\magnification=\magstep1
\centerline{MATH 133, section 1 -- Calculus With Fundamentals 1}
\centerline{Exam 3 Practice Problems}
\centerline{November 4, 2015}
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\noindent
I. Do not use the differentiation rules from Chapter~3 in this
  question.
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\item{A)} State the limit definition of the
  derivative~$f'(x)$.
\item{B)} Use the definition to
    compute the derivative function of $f(x) = {1\over 3x}$.
\item{C)} Find the equation of the line tangent to
    the graph $y = {1\over 3x}$ at $x=2$.
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\noindent
II.  Use the sum, product, quotient, and/or chain  
  rules to compute the following derivatives. You may use any correct
  method, but must show work for full
  credit.
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\item{A)}
$${d\over dx}\left(5x\sqrt{x}-{2 \over x^3}+ 11x- 4\right)$$
\item{B)}
$${d\over dt}\left({t^2e^{3t}\over t^4 + 1}\right)$$
\item{C)}
$${d\over dz} {z^2 - 2z + 4\over z^2 + 1}$$
\item{D)}
$${d\over dx}\left({\pi^2+\tan(e^\pi)-2x^e\over 4}\right)$$
\item{E)} 
$${d\over dx}\left( \sin(x)\left(x^7 - {4\over \sqrt{x}}\right)\right)$$
\item{F)} Find $y'$ (note this is just another way of asking the same question!)
$$y = (e^{2x} + 2)^3$$
\item{G)}  Find $y'$  
$$y = {x + 1\over 3x^4 - 1}$$
\item{H)}  Find $y'$
$$y = {\sin(x) \over 1 + \cos(x)} + x^2 \cos(x^3 + 3)$$
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\noindent
III. The total cost (in $ \$ $) of repaying a car loan at 
interest rate of $r \%$ per year is $C = f(r)$.  
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\item{A)} What is the meaning of the statement $f(7) = 20000$?
\item{B)} What is the meaning of the statement $f'(7) = 3000$? What are the units of $f'(7)$? 
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\noindent
IV.  The quantity of a reagent present in a chemical 
reaction is given by $Q(t) = t^3 - 3t^2 + t + 30$ grams at time $t$ seconds for all $t \ge 0$.
(Note:  For a question like this, I could also give you the plot of the function and ask
questions like those below.  In this case you need to start from the formula and compute
$Q'(t)$; if you were given the graph, you need to make the connection between slopes of 
tangent lines and signs of $Q'(t)$ visually.)
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\item{(a)}  Over which intervals with $t \ge 0$ is the amount increasing? (i.e. $Q'(t) > 0$) decreasing (i.e. $Q'(t) < 0$)?
\item{(b)}  Over which intervals is the rate of change of $Q$ increasing? decreasing?
decreasing? 
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\noindent
V.  A spherical balloon is being inflated at 20 cubic inches per
minute.  When the radius is $6$ inches, at what rate is the 
radius of the balloon increasing?  At what rate is the surface
area increasing?  (The volume of a
sphere of radius $r$ is $V = {4\pi r^3\over 3}$ and the surface
area is $A = 4\pi r^2$.)
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\noindent
VI.  Review problems 2 and 3 from Quiz 5.  
\bye
