Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$f ( x ) = 5 ( 2 ) ^ { x }$ on $[ 1,7 ]$ divided into 3 subintervals
A) 70
B) 280
C) 420
D) 140

Question

Quizplus · Accepted Answer

To estimate the average value of $f(x) = 5(2)^x$ on the interval $[1,7]$ using 3 subintervals, we first find the width of each subinterval: $\Delta x = \frac{7-1}{3} = 2$. The midpoints of these subintervals are at $x = 2, 4, 6$. Evaluating $f(x)$ at these midpoints gives us $f(2) = 5(2)^2 = 20$, $f(4) = 5(2)^4 = 80$, and $f(6) = 5(2)^6 = 320$. The sum of these values is $20 + 80 + 320 = 420$. To find the average value, we divide this sum by the number of subintervals: $\frac{420}{3} = 140$.

Use a ﬁNite Sum to Estimate the Average Value of the Function