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\sethandouttitle{From First Principles}
\allhandoutinfo{}
\begin{enumerate}
\item Find the equation of the tangent line to the curves
\begin{enumerate}
\item $y=4x-3x^2$ ar $(2,-4)$.
\vfill
\item $y = \sqrt{x}$ at $(1,1)$.
\vfill
\end{enumerate}
\item The displacement of a particle moving in a straight line is given by the equation of motion $s=1/t^2$.
Find the velocity of the particle at $t=a$, $t=1$, $t=2$ and $t=3$.
\vfill
\newpage
\item Each limit represents the derivative of some function $f$ at a number $a$. State such an $f$ and $a$ in each case.
\begin{enumerate}
\item $\displaystyle\lim_{h \rightarrow 0} \frac{ (1+h)^8 - 1 }{h}$.
\vfill
\item $\displaystyle\lim_{x \rightarrow 5} \frac{ 2^x - 32}{x-5}$.
\vfill
\item $\displaystyle\lim_{h \rightarrow 0} \frac{ \cos(\pi + h) + 1 }{h}$.
\vfill
\end{enumerate}
\newpage
\item The cost in dollars of producing $x$ units of a certain commodity is $C(x)= 500 + 10x + 0.05 x^2$.
\begin{enumerate}
\item Find the average rate of change of $C$ with respect to $x$ when the production level is changed
from $x=100$ to $x=105$.
\vfill
\item Find the average rate of change of $C$ with respect to $x$ when the production level is changed
from $x=100$ to $x=101$.
\vfill
\item Find the instantaneous rate of change of $C$ with respect to $x$ when $x=100$.
\vfill
\end{enumerate}
\end{enumerate}
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