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\centerline{Mathematics 136, section 2 -- Advanced Placement Calculus}
\centerline{Review Sheet for Exam 2}
\centerline{October 27, 2009}
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{\it General Information}
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The second exam for the course will be given at 7:00pm 
next Wednesday evening, November 4, 
as announced in the course syllabus. It will cover the material
from Chapters 4 and 5 in the text that we have discussed starting 
from immediately before the first exam through and including the material 
from Wednesday, October 28 on the table of integrals.  (This is also almost
exactly the material from Problem Sets 4, 5, and 6.)  There will be 4 or 
5 problems, some possibly with several parts. The format will be similar 
to that of the first exam. 
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{\it Topics to be Covered}
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\item{1.} Critical points, local maxima and minima, global maxima and 
minima on an interval.  First and Second Derivative Tests.  Using
$f'$ and $f''$ to sketch $y = f(x)$.  
\item{2.} Indeterminate form limits and L'Hopital's Rule.
\item{3.} Applied optimization problems.
\item{4.} Riemann sums, the definite integral of a function over an interval 
$[a,b]$.  The definite integral as ``signed area.''  
Be prepared to compute a Riemann sum for a particular function and 
a given number of subdivisions (small, of course!) Also know how to derive the 
exact value of a definite integral of a function $ax^2 + bx + c$ by taking the limit of 
the Riemann sums.  The summation formulas for $\sum_{i=1}^n i$ and $\sum_{i=1}^n i^2$ 
will be provided.  
\item{5.}  The Fundamental Theorem of Calculus (know the statement of this 
theorem -- both parts)
\item{6.}  Methods of integration, including substitution, integration by 
parts, trigonometric substitution, partial fractions, use of the table of integrals.  
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{\it Review Session}
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We will review for the exam in class on Tuesday, November 3.  
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{\it Suggested Practice Problems}
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\item{$\bullet$} From Review Problems for Chapter 4: 1,3,7,25,27,29,31,39,45,47,51,57,63
\item{$\bullet$} From Review Problems for Chapter 5: 1,2,9-34,39,41
\item{$\bullet$} From Section 5.7/15,17,21,23,25,27.
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{\it Sample Exam} -- Note the real exam will contain different questions,
and possibly different types of questions.  The actual exam will be about this length.
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I. (15)  In an alternate universe, the ``super-attractive'' force
exerted by two objects of mass $m_1,m_2$ 
is {\it directly} proportional to the square of the distance between
them: $F = S m_1m_2 r^2$, where $S$ is a constant and $r$ is the 
distance between the masses.  Two 1kg masses are fixed 
at points $x = -1$ and $x = 3$ respectively along a straight line
in this alternate universe.  At what location $x$ along that line should
a third 2kg mass be placed to {\it minimize} the sum of the super-attractive
forces exerted on it by the two unit masses?
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II. (15) Evaluate $\lim_{x\to\infty} \left(1 - {3\over x}\right)^x$. 
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III. (10)  In a car moving at 90 ft/sec, the driver 
suddenly saw an obstacle 400 feet ahead and braked
to a stop in 10 seconds.  The car's velocity was recorded
by a sensor in its onboard computer every two seconds:
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$$\matrix{t\ (sec) & 0 & 2 & 4 & 6 & 8 & 10\cr
     v(t)\ (ft/sec) & 90 & 75 & 50 & 25 & 5 & 0 \cr}$$
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Can you say for sure from the information
here whether the car hit the obstacle or not?  Explain, using 
the left- and right-hand Riemann sums for $v(t)$.
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IV. 
\item{A)} (10) Using the definition of the definite integral (not the 
Evaluation Theorem), determine $\int_1^2 x^2 + 3x + 1\ dx$.  
(Possibly useful information:
$$\sum_{i=1}^n i = {n(n+1)\over 2} \qquad \sum_{i=1}^n i^2 = {n(n+1)(2n+1)\over 6}.)$$ 
Check your answer using the Evaluation Theorem.
\item{B)} (10) What is ${d\over dx} \int_1^{x^2} {\sin(t)\over t}\ dt$?
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V.  Methods of integration. In B,C,D you may use the table of integrals provided.
\item{A)} (10) Using integration by parts, show that
$$\int x^n \cos(ax)\ dx = {x^n\over a} \sin(ax) - {n\over a}\int x^{n-1} \sin(ax)\ dx.$$ 
\item{B)} (10)  $\int \tan(\sqrt{x})/\sqrt{x}\ dx$
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\item{C)} (10)  $\int (3x + 2)/(x^2 + 8x + 7)\ dx$
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\item{D)} (10) $\int (4 - x^2)^{-3/2}\ dx$
\bye 


