overestimation of Area

When using the midpoint Riemann sum to approximate the area under a concave down function, it often results in an overestimation. But why does this happen? It takes us back to how the area is estimated using rectangles and why the shape of the curve is so crucial.

• The midpoint Riemann sum uses the value of the function at the middle of each interval to decide the height of each rectangle.
• Since the function is concave down, each rectangle will inadvertently peak above the actual curve at its midpoint.
• This means, along the interval, there's always additional area being measured that extends beyond the curve itself.

Think of tracing a hand over a dome-shaped object with a straight stick. The stick never quite touches the entire surface, staying elevated in the middle. This extra space between the straight line and the dome represents the overestimated area. Thus, for a concave down function, midpoint Riemann sums tend to cover more area than what exists below the curve.

Riemann Sums Analysis

Riemann sums are a critical tool in calculus used to approximate the integral of a function. Understanding their analysis is key to grasping how they estimate area under a curve.

• In Riemann sums, the area under a curve is approximated by summing up areas of multiple rectangles that fit under or over the curve.
• The height of each rectangle is determined based on the function’s value at specific points – left endpoints, right endpoints, or midpoints.
• Midpoint Riemann sums specifically use the value at the midpoint of each subinterval to determine the height.

The efficacy of a Riemann sum can be estimated by analyzing the nature of the function, like whether it is concave up or down and increasing or decreasing. For example, with an increasing, concave down function, as explained earlier, the approximation tends to overestimate the actual area because the rectangles surpass the curve. Recognizing patterns like these helps predict the outcome of using different types of Riemann sums for varying functions.