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{\Large Worksheet}

\begin{enumerate}

\item  Find the following derivatives:(Challenge: find the second derivatives)
\begin{enumerate}
\item  \[ \frac{d}{dx}(\cos^3(x^2-2x)) \]
\item  \[ \frac{d}{dx}(\cot(3x+1)) \]
\item  \[ \frac{d}{dt}(\csc^2(\sin t)) \]
\item  \[ \frac{d}{dx} \left( \frac{\sin^2x}{2x^2+3\cos^22x} \right) \]
\item  \[ \frac{d}{dx} \left( (\sin x)(\tan^2 x) \right) \]
\item  \[ \frac{(x^2-\sin x)(\cot^2x)}{\sec x^2 - \tan(\sin x)} \]
\item  \[ \frac{d}{dx}(x^3+sin^2x-1) \]
\end{enumerate}

\item  You are fishing and your bobber is moving up and down (from the waves).  The position of your bobber at time $t$ is the function $\sin(3t)$.  Find the velocity and acceleration of your bobber when $t=\frac{\pi}{9}$.
Now an extremely large fish takes your worm and your bobber begin to move violently up and down with its position given as $3t\cos^3(12t)$.  Find the velocity and acceleration of the bobber when $t=\frac{\pi}{48}$.

\item  If a the radius of a sphere is growing at a rate of 3 cm per minute, find:
\begin{enumerate}
\item  The rate that volume is growing when the radius is 6 cm
\item  The rate the surface area is growing when the radius is 6 cm
\end{enumerate}

\item  Find the following limits:
\begin{enumerate}
\item \[  \lim_{t \rightarrow 0} \frac{\sin 3x}{25x} \]
\item \[  \lim_{t \rightarrow 0} \frac{x}{\sin 13x} \]
\item \[  \lim_{h \rightarrow 0} \frac{\sin^23h}{h(\cos h -1)} \]
\item \[  \lim_{c \rightarrow 0} \frac{1-\cos^2 3c}{\sin2c} \]
\end{enumerate}

\item  Find $f'(2)$ and $f''(2)$ given  $f(t)=\frac{t^2-3t+1}{2t+1}$

\item  You are outside and kick a basket ball straight into the air and the height (in feet) is given by: $h(t)=-16t^2+75t$.  What is is the intitial velocity of the ball?  How high does the ball go?  When does the ball hit the ground?  How fast is it going when it hits the ground again?

Suppose instead that you are indoors and the cieling is only 30 feet high.  Does the ball hit the cieling?  If so, when and how fast is it going when it does?



\end{enumerate}



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