\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}