<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\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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