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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Exam 4 Sample/Practice Problems}
\centerline{November 28, 2017}
\vskip 10pt
\noindent
{\it Disclaimer:}  As always, the actual exam questions may
be posed in different ways, may be formatted differently,
and may focus on slightly different aspects of the material we have
covered.  Also, the actual exam will be significantly shorter than 
this group of questions(!)
\vskip 10pt
\begin{enumerate}
\item[I.]  Find $y'$; don't simplify.
\vskip 10pt
\begin{enumerate}
\item[(A)]  $$y = \ln(x)\left(x^7 - \frac{4}{\sqrt{x}}\right)$$
\item[(B)]  $$y = \sin^{-1}(e^{2x} + 2)$$
\item[(C)]  $$y = \frac{\ln(x + 1)}{3x^4 - 1}$$
\item[(D)]  $$y = \frac{\sin(x)}{1 + \cos(x)}$$
\item[(E)]  $$y = \tan^{-1}(x^2+x)$$
\item[(F)]  Using implicit differentiation: $$xy^2 - 3y^3 + 2x^4 - 4xy = 2$$
\item[(G)]  Find the equation of the line tangent to the curve from (F) at $(x,y) = (1,0)$
\end{enumerate}
\vskip 10pt
\item[II.]  The quantity of a reagent present in a chemical 
reaction is given by $Q(t) = t^3 - 3t^2 + t + 30$ grams at time $t \ge 0$ seconds.
\begin{enumerate}
\item[(A)]  Over which intervals with $t \ge 0$ is the amount increasing? decreasing?
\item[(B)]  Over which intervals is the rate of change of $Q$ increasing? decreasing?
\end{enumerate}
\vskip 10pt
\item[III.]  A spherical balloon is being inflated at 20 cubic inches per
minute.  (The volume of a sphere of radius $r$ is $V = \frac{4\pi r^3}{3}$ and the 
surface area is $V = 4\pi r^2$.)
\begin{enumerate}
\item[(A)]  When the radius is $6$ inches, at what rate is the 
radius of the balloon increasing?
\item[(B)]  When the radius is $6$ inches, at what rate is the surface
area increasing?
\end{enumerate}
\vskip 10pt
\item[IV.]  A baseball diamond is a square with side of length 90 feet. After
hitting the ball, a player leaves home plate and runs toward first base at 15 ft/sec.
(Assume the runner is running straight along the base path -- this is a bit 
unrealistic, of course, but let's
keep it simple for the purposes of the problem!)  How fast is the runner's distance
from second base changing when he is half way to first base? 
\vskip 10pt
\item[V.]   All parts of this question refer to $f(x) = 4x^3 - x^4$.
\vskip 10pt
\begin{enumerate}
\item[(A)]  Find and classify all the critical points of $f$
using the First Derivative Test.
\item[(B)]  Over which intervals is the  graph $y = f(x)$ concave
up?  concave down?
\item[(C)]  Sketch the graph $y = f(x)$.  
\item[(D)]  Find the absolute maximum and minimum of $f(x)$
on the interval $[1,4]$.  
\end{enumerate}
\vskip 10pt
\item[VI.]  All three parts of this question refer to the function $f(x)$
\emph{whose derivative is plotted in Figure 1.  NOTE:  This is the graph $y = f'(x)$
not $y = f(x)$.}
\begin{figure}
\begin{center}
\includegraphics[width=100mm]{Ex4SamplePlot.png}
\caption{Plot of $y = f'(x)$ for Problem VI}
\end{center}
\end{figure}
\begin{enumerate}
\item[(A)]  Give approximate values for all the critical points of $f(x)$
in the interval shown, and say whether $f$ has a
local maximum, a local minimum, or neither at each.
\item[(B)]  Find approximate values for all the inflection points of $f(x)$.
\item[(C)]  Over which intervals is $y = f(x)$ concave up? concave down?
\end{enumerate}
\vskip 10pt
\item[VII.]  Optimization problems.  
\begin{enumerate}
\item[(A)]  A rectangular box with no top is created out of a rectangular piece of 
cardboard by cutting equal squares out of the four corners and folding up the sides. 
If the original piece of cardboard was 20 inches by 15 inches, what is the largest
volume possible for the resulting box?  (Hint: Let $x$ be the side of the four squares
cut out of the corners.  The volume is length times width times height.)
\item[(B)]  A rectangular poster is to be created with 400 square inches of printed material
surrounded by 2 inch margins on the top and bottom and the left and right edges.  
What should the dimensions of the poster be to minimize the total area (printed material
plus margins)?  
\item[(C)]  A billboard 20 feet tall is mounted 10 feet above eye level on the wall 
of a building.  How far should a person stand from the wall in order to maximize the 
angle $\theta$ subtended by the billboard at the person's eye.  (Hint:  Draw a diagram first;
see the top diagram in Figure 30 on page 248 of the text if you cannot figure out what this
means.)
\item[(D)]   A window has the shape of a rectangle surmounted by a semicircle (see
Figure 10 on page 245 of our textbook if you don't understand what this means).  The 
total perimeter of the window is 600 cm.  What should the dimensions be to make the 
area of the window be as large as possible (so that it will admit the most light possible)?
\end{enumerate}
\end{enumerate}
\end{document}
