Question

FAN AND MEDAL!!

A volleyball playoff competition has eliminations each round. The table below shows the number of teams in each round of the tournament:


Round (x) 1 2 3 4
Teams f(x) 16 8 4 2


Compute the average rate of change of f(x) from x = 1 to x = 4 and identify the meaning of that rate.
negative three over fourteen; on average, there were 3 fewer teams every 14 rounds
negative fourteen over three; on average, there were 14 fewer teams every 3 rounds
3; there were 3 rounds between rounds 1 and 4
14; there were 14 fewer team between rounds 1 and 4

Answer 1

@loser66

Answer 2

@paki

Answer 3

@whpalmer4

Answer 4

@wio

Answer 5

@freckles

Answer 6

Here's our data table:

\[
\begin{array}{cc}
x&f(x)\\\hline 1 & 16 \\
2 & 8 \\
3 & 4 \\
4 & 2 \\
\end{array}
\]

Answer 7

We want to compute the average rate of change of \(f(x)\) from \(x=1\) to \(x=4\).

The average rate of change of a function over an interval is simply the difference between the starting value of the function (at the left endpoint of the interval) and the final value of the function (at the right endpoint of the interval), divided by the length of the interval.

If you think of the average rate of change as the slope of a line connecting the two endpoints, you can see that the slope of that line is just the rise over the run, or the difference in height divided by the difference in position on the \(x\)-axis.

Answer 8

If you have $10 in your wallet, and every day your parents give you $5 for your allowance, the average rate of the balance in your wallet is $5 / day. If you look on Monday and you have $10, and by Friday you have not spent any money, your balance will now be the $10 you started with, plus the average rate of change ($5/day) * the number of days (5), giving you \[10+5*5 = 35\]dollars in the wallet.

What we are doing is the opposite of that. We know how much was in the wallet on Monday ($10) and how much was in the wallet 5 days later ($35), and we are using that information to find the average rate of change.

Answer 9

To find the average rate of change of \(f(x)\) from \(x=a\) to \(x=b\), you evaluate the following expression:

\[\frac{f(b) - f(a)}{b-a}\]
That's all there is to it...