Calculate R8 for f(x)=8−x over [3,5].
R8=
To calculate R8 for the function f(x) = 8 - x over the interval [3, 5], we need to divide the interval into subintervals of equal length and approximate the area under the curve using a right endpoint Riemann sum with 8 rectangles.
First, let's find the length of each subinterval. In this case, the interval [3, 5] has a length of (5 - 3) = 2. To create 8 subintervals, we divide the interval length by the number of subintervals: 2 / 8 = 0.25.
Next, we will determine the x-values at the right endpoints of each subinterval. Starting from the left endpoint of the interval [3, 5], we add the subinterval length successively to get the x-values for the right endpoints:
3.00, 3.25, 3.50, 3.75, 4.00, 4.25, 4.50, 4.75, 5.00
Now, we evaluate the function at each right endpoint to get the corresponding y-values:
f(3.00) = 8 - 3.00 = 5.00
f(3.25) = 8 - 3.25 = 4.75
f(3.50) = 8 - 3.50 = 4.50
f(3.75) = 8 - 3.75 = 4.25
f(4.00) = 8 - 4.00 = 4.00
f(4.25) = 8 - 4.25 = 3.75
f(4.50) = 8 - 4.50 = 3.50
f(4.75) = 8 - 4.75 = 3.25
f(5.00) = 8 - 5.00 = 3.00
Now, we multiply each y-value by the subinterval length (0.25) to calculate the area of each rectangle:
5.00 * 0.25 = 1.25
4.75 * 0.25 = 1.1875
4.50 * 0.25 = 1.125
4.25 * 0.25 = 1.0625
4.00 * 0.25 = 1.00
3.75 * 0.25 = 0.9375
3.50 * 0.25 = 0.875
3.25 * 0.25 = 0.8125
3.00 * 0.25 = 0.75
Finally, we sum up the areas of all the rectangles to find R8:
1.25 + 1.1875 + 1.125 + 1.0625 + 1.00 + 0.9375 + 0.875 + 0.8125 + 0.75 = 8.00
Therefore, R8 for the function f(x) = 8 - x over the interval [3, 5] is equal to 8.
To calculate R8 for the function f(x) = 8 - x over the interval [3,5], you need to partition the interval into subintervals of equal width and then find the maximum value of f(x) within each subinterval.
First, determine the width of each subinterval. In this case, we have the interval [3,5] which has a width of 5 - 3 = 2. To calculate R8, we need to divide this interval into 8 equal subintervals. Therefore, the width of each subinterval will be (2/8) = 0.25.
Next, we need to consider each subinterval and find the maximum value of f(x) within each subinterval. Starting with the first subinterval [3, 3.25], we evaluate f(x) at the endpoints and find the maximum:
f(3) = 8 - 3 = 5
f(3.25) = 8 - 3.25 = 4.75
In this subinterval, the maximum value of f(x) is 5.
Continue this process for each subinterval to find the maximum value within each one:
[3.25, 3.5]: f(3.25) = 4.75, f(3.5) = 4.5, maximum = 4.75
[3.5, 3.75]: f(3.5) = 4.5, f(3.75) = 4.25, maximum = 4.5
[3.75, 4]: f(3.75) = 4.25, f(4) = 4, maximum = 4.25
[4, 4.25]: f(4) = 4, f(4.25) = 3.75, maximum = 4
[4.25, 4.5]: f(4.25) = 3.75, f(4.5) = 3.5, maximum = 3.75
[4.5, 4.75]: f(4.5) = 3.5, f(4.75) = 3.25, maximum = 3.5
[4.75, 5]: f(4.75) = 3.25, f(5) = 3, maximum = 3.25
Finally, the value of R8 is the maximum of all these maximum values within each subinterval. In this case, the maximum value is 5. Therefore, R8 for f(x) = 8 - x over the interval [3,5] is 5.