\noindent
Multivariable Content Post Test \hspace{2.01in} {\it Raising Calculus to the Surface}
\vspace{.1in}
\noindent
\hspace{4.8in} {\it Code: } \underline{\hspace{1in}}
\vspace{1in}
\begin{tabular}{ll}
{\bf Name:} & \underline{\hspace{2.5in}} \\[.75in]
{\bf Course:} & \underline{\hspace{2.5in}} \\[.75in]
{\bf Instructor:} & \underline{\hspace{2.5in}} \\[0.75in]
{\bf Institution:} & \underline{\hspace{2.5in}} \\[0.75in]
{\bf Term:} & \underline{\hspace{2.5in}} \\[0.75in]
{\bf Year:} & \underline{\hspace{2.5in}} \\[0.75in]
\end{tabular}
\newpage{}
\noindent
Multivariable Content Post Test \hspace{2.01in} {\it Raising Calculus to the Surface}
\vspace{.1in}
\noindent
\hspace{4.8in} {\it Code: } \underline{\hspace{1in}}
\vspace{0.25in}
\begin{enumerate}
\item Consider the function $f(x,y) = x+y^2$. Plot each of the following situations and label the two axes for each plot.
\begin{center}
\begin{tabular}{|c|c|}
\hline
\hspace{2in} & \hspace{2in} \\[1.75in]
\hline
$f(x,y)$ when $x=3$ & $f(x,y)$ when $f=5$ \\
\hline
\end{tabular}
\end{center}
\vfill
\item The yield of candy $C$ (in $kg$) from a cooking process depends on
the ambient air temperature $T$ (in $^\circ C$) and the amount of sugar $S$ (in $L$).
Thus, $C$ is a function of $T$ and $S$, given by $C = C(T,S)$ as shown in the contour plot below. Plot each of the following situations and label the two axes for each plot.
\hspace{-.85in}
\begin{minipage}[]{3.25in}
\vspace{0pt}
\resizebox{3.25in}{!}{\includegraphics{../images/assessment_images/volumeTempContourPlot2}}
\end{minipage}
\hspace{.125in}
\begin{minipage}[]{2.5in}
\begin{tabular}{|c|}
\hline
\hspace{2.25in} \\[1.65in]
\hline
Plot the yield of candy when $S= 56L$\\
\hline
\hline
\hspace{2.25in} \\[1.65in]
\hline
Plot sugar and temperature when $C=35kg$ \\
\hline
\end{tabular}
\end{minipage}
\end{enumerate}
\newpage{}
\hspace{-.75in}
Multivariable Content Post Test \hspace{2.01in} {\it Raising Calculus to the Surface}
\vspace{.1in}
\noindent
\hspace{4.8in} {\it Code: } \underline{\hspace{1in}}
\vspace{0.25in}
\begin{enumerate}
\item The function $g(x,y) = \cos(x) + y^2 - \ln(xy)$. Find the following:
\vspace{.1in}
\begin{enumerate}
\item $\frac{\partial g}{\partial y}(5,2)$
\vfill
\item $\frac{\partial g}{\partial x}(5,2)$
\vfill
\end{enumerate}
\item The function $t=t(h,s)$ relates temperature to height and speed of an airplane, where $h$ is the height above ground (in km) and $s$ is the speed of flight (in km/hr). The temperature is given in degrees Celsius.
\vspace{.1in}
\begin{enumerate}
\item What does $t(23,140)=12$ mean in words?
\vfill
\item What does $t(h, 180)$ mean in words?
\vfill
\item What does $\frac{\partial t}{\partial h}$ mean in words?
\vfill
\item What does $\frac{\partial t}{\partial s}(6,215)=8$ mean in words?
\vfill
\end{enumerate}
\newpage{}
\end{enumerate}
\newpage{}
\hspace{-.75in}
Multivariable Content Post Test \hspace{2.01in} {\it Raising Calculus to the Surface}
\vspace{.1in}
\noindent
\hspace{4.8in} {\it Code: } \underline{\hspace{1in}}
\vspace{0.25in}
\begin{enumerate}
\item Let $f(x,y) = x \; \ln(y)$.
\vspace{.1in}
\begin{enumerate}
\item Find $\vec{\nabla} f (x,y)$.
\vspace{1.25in}
\item Find the directional derivative of $f$ at the point $(2,5)$ in the direction $\vec{v} = 2\hat{\imath} - \hat{\jmath}$.
\vspace{1.25in}
\end{enumerate}
\item The function $g$ gives the density of a thin metal plate (in $\frac{g}{cm^2}$) at a point $(x,y)$. The plot below shows the gradient vector $\vec{\nabla} g$ at point $a$, which has magnitude $8\frac{g}{cm^3}$. Assume $x$ and $y$ are measured in $cm$.
\hspace{-.85in}
\begin{minipage}[]{3in}
\vspace{0pt}
\fbox{\resizebox{3in}{!}{\includegraphics{../images/assessment_images/gradientSketch1}}}
\end{minipage}
\hspace{.1in}
\begin{minipage}[]{3.35in}
\vspace{.25in}
\begin{enumerate}
\item Sketch the level curve through point $a$ on the plot.
\vspace{.35in}
\item If $g(a) = 80$ and $\left| \vec{\nabla} g (a) \right| = 8$ (with appropriate units), estimate $g(b)$.
\vspace{.75in}
\item Estimate the rate of change from point $a$ towards the origin.
\vspace{.75in}
\item Estimate the rate of change from point $a$ in the $\hat{j}$ direction.
\vspace{.65in}
\end{enumerate}
\end{minipage}
\end{enumerate}
\newpage{}
\hspace{-.75in}
Multivariable Content Post Test \hspace{2.01in} {\it Raising Calculus to the Surface}
\vspace{.1in}
\noindent
\hspace{4.8in} {\it Code: } \underline{\hspace{1in}}
\vspace{0.25in}
\begin{enumerate}
\item Set up an integral to find the volume enclosed between the two surfaces given by $z = x^2 + y^2$ and $z = 10 - x^2 - y^2$.
\vspace{2in}
\item The graphs of two functions $f$ and $g$ both have $f(0,0) = 100 = g(0,0)$. The functions have the same average value on the domain $D$, a disk (with boundary) of radius $2$ centered at the origin. However, $\left| \vec{\nabla} f \right| < \left| \vec{\nabla} g \right|$ at every point on $D$.
\vspace{.1in}
\begin{enumerate}
\item Which function's graph ($f$ or $g$) has less surface area over $D$? Explain.
\vfill
\item Which function's graph ($f$ or $g$) has a larger volume over $D$? Explain.
\vfill
\item Which function ($f$ or $g$) has a larger maximum value on $D$? Explain.
\vfill
\item Which function ($f$ or $g$) has a smaller minimum value on $D$? Explain.
\vfill
\end{enumerate}
\end{enumerate}
i