Question

5. The function is continuous on the interval [10, 20] with some of its values given in the table above. Estimate the average value of the function with a Right Hand Sum Approximation, using the intervals between those given points. (4 points)
x 10 12 15 19 20
f(x) -2 -5 -9 -12 -16


A -9.250
B -10.100
C-7.550
D-6.700

Answer 1

The average value of $f(x)$ over the given interval is

$\dfrac1{20-10}\displaystyle\int_{10}^{20}f(x)\,\mathrm dx$

You're given five points, but only four will contribute to the sum as there are four subintervals that you can work with. Because you are finding the right-endpoint approximation, the height of the rectangles used in the Riemann sum will be determined by the points $x\in\{12,15,19,20\}$, where the height of each rectangle is the corresponding value of $f(x)$ and the width is the length of the subinterval. In this case, the lengths would be (left to right and respectively) $\Delta x\in\{2,3,4,1\}$, which is simply the set of forward differences of the previous set.

So the average value is approximated by

$\dfrac1{20-10}\displaystyle\sum_{x\in\{12,15,19,20\}}f(x)\Delta x=\frac1{10}\left(2f(12)+3f(15)+4f(19)+f(20)\right)=-\dfrac{101}{10}=-10.1$

Answer 2

Answer:

-10.100

Step-by-step explanation:

I just took this and the answer calculated above is correct