Question

(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

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.

Answer:

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.