
% Time-stamp: <11/21/98 at 16:51:31 by rosen on euler.math.nd.edu>
 % Dear Andrew,
 % Here an update on the draft for Test 3. I essentially did
 % incorporate your suggestions. Instead of making the test
 % easier I suggest that we agree on a set of difficulties we all 
 % go over in class. (For example questions 2 and 3 of the
 % practice test (which corresponds to 3,4 of the test), then we
 % compute in class the area of the cardioid which will cover
 % problem number 10.
 % What do you think, 
 % Joachim.
\documentclass[11pt]{letter}
\textwidth 7in\oddsidemargin -0.15in
\textheight 9in\topmargin -0.5in
\begin{document}

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%% Was Homework 9.5 no. 19
1. Find the length of the curve:
$$
(x(t),y(t))=\left(\frac{t^2}{2},\frac{1}{3}(2t+1)^\frac{3}{2}\right),\ 
0\leq t\leq 4.
$$

(A) 12 \hfill   
(B) 13 \hfill   
(C) 14 \hfill   
(D) 15 \hfill   
(E) 16

\vfill

2. For what values of $x$ does the series
$\displaystyle \sum_{n=0}^\infty (4x^2)^n$ converge absolutely.

(A) $|x|<\frac{1}{2}$ \hfill   
(B)  $|x|< 1$ \hfill   
(C) $|x|< 4$  \hfill   
(D) $|x|<2$ \hfill   
(E) $|x|<\frac{1}{4}$

\vfill

\newpage
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3. The degree 5 term of the Maclaurin series for
$\cos(x)\cos(x^2)$ is:\\ 
(Hint: Expand first the factors as a Maclaurin series).

(A) $0$ \hfill   
(B)  $-\frac{1}{720}$ \hfill   
(C) $\frac{1}{720}$  \hfill   
(D) $\frac{7}{720}$ \hfill   
(E) $-\frac{7}{720}$


\vfill

%% Like on Practice Test
4. Suppose we compute an approximation value for
$(1.2)^\frac{7}{2}$ by using the second order Taylor polynomial
for $f(x)=x^\frac{7}{2}$ at $a=1$. According to Taylor's theorem
the size of the error in the approximation is:

(A) $\frac{7}{400}\sqrt{c}$\ where \ $1\leq c \leq 1.2$ \hfill   
(B)  $\frac{21}{200}\sqrt{c}$\ where \ $1\leq c \leq 1.2$ \hfill   
(C)  $\frac{7}{400}\sqrt{c}$\ where \ $0\leq c \leq 0.2$ 
  
(D) $\frac{21}{200}\sqrt{c}$\ where \ $0\leq c \leq 0.2$  \hfill   
(E) $\frac{21}{200} c^{1.5}$\ where \ $0\leq c \leq 0.2$ \hfill \hspace{1cm}

\vfill

\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Similar to Homework 9.5 no. 23
5. Find the area of the surface generated by revolving the
circle\\
$(x(t),y(t))=(\cos(t),5+\sin(t))$, $0\leq t\leq 2\pi$, around the
$x$-axis.

(A) $20 \pi^2$ \hfill   
(B)  $5 \pi^2$ \hfill   
(C)  $10 \pi^2$ \hfill   
(D)  $ \pi^2$ \hfill   
(E)  $4 \pi^2$


\vfill

6. Compute the Eccentricity of the Hyperbola
$\frac{x^2}{16}-\frac{y^2}{9}=1$

(A)  $\frac{5}{4}$\hfill   
(B)   $5$ \hfill   
(C)   $\frac{\sqrt{7}}{4}$ \hfill   
(D)   $\frac{\sqrt{7}}{3}$\hfill   
(E)  $\frac{5}{3}$



\vfill

\newpage
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7. The polar equation $r=-8\cos(\theta)$, where $r\geq 0$ and
$0\leq \theta\leq 2\pi$, is the same as the cartesian equation:

(A) $x^2+8x+y^2=0$ \hfill   
(B)  $(x-4)^2+y^2=16$ \hfill   
(C) $(x+4)^2+y^2=8$  

(D) $(x-4)^2+y^2=8$ \hfill   
(E) $x^2-8x+y^2-8=0$\hfill \hspace{1cm}


\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Homework 8.11 number 23.

8. (12 pts) Find a series solution for the initial value problem:
$$
(1-x)y^\prime-y=0,\ \ y(0)=2.
$$


\newpage
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9. (A) (6 pts) Write down the Maclaurin series for
$f(x)=\cos\sqrt{x}$.

\vfill

(B) (6 pts) Find a series exression for the definite integral
$\displaystyle\int_0^1\cos\sqrt{x}\ dx.$

\vfill


\newpage
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10. (12 pts) Find the area of the region in the plane enclosed by
the cardioid $r=2(1+\cos(\theta))$.


\newpage
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11. (15 pts) Cosider the ellipse $\displaystyle
\frac{x^2}{25}+\frac{y^2}{9}=1$. Sketch in the following graph
the ellipse and indicate the focal points and the lines of
Directrix. On the bottom provide your computed results:

\begin{center}
  \setlength{\unitlength}{15mm}
\begin{picture}(10,8)(0,0)
 \put(0,4){\vector(1,0){10}}
 \put(5,0){\vector(0,1){8}}
 \put(5.2,7.8){\makebox(0,0){$y$}}
 \put(9.7,3.7){\makebox(0,0){$x$}}
 \multiput(0.8,3.9)(0.6,0){15}{\line(0,1){0.2}}
 \multiput(4.9,0.4)(0,0.6){13}{\line(1,0){0.2}}
\end{picture}
\end{center}

\vfill


(A) Focal Points:  

(B) Eccentricity:

(C) Lines of Directrix:



\end{document}
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Joachim Rosenthal              Voice:   (219)-631-8370   %
%  Department of Mathematics            or (219)-631-7245   %
%  University of Notre Dame        or home (219)-289-6293   %
%  Notre Dame, IN 46556-5683      Fax:     (219)-631-6579   %
%  http://www.nd.edu/~rosen/      e-mail: Rosenthal.1@nd.edu%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

