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    \noindent\the\probno . }
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\def\true{\par\noindent\smallskip\kern -.7in \parbox{.65in}{T or F}}
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\begin{document}


\def\header{{\hfill Exam 1\hfill A. Miller \hfill Spring 96
                                       \hfill Math 222\hfill}}
\markboth\header\header
\header
\putanswer \bigskip Exam 1

\prob Find the integral $\int{3x^2+3x+1\over 2x^3+2x^2+x}dx$.
\answer Partial fractions.
${1\over 4}\ln|2x^2+2x+1|+{1\over 2}\arctan(2x+1)+\ln|x|+C$

\prob Find the integral $\int {{x^2}\over (1+x^2)^{3\over 2}}dx$.
\answer Substitute $x=\tan(\theta)$.
$\ln|x+\sqrt{1+x^2}|-{x\over\sqrt{1+x^2}}dx + C$

\prob Find the integral $\int x^5\cos(x^3)dx$.
\answer Substitute $u=x^3$ then do by parts.
${1\over 3}\cos(x^3)+ {1\over 3}x^3\sin(x^3)+ C$

\prob Find the integral $\int {dx\over 1+\sin(x)}$.
\answer Multiply by ${1-\sin(x)}\over {1-\sin(x)}$ and simplify.
$\tan(x)-\sec(x)+C$

\prob Find the integral $\int\sqrt{x}\ln(x)dx$.
\answer Substitute $u^2=x$ then by parts.
${2\over 3}x^{3\over 2}\ln|x|-{4\over 9}x^{3\over 2}+C$

\prob Find the integral $\int\sin(2x)cos(x)dx$.
\answer $-{2\over 3}\cos^3(x)+C$

\prob Find the general solution of ${dy\over dx}={\sec^3(y)\over 4-x^2}$.
(You do not need to solve for $y$ explicitly.)
\answer $\sin(y)-{1\over 3}\sin^3(y)=
{1\over 4}(-\ln|2-x|+\ln|2+x|)+C$.

\prob Find the arclength of the curve $y=2 e^{x\over 2}$ for
$x=0$ to $x=1$.
\answer Substitute $u^2=1+e^x$ in the integral
$\int^1_0\sqrt{1+e^x}dx$.
$$(2{\sqrt{1+e}}+\ln({\sqrt{1+e}}-1)-\ln({\sqrt{1+e}}+1))
-(2{\sqrt{2}}+\ln({\sqrt{2}}-1)-\ln({\sqrt{2}}+1))$$


\prob Determine whether the improper integral
$\int^{\infty}_1 {x\over \sqrt{x^3+1}}dx$ is convergent
or divergent and explain why.
\answer It diverges because
$${x\over \sqrt{x^3+1}}\geq{x\over \sqrt{x^3+x^3}}={1\over\sqrt{2}}
{1\over x^{1\over 2}}$$ and
$$\int^{\infty}_1 {1\over x^{1\over 2}}dx=\lim_{b\to\infty}2b^{1/2}-2=\infty$$

\prob If the curve $y={x^3\over 3}$ for $x$ between $0$ and $1$
is rotated about the $x$-axis, what is the resulting surface area?
\answer $\int^1_0 2\pi {x^3\over 3}\sqrt{1+x^4}dx=
{\pi\over 9}(2^{3\over 2}-1)$.

Bonus Extra Credit 10 point Problem.  Work only if you have done
the problems 1-10 completely.   Show how to derive Simpson's (local)
rule: $$\int_x^{x+2h}f(t)dt\approx {h\over 3}(f(x)+4f(x+h)+f(x+2h))$$

\newpage
\probno=0

\def\header{{\hfill Exam 2\hfill A. Miller \hfill Spring 96
                                       \hfill Math 222\hfill}}
\markboth\header\header
\putanswer \bigskip Exam 2

\prob Find ${dy\over dx}$ and ${d^2y\over dx^2}$ as expressions
of $t$ for the parameterized curve
$$x=t^2+t$$ $$y=t^2+1$$
\answer ${dy\over dx}={2t\over 2t+1}$, ${d^2y\over dx^2}={2\over(2t+1)^3}$.

\prob Find the length of the curve
$$x=e^t-t$$
$$y=4e^{t/2}$$
$$0\leq t\leq 1$$
\answer $e$

\prob Find the area enclosed by one loop of the (polar coordinate)
curve $$r=\sin(5\theta)$$
\answer $\pi\over 20$.

