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\noindent{\bf Projects Math 112 \hfill Prelim 1 \hfill October
2, 2001 \hfill Fall 2001}

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\noindent{\bf Instructions:} \newline
\noindent 1) Write your name, your instructor's
name, and your section number on the front cover of the exam booklet.
Sign the academic integrity statement.
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\noindent 2) Show all your work on all questions and explain your
answers carefully.\newline
A correct answer without justification gets no credit.
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\noindent 3) Do not simplify your answers.
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\noindent 4) There are 6 problems worth a total of 100 points.  Do
them in whatever order you wish, but label each solution clearly.
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\noindent 5) No calculators are allowed on this exam.

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\noindent{\bf Problems:}

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\noindent{\bf 1.} (26 points) Find the following definite or
indefinite integrals:  
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a) $\displaystyle\int \cos x \; \sin^{2}\!x\; dx$
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b) $\displaystyle\int \displaystyle\frac{x \ dx}{(x-3)(x+5)}\; $
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c) $\displaystyle\int \cos\,\sqrt{x}\; \, dx$
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d) $\displaystyle\int_{-1}^{2} | x e^{x} | \, dx.$

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\noindent{\bf 2.} (20  points) The following is a sketch of a function
$f(t)$, for $0 \leq t \leq 3$.

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Let $\displaystyle g(x) = \int_{0}^{x} f(t)\, dt$, for $0\leq x< 3$.
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a) Find the interval(s) where $g(x)$ is increasing.   
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b) Find the interval(s) where $g(x)$ is decreasing.
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c) Find the $x$-value(s) where $g(x)$ has a global minimum.
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d) Find the $x$-value(s) where $g(x)$ has a global maximum.
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e) Find $g'(1)$.

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\phantom{abc } \hskip 85mm PLEASE TURN OVER
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\noindent{\bf 3.} (15 points) a) Estimate
$\displaystyle \int_{0}^{4}x^{2}\,
dx$ using the left and right endpoint approximations with $n = 4$.
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b) Compute $\displaystyle \int_{0}^{4}x^{2}\, dx$.
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c) Which choice of endpoints (left or right) in part a) gave a more
accurate approximation?  Explain why that same choice will give a more
accurate approximation for every value of $n$.  Give specific
reasons.   (Hint: Draw a picture.)

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\noindent{\bf 4.} (14 points) Determine by comparison whether the
following improper integrals are convergent or divergent.
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a) $\displaystyle \int_{1}^{\infty}\displaystyle
\frac{dx}{x^{2}+\sqrt{x}}$
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b) $\displaystyle \int_{0}^{\infty}xe^{\sqrt{x}}\, dx$.

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\noindent{\bf 5.} (12 points) Sketch the region bounded by the line
$x+y = 0$ and the curve $y = x - x^{2}$. Find the area of this region.


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\noindent{\bf 6.} (13 points) Set up (but do not evaluate) an integral
expressing the volume of the solid of revolution obtained by rotating the
area in problem 5 about the line $y= -3$.
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