(a) use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to…

(a) use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 24. (i) l6 = sample points are left - endpoints. (ii) r6 = sample points are right - endpoints. (iii) m6 = sample points are midpoints. (b) is l6 an underestimate or overestimate of the true area? o underestimate o overestimate (c) is r6 an underestimate or overestimate of the true area? o underestimate o overestimate

(a) use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 24. (i) l6 = sample points are left - endpoints. (ii) r6 = sample points are right - endpoints. (iii) m6 = sample points are midpoints. (b) is l6 an underestimate or overestimate of the true area? o underestimate o overestimate (c) is r6 an underestimate or overestimate of the true area? o underestimate o overestimate

Answer

Explanation:

Step1: Determine the width of each sub - interval

The interval is from (x = 0) to (x=24). We are using (n = 6) rectangles. The width of each sub - interval (\Delta x=\frac{b - a}{n}), where (a = 0), (b = 24) and (n = 6). So (\Delta x=\frac{24-0}{6}=4). The sub - intervals are ([0,4],[4,8],[8,12],[12,16],[16,20],[20,24]).

Step2: Calculate (L_6) (left - endpoint estimate)

For a left - endpoint estimate (L_n=\sum_{i = 0}^{n - 1}f(x_i)\Delta x). Here (n = 6) and (\Delta x=4). We need to find the value of the function (y = f(x)) at the left - endpoints of each sub - interval: (x_0=0,x_1 = 4,x_2=8,x_3 = 12,x_4=16,x_5=20). Let (y_i=f(x_i)). Then (L_6=4(f(0)+f(4)+f(8)+f(12)+f(16)+f(20))).

Step3: Calculate (R_6) (right - endpoint estimate)

For a right - endpoint estimate (R_n=\sum_{i = 1}^{n}f(x_i)\Delta x). Here (n = 6) and (\Delta x = 4). The right - endpoints are (x_1=4,x_2=8,x_3 = 12,x_4=16,x_5=20,x_6=24). Then (R_6=4(f(4)+f(8)+f(12)+f(16)+f(20)+f(24))).

Step4: Calculate (M_6) (mid - point estimate)

The mid - points of the sub - intervals ([0,4],[4,8],[8,12],[12,16],[16,20],[20,24]) are (x_1^=2,x_2^=6,x_3^* = 10,x_4^=14,x_5^=18,x_6^=22). For a mid - point estimate (M_n=\sum_{i = 1}^{n}f(x_i^)\Delta x), so (M_6=4(f(2)+f(6)+f(10)+f(14)+f(18)+f(22))).

Step5: Determine if (L_6) is an over - or under - estimate

If the function (y = f(x)) is increasing on the interval ([a,b]), then the left - endpoint estimate (L_n) is an underestimate and the right - endpoint estimate (R_n) is an overestimate. If the function (y = f(x)) is decreasing on the interval ([a,b]), then the left - endpoint estimate (L_n) is an overestimate and the right - endpoint estimate (R_n) is an underestimate. Since the function (y = f(x)) in the graph is increasing, (L_6) is an underestimate.

Step6: Determine if (R_6) is an over - or under - estimate

Since the function (y = f(x)) is increasing, (R_6) is an overestimate.

(a) (L_6=4(f(0)+f(4)+f(8)+f(12)+f(16)+f(20))) (R_6=4(f(4)+f(8)+f(12)+f(16)+f(20)+f(24))) (M_6=4(f(2)+f(6)+f(10)+f(14)+f(18)+f(22))) (b) (L_6) is an underestimate because the function (y = f(x)) is increasing and we are using left - endpoints. (c) (R_6) is an overestimate because the function (y = f(x)) is increasing and we are using right - endpoints.