\prob Find the value of $$\sum_{n=0}^\infty 4({2\over 3})^n$$
or show that it diverges and explain why.
\answer $12$

\prob Determine whether the series
$$\sum_{n=1}^\infty {{n^2+1}\over {n^4+1}}$$
converges or diverges.  Explain why.
\answer Converges, explain why.

\prob Determine whether the series
$$\sum_{n=2}^{\infty}(-1)^n {\ln(n)\over n}$$
converges absolutely, converges conditionally, or diverges.
Explain why.
\answer  Converges conditionally, explain why.

\prob Determine whether the series
$$\sum_{n=0}^{\infty}(-1)^n(1-{n\over {n+1}})^2$$
converges absolutely, converges conditionally, or diverges.
Explain why.
\answer Converges absolutely, explain why.

\prob Find the radius of convergence and interval of convergence
of the series:
$$\sum_{n=0}^{\infty}{3^n(x+1)^n\over {n+1}}$$
\answer $1\over 3$, $[-{4\over 3}, -{2\over 3})$

\prob Find the power series ($\sum_{n=0}^\infty a_nx^n$) representation for
the function
$$g(x)={1\over 1+9x^2}$$
and determine the interval of convergence.
\answer $1\over 3$, $(-{1\over 3}, {1\over 3})$

\prob Find the power series ($\sum_{n=0}^\infty a_nx^n$) representation for
the function
$$f(x)=x e^{-x}$$
What is $f^{(100)}(0)$?  (This is the 100$^{th}$ derivative at zero.)
\answer $\sum_{n=1}^\infty {{(-1)^{n-1}}\over {(n-1)!}}x^n$, $-100$.




%%%% start of final %%%%%
\newpage
\probno=0
\def\header{{\hfill Final\hfill A. Miller \hfill Spring 96
                                       \hfill Math 222\hfill}}
\markboth\header\header
\noindent  \header
\putanswer \newpage Final

\prob Find $\int \sec^3(x)dx$.  Show all details.
\answer ${1\over 2}(\sec(x)\tan(x)+\ln|\sec(x) + \tan(x)|)+C$

\prob Find $\int {x^3\over x^2-1}dx$.
\answer ${1\over 2}(x^2+\ln|x^2-1|)+C$

\prob Find the values of $p$ for which the series
$$\sum_{n=2}^\infty {1 \over  n(\ln(n))^p}$$
is convergent.
\answer $p>1$

\prob Find an equation $ax+by+cz=d$ for the plane consisting of all
points which are equidistant from the points $(1,-1,1)$ and $(0,1,3)$.
\answer $-2x+4y+4z=7$


\prob Find the volume of the parallelopiped determined by
the vectors $\i+\j$, $\i-\k$ and $\j+2\k$.
\answer 1

\prob Find the velocity, acceleration, and speed of a
particle with the position function:
$${\bf r}(t)=e^{t}\i+e^{-t}\j+t\k$$
\answer ${\bf v}=e^t\i-e^{-t}\j + \k$, ${\bf a}=e^t\i+e^{-t}\j$,
$\sqrt{e^{2t}+e^{-2t}+1}$

\prob
Which of the following are always true, and which are at least sometimes
false?  ($\{a_{n}\}$ denotes an arbitrary sequence of
real numbers. $\sum a_n$ denotes an arbitrary series.)

