Preview Activity 1.4.1.
(a)
Let \(y = f(x) = 7 - 3x\text{.}\) Determine \(AV_{[-3,-1]}\text{,}\) \(AV_{[2,5]}\text{,}\) and \(AV_{[4,10]}\) for the function \(f\text{.}\)
(b)
Let \(y = g(x)\) be given by the data in the following table.
| \(x\) | \(-5\) | \(-4\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
| \(g(x)\) | \(-2.75\) | \(-2.25\) | \(-1.75\) | \(-1.25\) | \(-0.75\) | \(-0.25\) | \(0.25\) | \(0.75\) | \(1.25\) | \(1.75\) | \(2.25\) |
Determine \(AV_{[-5,-2]}\text{,}\) \(AV_{[-1,1]}\text{,}\) and \(AV_{[0,4]}\) for the function \(g\text{.}\)
(c)
Consider the function \(y = h(x)\) defined by the graph in the following figure.
Determine \(AV_{[-5,-2]}\text{,}\) \(AV_{[-1,1]}\text{,}\) and \(AV_{[0,4]}\) for the function \(h\text{.}\)
(d)
What do all three examples above have in common? How do they differ?
(e)
For the function \(y = f(x) = 7 - 3x\) from (a), find the simplest expression you can for
\begin{equation*}
AV_{[a,b]} = \frac{f(b)-f(a)}{b-a}
\end{equation*}
where \(a \ne b\text{.}\)
