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\centerline{MATH 131, section 1 -- Calculus for Physical and Life Sciences}
\centerline{Review Sheet -- Exam 3}
\centerline{November 16, 2007}
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{\it General Information}
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\item{$\bullet$}  The third exam of the semester will be given on 
the Wednesday of the week after Thanksgiving break, 
November 28, from 6:00pm to 7:30pm {\it in Haberlin 103}.  If you have a 
conflict at that time, we will run the ``second seating'' at 8:00pm again.
\item{$\bullet$}  {\it Bring a photo ID to the exam.}
\item{$\bullet$}  The exam will be designed to take an hour but you will have an extra 
30 minutes to work and check your solutions.
\item{$\bullet$}  You will be given a TI-30 scientific calculator for the exam which 
does NOT have graphing capabilities so be prepared to answer questions without 
your personal calculator.  (Note: Some of you may have one of these calculators
purchased for use in Chemistry courses here.  That is also OK.)
\item{$\bullet$} Use of cell phones, I-pods, and all other electronic devices
{\it is not allowed} during the exam.  Please leave such devices in your room 
or put them away in your backpack (make sure cell phones are {\it turned off}).
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{\it What will be covered}
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The third exam covers sections 3.3, 3.4, 3.5, 3.6, and 3.7 in Chapter 3, plus
sections 4.1, 4.2, 4.3, 4.5, and 4.6 in Chapter 4.
This includes specifically:
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\item{1.}  Rates of change in the sciences (section 3.3)
\item{2.}  Derivatives of trigonometric functions (section 3.4)
\item{3.}  The chain rule for derivatives of compositions (section 3.5)
\item{4.}  Implicit differentiation (section 3.6), including tangent lines
to parametric curves
\item{5.}  Derivatives of inverse trigonometric and logarithm functions via
the chain rule (section 3.7).  The method of logarithmic differentiation for
functions of the form $u(x)^{v(x)}$ and other forms.
\item{6.}  Related rates problems (section 4.1)
\item{7.}  Critical numbers, maximum and minimum values of $f(x)$on $[a,b]$,
the first and second derivative tests, applications to curve sketching (sections 4.2 and 4.3)
\item{8.}  Indeterminate form limits and L'Hopital's Rule (section 4.5)
\item{9.}  Optimization (max/min) problems (section 4.6)
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{\it How to prepare}
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You should go over the homework problems as well as the text and your class notes. 
Many of the problems and questions we discuss in class are excellent examples of 
test questions. I have also listed some sample problems from the text below.  
Answers can be found in the text.

We will review for the exam in class on Tuesday, November 27. 
Come prepared with specific questions if there are things you want
to discuss.
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{\it Some suggested additional review problems}
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Chapter 3 Review Problems
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\item{$\bullet$} 1 - 35 odd numbers. (Note that all the different functions and 
methods of differentiation that we discussed are included in these, {\it often combined
in various ways}.  There will be several similar functions to differentiate
on the actual exam and you will need to be able to determine which method to use 
just by looking at the form of the function.)
\item{$\bullet$} 39, 41, 47 - 57 odds (be sure you understand these)
\item{$\bullet$} 64 ({\it Answers:}  (a)  $v(t) = 3t^2 - 12$
and $a(t) = 6t$.  (b)  upward when $t > 2$, downward when $0 \le t < 2$.  (c)  Total distance
is $23$ ($7$ up and $16$ down).  (e)  Speeding up for $t > 2$, slowing down for $0 < t < 2$.), 
\item{$\bullet$} 65.
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Chapter 4 Review Problems
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\item{$\bullet$} Graphing, maxima and minima, etc. : 1, 3, 5, 7 (be careful about the domain!), 
11, 23
\item{$\bullet$} L'Hopital's Rule:  25, 27, 29, 31
\item{$\bullet$} Related rates:  34 ({\it Answer:}  ${8\over 9\pi}$ cm/sec when $h = 5$ cm), 35
\item{$\bullet$} optimization problems:  37, 39, 59
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{\it Some Sample Exam Questions}
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\noindent
{\it Disclaimer:}  As always, the actual exam questions may
be posed in different ways, may be formatted differently,
and may focus on somewhat different aspects of the material we have
covered.
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I.  Find $y'$ and simplify.
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\item{(a)}  $$y = \ln(x)\left(x^7 - {4\over \sqrt{x}}\right)$$
\item{(b)}  $$y = (e^{2x} + 2)^3$$
\item{(c)}  $$y = {x + 1\over 3x^4 - 1}$$
\item{(d)}  $$y = {\sin(x) \over 1 + \cos(x)}$$
\item{(e)}  $$y = \tan^{-1}(e^{5x})$$
\item{(f)}  $$xy^2 - 3y^3 + 2x^4 = 2$$
\item{(g)}  $$y = \cos(x)^{x^3}$$
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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 $t$ seconds for all $t \ge 0$.
\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?
decreasing? 
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III.  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 $4\pi r^2$.)
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IV. All parts of this question refer to $f(x) = 4x^3 - x^4$.
\item{(a)}  Find and classify all the critical numbers 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 $[1,4]$.  
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V.  All three parts of this question refer to the function
$f(x) = x^{2/3} - {1\over 5}x^{5/3}$.
\item{(a)}  Find all the critical numbers of $f(x)$.
\item{(b)}  Find all the inflection points of $f(x)$.
\item{(c)}  For which of the critical numbers here 
is the Second Derivative Test applicable?  Why?  Determine
the type of each such critical number using the Second
Derivative Test.
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VI. 
\item{(a)} A rectangular pen is to be constructed with one
side along an existing long stone wall (so no fence is
needed along that side), and the other three sides
made of new fencing.  The pen must enclose $1000$ square feet
of space.  What is the smallest total length of fencing that
can be used to enclose the pen?  What are the dimensions
of the pen that will use the smallest total length of fencing.
\item{(b)} Consider all rectangles with one side along the $x$-axis
that are inscribed under the curve $y = \sqrt{x}$, extending
to $x = 4$ on the right.  Find the dimensions of the rectangle
of this type of largest area.
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VII.  Evaluate the following limits.
\item{(a)} $$\lim_{x\to 0} {1 - \cos(x)\over x^2}$$
\item{(b)} $$\lim_{x\to \infty} {\ln(1 + e^x)\over x}$$ 
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