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\def\header{{\hfill All Problems \hfill A. Miller \hfill Spring 2001
                                       \hfill Math 217\hfill}}
\markboth\header\header
\newcount\probno
\newcount\examno

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    \noindent\the\examno .\the\probno . }

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\begin{document}

\exam

\prob{10} % from M171 example 3 page 143
A square sheet of tin $12$ inches on a side is used to make an open-top box
by cutting a small square of tin from each corner and bending up the sides. 
How large a square should be cut from each corner in order that the box
shall have as large a volume as possible?

Let $x$ be the lenght of the side of the small square.  Give
your answer in the form $x=\ldots$.
\answer The volume of the box is $V=x(12-2x)^2$ and is a maximum on
$[0,12]$ when $x=2$.


\prob{10} % 4.2-6
Solve the following differential equation:
$${dy\over dx}=3x^2y^2$$
Give your answer in the form $y=\ldots$
\answer $y={-1\over{x^3+C}}\;\;\;$  Remark: there is also the constant solution
$y\equiv 0$.

\prob{10} % 4.4-18
Evaluate the following indefinite integral:
$$\int\sin^3(3x)\cos(3x)dx$$
\answer $\int\sin^3(3x)\cos(3x)dx={\sin^4(3x)\over 12}+C$

\prob{10} % 4.6-6
Establish the formula
$$\sum_{k=1}^n(2k-1)=1+3+5+\cdots+(2n-1)= n^2$$
for every positive integer $n$, by
showing
\bigskip
\par (a) the formula is correct for $n=1$, and
\bigskip
\bigskip
\par (b) if true for $n$, the formula is also true for $n+1$.
\answer (a) $\sum_{k=1}^1(2k-1)=2\cdot 1 -1=1=1^2$
\par\bigskip (b) $\sum_{k=1}^{n+1}(2k-1)= \sum_{k=1}^{n}(2k-1)\;\;+(2(n+1)-1)=
n^2+(2(n+1)-1)=n^2+2n+1=(n+1)^2$

\prob{10} % 4.7-12
Find the area $A$ between the curve $y=9-x^2$ and the $x$-axis.
Give your answer in the form $A=\ldots$.
\answer $A=36$

\prob{10}  % 4.8
Prove the following part of the fundamental theorem of calculus:

\bigskip
Suppose $f(x)$ is continuous on the interval $[a,b]$,
$F(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$ with
the $F^\prime(x)=f(x)$ for all $x$ in $(a,b)$.  Let
$n$ be a large integer.

\bigskip
(a) Define $\triangle x=$

\bigskip
(b) Define $x_k=$

\bigskip
(c) Show there exists $c_k$ with $x_{k-1}\leq c_k\leq x_k$
for $k=1,2,\ldots,n$ and such that
$$F(b)-F(a)=\sum_{k=1}^n \; f(c_k)\triangle x$$
\answer (a)$\triangle x={b-a\over n}$
\par (b) $x_k=a+k\triangle x$
\par (c) Using the Mean Value Theorem we have that for every
$k$ there exists $c_k$ with $x_{k-1}<c_k<x_k$ and
$$F^\prime(c_k)={{F(x_{k})-F(x_{k-1})}\over{x_{k}-x_{k-1}}}$$
Noting that $F^\prime=f$ and $\triangle x =x_{k}-x_{k-1}$
we have that
$$f(c_k)=F^\prime(c_k)={{F(x_{k})-F(x_{k-1})}\over{x_{k}-x_{k-1}}}
={{F(x_{k})-F(x_{k-1})}\over{ \triangle x}}  $$
and hence
$$f(c_k)\triangle x= F(x_{k})-F(x_{k-1})$$
Hence,
$$\sum_{k=1}^n f(c_k)\triangle x= \sum_{k=1}^n F(x_{k})-F(x_{k-1})$$
$$=(F(x_{1})-F(x_{0}))+ (F(x_{2})-F(x_{1}))+ (F(x_{3})-F(x_{2}))+\cdots+
(F(x_{n})-F(x_{n-1}))$$
This is a telescoping sum in which everything cancels except
$-F(x_{0})$ and $F(x_{n})$
$$=F(x_{n})-F(x_{0})=F(b)-F(a)$$
Noting that
$x_0=a$ and $x_n=b$ finishes the proof.
$$\sum_{k=1}^n f(c_k)\triangle x= F(b)-F(a)$$

