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\begin{center} Test 2A, Math 1410
\end{center}

\normalsize

\textbf{Name:} \underline{\hspace{3in}}

\textbf{PID Number:} \underline{\hspace{3in}}

\begin{center}

I pledge that I have neither given nor received any unauthorized
assistance on this exam.

\vspace{.4in}

$\overline{\hspace{2.4in} (signature) \hspace{2.4in}}$

\vspace{1in}

\Large \textbf{DIRECTIONS} \normalsize

\end{center}

\begin{enumerate}

\item Show all of your work. A correct answer with insufficient
work will lose points, as will an answer with incorrect notation.

\item Clearly indicate your answer by putting a box around it.

\item Calculators are allowed on this exam.

\item Give all answers in exact form, not decimal form (that is,
put $\pi$ instead of 3.1415, $\sqrt{2}$ instead of 1.414, etc) unless otherwise stated.

\item Make sure you sign the pledge and write your ID on both pages.

\item Number of questions = 10.  Total Points = 100.

\end{enumerate}

\newpage

\normalsize
\hspace{.4in}
\textbf{PID Number:} \underline{\hspace{3in}}\\
\begin{enumerate}


\item (8  points) Calculate $y'$ if $$y=\tan\left[\ln\left(\frac{1}{x^3}\right)\right]$$

\newpage


\item (10  points) The law of Graham-Squire states that the following relationship holds between $P$ (Pressure), $V$ (Volume), and $T$ (Temperature) of air inside a sphere.  (Assume that $P$, $V$, and $T$ are all positive numbers.) $$\frac{PV^3}{T^2}=K$$Where $K$ is some constant.

(a) Assuming that the pressure is fixed (i.e. $P$ is held constant), find the rate of change of the volume with respect to the temperature.

\vspace{4in}

(b) Assuming that $P$ is held constant, \underline{use your answer from (a)} to explain why an increase in temperature will result in an increase in the volume. Hint: It may help to think about the volume function.

\newpage


\item (14  points) For the equation $x^2+xy+\sin ^2 y=1$, find $\d{\frac{dy}{dx}}$ \underline{and} the equation for the tangent line at the point (1,0).

\newpage


\item (12  points) Use calculus to find the $x$-coordinate(s) where $\d{y=\frac{x^2-3}{x-2}}$ has a horizontal tangent line.

\newpage


\item (8  points) Find $\frac{d}{dx}(\arccos(x)\sqrt{1-x^2})$.  Simplify your answer.

\vspace{4.5in}

\item (4 points) State if the following is true or false.  Give a brief explanation to justify your answer.

``The graph of $9x^2+4y^2=36$ is continuous and has no sharp corners or cusps, therefor the derivative $\d \frac{dy}{dx}$ exists for all points on the graph."

\newpage

\item (10 points)  Using only the derivatives for $\sin x$ and $\cos x$, as well as either the chain rule or the quotient rule, prove that 
\[\frac{d}{dx}(\sec x) = \sec x\tan x\]
\newpage

\item (10  points) \underline{Use differentials} to approximate the change in the area of a circle as the radius changes from 5 cm to 5.3 cm.

\newpage


\item (12  points) Let $A(x)=\d \frac{f(g(x))}{h(x)}$.  Find $A'(0)$ if the following values hold for $f$, $g$, and $h$:
\[ \begin{tabular}{ccc}
	$f(0)=4 $&$ g(0)=1 $&$ h(0)=-2$ \\
	$f(1)=-1$ & $g(1)=6$ & $h(1)=7$ \\
	$f'(0)= -7$ & $g'(0)=3$ & $h'(0)=2$ \\
	$f'(1)= -3 $& $g'(1)=-5$ & $h'(1)=0$
\end{tabular} \]Simplify your answer.  Note: You may not need all of the values above.
\newpage


\item (12  points) Use logarithmic differentiation to calculate $f'(x)$ for 
\[ f(x)=(\cot x)^{1/x}  \]Simplify your answer.

\vspace{6.5in}

Extra Credit (2 points): Is the following equation True or False?  If true briefly explain why.  If false, either explain why or give a counterexample. $$\frac{d}{dx}[f(\sqrt{x})g(x)]=[f'(x)\cdot \frac{1}{2}x^{(-1/2)}]g(x)+g'(x)f(x)$$


\end{enumerate}
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