\bigskip\par\noindent Circle either T or F. \bigskip

\true If $a_{n} > 0$ for all $n$ and $a_{n} \rightarrow L$, then $L > 0$.
\true
If $a_{n} \geq 0$ for all $n$ and $a_{n} \rightarrow L$, then $L \geq 0$.
\true
If $\left\{a_{n}\right\}$ is bounded, then it converges.
\true
If $\left\{a_{n}\right\}$ is not bounded, then it diverges.
\true
If $\left\{a_{n}\right\}$ is decreasing, then it converges.
\true
If $\left\{a_{n}\right\}$ is decreasing and $a_{n} > 0 $ for all $n$,
then it converges.
\true
If $\left\{a_{n}\right\}$ is neither increasing nor decreasing,
then it diverges.
\true If $a_{n} > 0$ for all $n$ and $(-1)^{n} a_{n} \to L$, then $L=0$.
\true If $\sum a_{n}$ converges, then $a_{n} \rightarrow 0$.
\true If $a_{n} \rightarrow 0$, then $\sum a_{n}$ converges.
\true If $\sum a_{n}$ and $\sum b_{n}$ diverge, then $\sum (a_{n} + b_{n})$
  diverges.
\true If $\sum a_{n}$ converges and $\sum b_{n}$ diverges, then
  $\sum (a_{n} + b_{n})$ diverges.
\true If $\sum a_{n}$ and $\sum b_{n}$ converge, then $\sum(a_{n}b_{n}) =
  \left(\sum a_{n}\right)\left(\sum b_{n}\right)$.
\true If $\left\{a_{n}\right\}$ is monotonic and bounded, then
  $\sum a_{n}$ converges.
\true If the partial sums of $\sum a_{n}$ are bounded, then $\sum a_{n}$
  converges.
\true If $a_{n} \geq 0$ and the partial sums of $\sum a_{n}$ are bounded, then
  $\sum a_{n}$ converges.
\true $\sum a_n$ converges if and only if $\sum_{n=N}^\infty a_n$ does
  for some finite $N$.
\true $\sum_{k=0}^{\infty} x^{k} = \frac{1}{1-x}$ whenever $x \neq 1$.
\true If $f(x)=\sum_{n=0}^\infty a_nx^n$ has a nonzero radius of convergence,
  then $f$ is differentiable and $f^\prime(x)=\sum_{n=1}^\infty na_nx^n$.
\true If $\sum a_nx^n$ converges when $x=1$, then it must also
  converge when $x=-{1\over 2}$.
\true If $a_n>0$ and both limits
  $\lim_{n\to\infty}{a_{n+1}\over a_n}=L_1$ and
  $\lim_{n\to\infty}{^{n}\sqrt{a_n}}=L_2$ exist, then $L_1=L_2$.
\true If $f(x)$ is a continuous positive function on the interval
  $[1,\infty)$ and $$\lim_{x\to\infty}f(x)=0,$$
  then $\int_1^\infty f(x)dx$ converges if and only
  if $\sum_{n=1}^\infty f(n)$ converges.
\answer{ftftf tfttf ftfff ttfft tf}

\prob Circle all of the following vectors which are orthogonal
to the vector $<1,-1,2>$
\pick $<-1,1,-2>$
\pick $2\i+2\j$
\pick $\i-\j$
\pick $<0,2,1>$
\answer $2\i+2\j$  $<0,2,1>$

\prob Circle all the vector equations of a line below which contain
the two points $(0,1,1)$ and $(1,-1,1)$.
\pick ${\bf r}(t)=(\j+\k) + t(\i-\j+\k)$
\pick ${\bf r}(t)=(\i-\j+\k) + t(\j+\k)$
\pick ${\bf r}(t)=(\j+\k) + t(\i-2\j)$
\pick ${\bf r}(t)=(\i-\j+\k) + t(\i-2\j)$
\pick ${\bf r}(t)=(\i+2\k) + t(-2\i+4\j)$
\answer ${\bf r}(t)=(\j+\k) + t(\i-2\j)$
${\bf r}(t)=(\i-\j+\k) + t(\i-2\j)$

\prob For each of the following series  fill in the blank with
exactly one of the answers a,b,c,d or e which stand for:
\par\noindent a) divergent \hfill b) conditionally convergent
\hfill c) absolutely convergent
\bigskip
\par\noindent\medskip
$\Sigma {1\over n}$ is \blankline.
\hfill $\Sigma {(-1)^n\over n}$ is \blankline.
\par\noindent\medskip
$\Sigma {1\over n^2}$ is \blankline.
\hfill $\Sigma {(-1)^n\over n^2}$ is \blankline.
\par\noindent\medskip
$\Sigma{1 \over \sqrt{n}}$ is \blankline.
\hfill $\Sigma{(-1)^n \over \sqrt{n}}$ is \blankline.
\answer column 1: aca, column 2: bcb

In the remaining problems fill in the blank.

\prob
Suppose $f(x)=\sum_{n=0}^{\infty} {n\over {n+1}}x^n$.   Then
$f^{(13)}(0)$=\bl
\answer $5782233600$

\prob The distance between the points $(1,1,-2)$ and
$(0,2,0)$ is \bl
\answer $\sqrt{6}$

\prob If $r=3$ and ${\bf v}=\langle 1,-2,3\rangle$ then $r{\bf v}$ is \bl
\answer $\langle 3, -6, 9\rangle$

\prob $|\langle 1,2,-3\rangle|=$ \bl
\answer $\sqrt{14}$

\prob The angle between the planes
$x+2y+z=4$ and $x-2y+2z=5$ is \bl
\answer 82.2 degrees or 97.8 degrees





\printanswers
\end{document}