\exam

\prob{10} % 5.2-10
The area bounded by the curves $y=x^3$, $y=0$, and $x=c>0$
has area $1$. What is $c$?
\answer $c=\sqrt{2}$

\prob{10} % 5.4-14
The base of a certain solid is the region bounded by
the curves $y=x^2$ and $y=1$.  Each plane section of the
solid cut out by a plane perpendicular to the $x$-axis is
a square with one edge of the square in the base of the
solid.   Find the volume of the solid.
\answer $16\over 15$


\prob{10} % 5.5-18
The area bounded by the curves $y=\sqrt{x}$ and $y=x$
is rotated about the $y$-axis.  Find the volume.
\answer $2\pi\over 15$

\prob{10} % 5.7-3
Find the lenght of the curve $x^2=4y^3$ from $(0,0)$ to $(2,1)$.
\answer ${2\over 27}(10^{3/2}-1)$


\prob{10} % 5.8-2
Find the surface area of the surface generated by rotating about the
line $y=1$ the arc of the curve $y={{x^3}\over 3}+1$
between $x=0$ and $x=1$.
\answer ${\pi\over 9}(2^{3/2}-1)$


\prob{10}  % 4.8
Prove the following part of the fundamental theorem of calculus:
\par \bigskip
Suppose $f(x)$ is continuous on the interval $[a,b]$,
$F(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$ with
the $F^\prime(x)=f(x)$ for all $x$ in $(a,b)$.  Let
$n$ be a large integer.
\par Define $\triangle x={{b-a}\over n}$.
\par Define $x_k=a+k\triangle x$ for $k=0,1,\ldots,n$.
\par\bigskip
  Show there exists $c_k$ with $x_{k-1}\leq c_k\leq x_k$
for $k=1,2,\ldots,n$ and such that
$$F(b)-F(a)=\sum_{k=1}^n \; f(c_k)\triangle x$$
\answer See Exam 1 problem 6.

\exam

\def\deg{\circ}

\prob{10} % 7.2-7
Find the value of
$$\cos(12^\deg)\cos(-18^\deg)\;+\; \sin(18^\deg)\sin(-12^\deg)$$
and simplify it.
\answer $\cos(30^\deg)={\sqrt{3}\over 2}$

\prob{10} % 7.3-25
Find $\sin^2({\theta\over 2})$ given that $\cos(\theta)={{4\over 5}}$.
\answer $1\over 10$

\prob{10} % 7.5-27
Find  all $\theta$ which solve the equation:
$$2\sin(\theta)\cos(\theta)\;+\;2\sin(\theta)\;+\;\cos(\theta)\;+\;1=0$$
\answer \par
\par $\theta=\pi + 2\pi n$ or
\par $\theta=-{\pi\over 6}+2\pi n$ or
\par $\theta=-{5\pi\over 6}+2\pi n$
\par for $n=0,1,-1,2,-2,\ldots$.

\prob{10} % 6.1
Suppose $z=\tan(\theta)\sec(\theta)$.
\par\medskip Find $dz\over d\theta$.
\par\medskip Simplify and express in terms of $\sec(\theta)$ only.
\par\medskip \answer $2\sec^3(\theta)-\sec(\theta)$

\prob{10} % 6.3
Find the value of
$$\int_0^2 {dx\over{1+{x^2\over 4}}}$$
and simplify.
\answer $\pi\over 2$


\prob{10}
Prove Pythagorus's Theorem.  In a right triangle with
hypotenus $c$ and sides $a$ and $b$, we have that

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\answer After removing the triangles the remaining areas must be equal.
(I would have shaded the triangles but my computer program wasn't up
to it.)
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\exam

\prob{10} % Exam 1 problem 4
Establish the formula
$$\sum_{k=1}^n 2k=2+4+\cdots+2n= n + n^2$$
for every positive integer $n$, by
showing
\bigskip
\par (a) the formula is correct for $n=1$, and
\bigskip
\bigskip
\par (b) if true for $n$, the formula is also true for $n+1$.

