Calculate R8 for f(x)=8−x over [3,5].
R8=
To calculate R8, we need to first divide the interval [3,5] into subintervals with equal width. Since we want to find R8, we need to divide the interval into 8 subintervals.
The width of each subinterval can be calculated by taking the difference between the endpoints of the interval and dividing it by the number of subintervals. In this case, the width would be:
Width = (5 - 3) / 8 = 2 / 8 = 0.25
Next, we need to choose a sample point within each subinterval. Since we want to find R8, we will choose the right endpoint of each subinterval as the sample point.
The right endpoint of each subinterval can be calculated by adding the width to the left endpoint of the subinterval. For example, for the first subinterval [3, 3 + 0.25], the right endpoint would be 3 + 0.25 = 3.25.
Now, we can calculate the sum of the function evaluated at each sample point multiplied by the width of each subinterval:
R8 = (f(3.25) x 0.25) + (f(3.5) x 0.25) + (f(3.75) x 0.25) + (f(4) x 0.25) + (f(4.25) x 0.25) + (f(4.5) x 0.25) + (f(4.75) x 0.25) + (f(5) x 0.25)
To calculate each term, we need to substitute the value of x into the function f(x) = 8 - x:
R8 = [(8 - 3.25) x 0.25] + [(8 - 3.5) x 0.25] + [(8 - 3.75) x 0.25] + [(8 - 4) x 0.25] + [(8 - 4.25) x 0.25] + [(8 - 4.5) x 0.25] + [(8 - 4.75) x 0.25] + [(8 - 5) x 0.25]
Now, we can simplify the expression and calculate the result:
R8 = (4.75 x 0.25) + (4.5 x 0.25) + (4.25 x 0.25) + (4 x 0.25) + (3.75 x 0.25) + (3.5 x 0.25) + (3.25 x 0.25) + (3 x 0.25)
= 1.1875 + 1.125 + 1.0625 + 1 + 0.9375 + 0.875 + 0.8125 + 0.75
= 7.75
Therefore, R8 for the function f(x) = 8 - x over the interval [3,5] is 7.75.