Question

Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 12. L6 (sample points are left endpoints) R6 (sample points are right endpoints) M6 (sample points are midpoints) Is U an underestimate or overestimate of the true area? Is an underestimate or overestimate of the true area? Which of the numbers L6, R6, or M6 gives the best estimate? Explain.

Answer 1

The estimate of the area under the given graph using left endpoints (L6) is an underestimate, while the estimate using right endpoints (R6) is an overestimate. The midpoint estimate (M6) provides the best approximation of the true area.

Step-by-step explanation:

In approximating the area under the graph using rectangles, the choice of sample points significantly influences the accuracy of the estimate. The left endpoint estimate (L6) involves using the leftmost point of each subinterval to determine the height of the rectangle. Since the function is increasing, this method tends to underestimate the true area.

Conversely, the right endpoint estimate (R6) employs the rightmost point of each subinterval, resulting in an overestimate since the function is ascending. The midpoint estimate (M6) strikes a balance by using the midpoint of each subinterval, providing a more accurate representation of the area under the curve.

Mathematically, if we denote the width of each subinterval as Δx and the function values at the left, right, and midpoint of each subinterval as f_i, f_{i+1}, and f_{(i+1/2)}, respectively, the estimates can be expressed as follows:

`\[ L6 = \sum_(i=1)^(6) f_i \cdot \Delta x \]`

`\[ R6 = \sum_(i=1)^(6) f_(i+1) \cdot \Delta x \]`

`\[ M6 = \sum_(i=1)^(6) f_((i+1/2)) \cdot \Delta x \]`

In this particular case, since the function is increasing, L6 < M6 < R6. Therefore, the midpoint estimate M6 provides the most accurate approximation of the true area under the graph.

Answer 2

To estimate the area under the graph, divide the interval into subintervals and use left endpoints, right endpoints, or midpoints as sample points. Compare the estimates to the true area to determine if they are overestimates or underestimates. The best estimate is the one with the smallest difference to the true area.

Step-by-step explanation:

The question asks to use six rectangles to estimate the area under the graph of function f from x = 0 to x = 12. The three types of estimates are left endpoints (L6), right endpoints (R6), and midpoints (M6). To find the estimate for each type, divide the interval from 0 to 12 into six equal subintervals, and use the sample points of the left endpoint, right endpoint, or midpoint of each subinterval as the x-coordinate of the rectangle.

For L6, the left endpoints are used as the sample points, meaning the rectangles touch the curve on the left side. For R6, the right endpoints are used, so the rectangles touch the curve on the right side. For M6, the midpoints of each subinterval are used as the sample points.

To determine whether the estimates are overestimates or underestimates, compare the estimates to the true area under the graph. If the estimate is smaller than the true area, it is an underestimate. If the estimate is larger than the true area, it is an overestimate.

To determine which estimate is the best, compare the differences between the estimates and the true area. The estimate that has the smallest difference (absolute value) is the best estimate.