\answer See Exam 1 problem 4.
\par\bigskip (a) $\sum_{k=1}^1 2k=2=1^2+1$
\par\bigskip (b) $\sum_{k=1}^{n+1}2k= \sum_{k=1}^{n}2k\;\;+2(n+1)=
n^2+n+2(n+1)=n^2+3n+2$
\par and $(n+1)^2 +(n+1)=n^2+2n+1+n+1=n^2+3n+2$  and so
$$\sum_{k=1}^{n+1}2k= (n+1)^2 +(n+1)$$

\prob{10} % Exam 3 problem 3
Find  all $\theta$ which solve the equation:
$$2\sin(\theta)\cos(\theta)\;-\;2\sin(\theta)\;-\;\cos(\theta)\;+\;1=0$$
\answer See Exam 3 problem 3.  $(2\sin\theta -1)(\cos\theta -1)=0$
\par $\theta=2\pi n$ or
\par $\theta={\pi\over 6}+2\pi n$ or
\par $\theta={5\pi\over 6}+2\pi n$
\par for $n=0,1,-1,2,-2,\ldots$

\prob{10} % P 4.3 27
Find $x$ if $\log_x(3x)={2\over 3}$
\answer $x={1\over 27}$

\prob{10} % P 4.5  42
Solve the logarithmic equation for $x$.
$$\ln(x)+\ln ({{x+2}})=\ln(3)$$
\answer  $x=1$


\prob{10} % C 6.9  14
Find $\int \; t \;2^{t^2}\;dt$.
\answer ${1\over{2\ln(2)}}2^{t^2}+C$

\prob{10} % C 6.10  9
Find the derivative of $y=10^{\log_2(x)}$
\answer ${\log_2(10)\over x}10^{\log_2(x)}$ or
${\ln 10\over \ln 2}{1\over x}e^{{\ln 10\over \ln 2}(\ln x) } $


\exam

\prob{10} % P 9-2
Sketch the ellipse with equation:
 $$ {x^2\over 25}+{y^2\over 9}=1 $$
Label all four  points where the ellipse intersects the axis.
\par\noindent Find, draw, and label the focal points.

\answer The focal points are at $(\pm 4,0)$.  The
ellipse intersects the axises at $(\pm 5,0)$ and $(0,\pm 3)$.

\prob{10} % C 5-10
Find the center of mass of a thin homogeneous plate covering
the first quadrant of the disk bounded by the circle $x^2+y^2=a^2$.

\answer $({4\over 3\pi}a, {4\over 3\pi}a)$

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\prob{10} % P4-2
The amount of \$4000 is invested at the interest rate
of 6\% compounded monthly.  How much will you have after
3 years and 4 months?

\answer $4000(1+{.06\over 12})^{40}=4000(1.005)^{40}=4883.1769$ dollars

\prob{10} % P4-6
A roasted Turkey is taken from an oven when its temperature has reached 170
degrees and placed on a table in a room where the temperature is 70 degrees.
 If the temperature of the Turkey is 150 degrees after 10 minutes, when will
its temperature be 140 degrees?

\answer $10{\ln(.7)\over\ln(.8)}=15.984103$ minutes


\prob{10} % Exam 3 problem 6
Prove Pythagorus's Theorem:  In a right triangle with
hypotenus $c$ and sides $a$ and $b$, we have that

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\answer See Exam 3 problem 6.

\prob{10}  % 4.8
(A) (Do either problem 6A or problem 6B.  {\bf Do not do both.})

Prove the following part of the fundamental theorem of calculus:
\par \bigskip
Suppose $f(x)$ is continuous on the interval $[a,b]$,
$F(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$ with
the $F^\prime(x)=f(x)$ for all $x$ in $(a,b)$.  Let
$n$ be a large integer.
\par Define $\triangle x={{b-a}\over n}$.
\par Define $x_k=a+k\triangle x$ for $k=0,1,\ldots,n$.
\par\bigskip
  Show there exists $c_k$ with $x_{k-1}\leq c_k\leq x_k$
for $k=1,2,\ldots,n$ and such that
$$F(b)-F(a)=\sum_{k=1}^n \; f(c_k)\triangle x$$

\bigskip
\noindent
5.6. (B) (Do either problem 6A or problem 6B.  {\bf Do not do both.})

Prove the following version of the fundamental theorem of
calculus:

\bigskip
Suppose $f(x)$ is a continuous function on the interval
$[a,b]$ and $f(x)\geq 0$ for all $x$ in $[a,b]$.  For
any $x\in [a,b]$ let $A(x)$ be the area under $f$ and above the
interval $[a,x]$ on the $x$-axis.

Prove
$${dA\over dx}=f(x)$$

\answer See Exam 1 problem 6 for 6A.  For 6B see pages 196-198
of your calculus book.



\printanswers
\end{document}