Let $f(t)=1 / t$ for $t \neq 0$

a. Find the average rate of change of $f$ with respect to $t$ over the intervals (i) from $t=2$ to $t=3,$ and (ii) from $t=2$ to $t=T$ .

b. Make a table of values of the average rate of change of $f$ with respect to $t$ over the interval $[2, T]$ , for some values of $T$ approaching $2,$ say $T=2.1,2.01,2.001,2.0001,2.00001,$ and $2.000001 .$

c. What does your table indicate is the rate of change of $f$ with respect to $t$ at $t=2 ?$

d. Calculate the limit as $T$ approaches 2 of the average rate of change of $f$ with respect to $t$ over the interval from 2 to $T$ . You will have to do some algebra before you can substitute $T=2$ .

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## Recommended Questions

Let \begin{equation}

g(x)=\sqrt{x} \text { for } x \geq 0

\end{equation}

\begin{equation}

\begin{array}{l}{\text { a. Find the average rate of change of } g(x) \text { with respect to } x \text { over }} \\ {\text { the intervals }[1,2],[1,1.5] \text { and }[1,1+h] .} \\ {\text { b. Make a table of values of the average rate of change of } g \text { with }} \\ {\text { respect to } x \text { over the interval }[1,1+h] \text { for some values of } h} \\ {\text { approaching zero, say } h=0.1,0.01,0.001,0.0001,0.00001,} \\ {\text { and } 0.000001 .}\end{array}

\end{equation}

\begin{equation}

\begin{array}{l}{\text { c. What does your table indicate is the rate of change of } g(x)} \\ {\text { with respect to } x \text { at } x=1 ?} \\ {\text { d. Calculate the limit as } h \text { approaches zero of the average rate of }} \\ {\text { change of } g(x) \text { with respect to } x \text { over the interval }[1,1+h] \text { . }}\end{array}

\end{equation}