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\centerline{Mathematics 132 -- Calculus for Physical and Life Sciences 2}
\centerline{Exam 1 -- Things To Know}
\centerline{February 16, 2005}
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\noindent
{\it General Information}
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As announced in the course syllabus, the first 
midterm exam of the semester will be given (``main seating'') 
at 6:00 p.m. on Wednesday, February 23.  You will have 90 minutes 
to work on the exam if you need that much time.
The format will be similar to that of the midterm exams
last semester.  The exam will cover the material since the start
of the semester, including the following material
from Chapters 6, and sections 1-4 of Chapter 7.
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\item{1)}  The Fundamental Theorem of Calculus:  If $F(x)$
is an antiderivative of a continuous function $f(x)$ on $[a,b]$, then 
$$\int_a^b f(x)\ dx = F(b) - F(a)$$
\item{2)}  Antiderivatives graphically and numerically
\item{3)}  Basic antiderivative rules: All rules coming from
basic derivative formulas:  Know $\int x^n\ dx$,
$\int a^x\ dx$, $\int \sin(x)\ dx$, $\int \cos(x)\ dx$,
$\int {1\over x^2 + 1}\ dx$, $\int {1\over \sqrt{1 - x^2}}\ dx$,
and so forth, plus the
sum, and constant multiple rules 
\item{4)}  Solving differential equations ${dy\over dx} = f(x)$
and applications to equations of motion problems
\item{5)}  Integrals by substitution
\item{6)}  Integrals by parts: Selecting appropriate
$u$ and $dv$, computing $du$ and $v$, using the parts formula
$\int u\ dv = uv - \int v\ du$, then finishing the integral on 
the right.  Recall that this might involve using parts again,
or another method such as substitution.
\item{7)}  Integrals by the table:  Recognizing the appropriate
entry, using reduction formulas repeatedly if necessary.
Be aware that a preliminary substitution might be necessary
to take an integral to one of the forms in the table.
\item{8)} Integration of rational functions by partial 
fractions (see summary distributed in class).
\item{9)} Integration by trigonometric substitution (see 
summary distributed in class).
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\noindent
Note:  Some problems may ask you to carry out a particular
integration method on a problem.  Others may leave the choice
up to you.  {\it Be prepared for both types of questions!}
There will be a review for the exam in class on Tuesday, February 22.
If necessary, we can also continue that on Wednesday, February 23 (the
day of the exam).    
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\noindent
{\it Review Problems}
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The Review Problems at the ends of Chapter 6 and 7 are great
for preparation for this exam.  It's not necessary to work
out every integral in Chapter 7 (there are over 100 of them!)
But you should try a good selection and practice choosing 
a method at least for most of them.
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\noindent
{\it Sample Exam}
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\noindent
Note:  This is somewhat longer than the actual exam will be, to
give you an idea of the range of types of questions we might ask!
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\noindent
I.  The following graph shows $y = F'(x)$, the {\it derivative}
of $F$.  
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\item{A)}  Assuming $F(0) = 0$, Compute $F(1), F(2), F(3), F(4)$ given 
the information in the graph.
\item{B)}  Where are the critical points of $F$?  Explain.
\item{C)}  Sketch the graph $y = F(x)$ if $F(0) = 0$,
and also if $F(0) = 2$.
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\noindent
III.  A child throws a baseball vertically upward with velocity 12 ft/sec 
from an initial height of 4 feet.
The acceleration of the baseball due to gravity is -32 ft/sec${}^2$, so
the velocity function $v$ satisfies the differential equation 
$${dv\over dt} = -32$$
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\item{A)} Find formula for the velocity as a function of time.
\item{B)}  When does the baseball reach its highest point?  When
does it hit the ground?
\item{C)}  Find the greatest height the ball reaches.
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\noindent
IV. 
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\item{A)}  Compute $\int 5x^4 - 3\sqrt{x} + e^x + {2\over x}\ dx$
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\item{B)}  Apply a $u$-substitution to compute $\int x(4x^2 - 3)^{3/5}\ dx$
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\item{C)} Apply a $u$-substitution to compute $\int_1^2 e^{\sin(\pi x)}\cos(\pi x)\ dx$
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\item{D)}  Do you need partial fractions to 
compute $$\int {t^2 + 1\over t^3 + 3t + 3}\ dt?$$
Explain, and give a simpler method.
\item{E)} Apply integration by parts to compute $\int x^2 e^{-2x}\ dx$
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\item{F)}  Apply partial fraction decomposition to 
compute $$\int {1\over x(x - 1)(x + 2)}\ dx$$
\item{G)}  Which trigonometric substitution would you apply
to compute $\int {1\over u \sqrt{a^2 - u^2}}\ du$?
What trigonometric integral do you get after making the substitution?
Which entry in the table applies to this form?
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\noindent
V.  Compute each of the integrals below using some combination of
basic rules, substitution, integration by parts, the table of integrals, 
partial fractions, and trigonometric substition. You must show all work
for full credit.  
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\item{A)}  $$\int {x^3 \over x^2 + 4x}\ dx$$
\item{B)} $$\int (x^2 + 2x)^{3/2}\ dx$$
\item{C)}  $$\int {e^{\sqrt{\sin(x)}} \cos(x) \over \sqrt{\sin(x)}}\ dx$$
\item{D)}  $$\int {dz \over z^2\sqrt{z^2 + 9}}$$
\bye
