Question

Consider the following. WebAssign Plot (a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 36. (i) Sample points are left endpoints. L6 = (ii) Sample points are right endpoints. R6 = (iii) Sample points are midpoints. M6 = (b) Is L6 an underestimate or overestimate of the true area? overestimate underestimate (c) Is R6 an underestimate or overestimate of the true area? overestimate underestimate (d) Which of the numbers gives the best estimate? R6 L6 M6

Answer 1

To estimate the area under the graph, we divide the x-axis into intervals and use left endpoints, right endpoints, or midpoints. L6 is an underestimate, R6 is an overestimate, and M6 gives the best estimate.

Step-by-step explanation:

To estimate the area under the graph, we divide the x-axis into 6 equal intervals. For left endpoints, we use the leftmost point of each interval. For right endpoints, we use the rightmost point. And for midpoints, we use the middle point. We then calculate the area of each rectangle, and sum them up to get the estimate of the total area.

(b) L6 is an underestimate of the true area because it only includes the left side of each interval.

(c) R6 is an overestimate of the true area because it only includes the right side of each interval.

(d) M6 gives the best estimate because it takes the midpoint of each interval.

Answer 2

Estimating the area under a graph using six rectangles involves calculating the areas with the function values at left endpoints, right endpoints, and midpoints. The L6 estimate may be an underestimate or overestimate depending on whether the function is decreasing or increasing. Similarly for R6, and M6 generally provides a better estimate since it averages out the other two methods.

Step-by-step explanation:

The student is asked to estimate the area under a graph of a function f from x = 0 to x = 36 using rectangles. To do this for L6 (using left endpoints), R6 (using right endpoints), and M6 (using midpoints), we divide the interval [0, 36] into six equal subintervals, each of width 6 (36/6 = 6). The height of each rectangle would be determined by the function's value at the sample point in that subinterval according to the method (left, right, or midpoint).

For L6, the rectangles' heights are determined by the value of f at the left endpoints of each subinterval. For R6, we use the right endpoints, and for M6, we use the midpoints. We sum the areas of all six rectangles to estimate the total area under the curve for each method.

To decide if L6 is an overestimate or an underestimate, we look at the behavior of the function. If the function is increasing, L6 tends to be an underestimate; if decreasing, an overestimate. Conversely, R6 would be an overestimate for an increasing function and an underestimate for a decreasing one. Finally, M6 usually provides the best estimate, especially when the function has no consistent trend of increasing or decreasing.