% Exams from M222 Fall 85 plain TeX

\newcount\probno
\def\prob{\par\smallskip\noindent\the\probno . \advance\probno by 1 }
\def\x{{\bf X}}
\def\0{{\bf 0}}
\def\c{{\bf C}}
\def\k{{\bf k}}
\def\ut{{\bf U_{\theta} }}
\def\ur{{\bf U_r }}
\def\i{{\bf i }}
\def\j{{\bf j }}
\def\a{{\bf A}}
\def\b{{\bf B}}
\def\o{{\bf 0}}


\probno=1
\noindent      Exam 1 \hfill A. Miller \hfill Fall 85  \hfill Math 222
\par\bigskip

\bigskip
\prob Prove that $\lim_{\theta\to 0}{\sin(\theta)\over\theta}=0$.
\medskip
\prob Find $\int^{\infty}_0{x \over (1+x^2)^2}dx$ or show it diverges.
\medskip
\prob Find $\int \arctan({1\over x})dx$.
\medskip
\prob Find $\int \sec^2(\theta)\sin^3(\theta) d\theta$.
\medskip
\prob Find $\int {\cos(x)\sin^3(x) \over \sqrt{1+\sin^2(x)}}dx$.
\medskip
\prob Find the area between the curves (given in polar coordinates)
$r=\sin(\theta)$ and $r=\cos(\theta)$ for $0\leq\theta\leq {\pi\over 4}$.
\medskip
\prob Find the length of the curve
$r=\sec(\theta)$ for $0\leq\theta\leq{\pi\over 4}$.

\bigskip
\probno=1
\noindent      Exam 2 \quad \quad A. Miller
\quad \quad Fall 85  \quad \quad Math 222
\bigskip

\prob Prove that $$1+b+b^2+\ldots +b^{n-1}={{1-b^{n}} \over {1-b}}.$$
\medskip
\prob Find $dy \over dx$ if $y=x^{ln(x)}.$
\medskip
\prob Find
 $$\lim_{n \to \infty} n^{\raise8pt\hbox{$\sqrt{1 \over n}$}}$$
\medskip
\prob Find $$\int {{2x^3 + 4x + 2}\over {x^4-1}} dx$$
\medskip
\prob Calculate the series:
$${{2\times3}\over{4}}+{{2\times3\times3}\over{4\times4}}
+{{2\times3\times3\times3}\over{4\times4\times4}}+\cdots$$
\medskip
\prob Find $$\int {{\sqrt x dx}\over{1 + \root 4 \of x}}$$
\medskip
\prob Define sequence.  Define what it means for a sequence to converge.
 Define what it means for a sequence to diverge.  What is the difference
 between a sequence and a series? ( Your answer to this question should be
 about four or five sentences.)


\probno=1

\bigskip
\noindent      Exam 3 \hfill A. Miller \hfill Fall 85  \hfill Math 222
\par\bigskip
\bigskip

\prob Are the following series absolutely convergent, conditionally
convergent, or divergent? Say why your answer is correct.
$$ \eqalign{
&a) \sum^{\infty}_{n=1} {{(-1)^n} \over {n(n+1)} } \cr
&b) \sum _{n=2}^{\infty} {{{(-1)^n} \over {n}}}
 ({{1} \over {log(n)}} )^{3 \over 2} \cr
 }  $$
\medskip
\prob Let $f(x) = f(0) + f^{(1)}(0)x + E(x)$ and $0 < x.$  Show that
there exists $\zeta$ with $0 < \zeta < x$ such that
$$ E(x)= { {{f^{(2)}(\zeta)} \over { 2! }} x^2  }. $$
\medskip
\prob Starting with the geometric series show how to obtain the Taylor series
of $f(x) = {\rm arctan}(x^3)$.  What is $f^{(47)}(0)$?
\medskip
\prob Find the intervals of convergence for the following series:
$$ \eqalign { &a)  \sum^{\infty}_{n=0} {{n} \over { 2^n3^{n+1}}}\, x^{2n} \cr
              &b)  \sum^{\infty}_{n=0} n^2 (x+1)^{n} \cr   }$$
\medskip
\prob Let  $\a  = i - 2j + k$ and $\b = -i + j. $ Find the following:
$$ \eqalign { &a) \a + 4 \b    \cr
              &b) \a \cdot  \b    \cr
              &c) ||\a|| \cr
              &d) {\rm the\,line\,containing}\,\a\,{\rm and}\, \b \cr
              &e) \a \times \b  \cr
              &f) {\rm the\,plane\,containing}\, \a,\,\b,\,{\rm and}\, \o \cr
              &g) {\rm the\,point\,on\,the\,line\,in\,d)\,
              closest\,to\,}\, \o \cr
} $$


\probno=1

\noindent      Final Exam \hfill A. Miller \hfill Fall 85  \hfill Math 222
\par\bigskip



\prob Find $$\int \sqrt{1+x^2} dx$$

\medskip

\prob Evaluate the limit
 $$ \lim_{x\mapsto\infty} { (\ln x)^2 \over\sqrt{x} }$$

\medskip

\prob Find $\underline{all}$ solutions of the differential equation
$$ {dy \over dx}=\sqrt{xy}$$

\medskip

\prob Use four terms of a Taylor series to approximate
$$\int^{1\over 2}_0 {dx\over 1+x^3}$$
Give an estimate for the error.  It is not neccessary for you to simplify
your answer by using a calculator.

\medskip

\prob What is the equation, in the form $ax+by+cz=d$,  of the plane
which contains the three points: $(1,0,1)$, $(-2,1,-3)$, and $(1,2,3)$?

\medskip

\prob Let $$f(x)=\sum^{\infty}_{n=0} {x^{n}\over (n!)^2}$$ What is the radius of
convergence?  What is the power series of the function $f^{(2)}(x)$?
What is the power series (up to a constant) of $\int f(x)\, dx $?
What is $f^{(48)}(0)$?

\medskip

\prob A 40 foot ladder is propped up against a 15 foot wall.  What
angle should the ladder make with the ground if the horizontal
 distance the ladder extends beyond the wall is to be a maximum?

\medskip


\prob Prove that if $\x \times {d\x \over dt}=\beta \k$ then
$r^2{d\theta \over dt}=\beta$, where $\beta$ is a scaler constant and
$\x= x\i + y\j$ and $ x=r\cos(\theta),\,y=r\sin(\theta) $.

\end