%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Answers to problem in the Koblitz Calc1  Notes
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter*{Answers}\
\setcounter{section}{0}

\section{Answers to Homework Problems}

\begin{small}
%\newlabel{homework:1}{{1}{5}}
\subsection{Answers to Selected Problems in Chapter 1}

%\begin{answers}{homework 1}

\noindent
1:
$m$, $b$ and distance are: (a) 1.5, $-3$, $\sqrt{13}$; (b) $-3$, 2,
$\sqrt{10}$; (c) 1, 0, $2\sqrt{2}$; (d) 0, 3, 6; (e) 1, 1, $5\sqrt{2}$; (f)
$-3$, $0.02$, $0.02\sqrt{10}$.~~~
%
2:
$y$- and $x$-intercepts are: (a) 2, $-1$; (b) 6, 6; (c) $1/2$, $-1$;
(d) $3/2$, none; (e) $-2$, $-3.$~~~
%
3:
(a) $x^2+y^2=9$, (b) $(x-5)^2+(y-6)^2=9$, (c) $(x+5)^2+(y+6)^2=9$, (d)
$x^2+(y-3)^2=9$ (or $x^2+y^2-6y=0$), (e) $x^2+y^2+6y=0$, (f)
$x^2-6x+y^2=0.$~~~
%
4:
radius and center are: (a) 5, $(0,-5)$; (b) 7, $(5,0)$; (c) 5, $(3,4)$.~~~

\noindent
5:
$y=1.8x+32$. ~~~
%
6:
$y=0.15x+10$.  ~~~
%
7:
$y=0.08x-3.2$.  ~~~
%
8(b); $y=3x+120$.~~~
%
9: (a) $y=0$ for $0\le x\le 100$, $y=0.1x-10$ for $100\le x\le 1000$,
$y=x-910$ for $x\ge 1000$; (b) he's introducing a ``negative income tax''
as a rather cheap way of getting good PR --- so that, for example, a
penniless person receives 10 gold coins from the King (since the
$y$-intercept is $-10$).~~~
%
10:(a) $P=-0.0001x+2$; (b) $x=-10000P+20000$.~~~
%
11:
$y=0.15x$ for $0\le x\le 19450$, $y=0.28x-2528.5$ for
$19450\le x\le 47050$, $y=0.33x-4881$ for $47050\le x\le 97620$.~~~

%\end{answers}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newlabel{homework:2}{{2}{5}}

%\begin{answers}{homework:2}

\subsection{Answers to Selected Problems in Chapter 2}
\noindent
1:
$[1.5,\infty)$.~~~
%
2:
$x\ne-1$.~~~
%
3:
$x\ne\pm 1$.~~~
%
4:
$(-\infty,0)$.~~~
%
5:
all $x$.~~~
%
6:
$[0,\infty)$.~~~
%
7:
$[h-r,h+r]$.
%
8:
$x$ negative or in $[1,\infty)$.~~~
%
9:
$\bigl(-\frac{1}{ 3},\frac{1}{ 3}\bigr)$.~~~
%
10:
$x$ nonnegative but $x\ne 1$.~~~
%
11:
$x$ nonnegative but $x\ne 1$.

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newlabel{homework:3}{{3}{5}}

%\begin{answers}{homework:3}

\subsection{Answers to Selected Problems in Chapter 3}
\noindent
1:
(a) $-5$, (b) $-2.47106$, (c) $-2.40679$, (d) $-2.40068$, (e) $-2.4$.
~~~
%
2:
(a) $-4/3$, (b) $-24/7$, (c) $7/24$, (d) $3/4$.~~~
%
%
3:
(a) 5, (b) 4.1, (c) 4.01, (d) 4.001; limit=4.~~~
%
4:
(a) $-10.29$, (b) $-9.849$, (c) $-9.8049$; $-9.8-4.9\Delta t$; $-9.8$
m/sec.~~~
%
5:
(a) $-0.107527$, (b) $-0.110742$, (c) $-0.111074$; $-1/\bigl(3(3+\Delta x)
\bigr)$; $f'(3)=-1/9$.~~~
%
6:
$(2+\Delta x)+1/(1+\Delta x)$; $f'(1)=3$.~~~
%
7:
(a) 3.31, (b) 3.003001, (c) 3.0000300001; $3+3\Delta x+(\Delta x)^2$;
$f'(1)=3$.~~~

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newlabel{homework:4}{{4}{5}}

%\begin{answers}{homework:4}
\subsection{Answers to Selected Problems in Chapter 4}
{\em (first two problems)}

\noindent
1:
$f'(x)=-x/\sqrt{169-x^2}$.~~~
%
2:
(a) $f'(t)=-9.8t$, (b) $f'(x)=2x+(1/x^2)$, (c) $f'(x)=2ax+b$, (d)
$f'(x)=3x^2$.

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newlabel{homework:5}{{5}{5}}

%\begin{answers}{homework:5}
\subsection{Answers to Selected Problems in Chapter 5}

\noindent
1:(a) 4.4, (b) 0.9, (c) 1.01, (d) 0.0099, (e) 2.0025, (f) 5.1,
(g) 4.9, (h) 2.001.  ~~~
%
2:
3.94 ~~~
%
3:
0.9433.~~~
%
4:
15 ft 2 in. ~~~ 
%
5:
208.9 ft. ~~~ 
%
6:
$25\pi\pm 4\pi(\Delta x)$. ~~~
%
7:
$\pm 30$, $\pm1\%$ and $\pm3\%$. ~~

\noindent
%
8:
(a) 5 sec, (b) $\pm 0.025$ sec, $\pm 0.5\%$, (c) $\pm\frac{1}{ 2}p\%$.
~~~
%
9:
(a) $r=C\,A^{1/2}$, where $C=1/\sqrt{\pi}$, (b) $\pm 0.5\%$,
(c) $\pm\frac{1}{ 2}p\%$.  ~~~
%
10:
$8\pm 0.15$.  ~~~
%
11:
$\pm 3p\%$.~~~  
%
12:
(a)$\pm\frac{1}{ 2}p\%$, (b) $\pm\frac{1}{ 3}p\%$. ~~~
%
13:
(a) $\pm p\%$, (b) $\pm 3p\%$, (c) $\pm 2p\%$. ~~
%
14:
$\pm\frac{2}{ 3}p\%$. ~~~
%
15:
(a) $\pm 2p\%$, (b) $\pm p\%$, (c) $\pm p\%$.
%
16:
$1\>83/84$.
%
17:
$\Delta x=-1/12$ cm.

%\end{answers}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%
%% answers to homework on rules for finding derivatives
%\newlabel{homework:6}{{6}{5}}

%\begin{answers}{homework:6}
\subsection{Answers to Selected Problems in Chapter 6}

\noindent
1:
(a) $4x^3-9x^2+x+7$,~~~ 
(b) $3x^2-4x+2/\sqrt{x}$, ~~~
(c)-(d) $6x(x^4+2x^2+1)$,~~~ 

\noindent
(e) $\sqrt{169-x^2}-x^2/\sqrt{169-x^2}$, ~~~
(f) $(2x-4)\sqrt{25-x^2}-x(x^2-4x+5)/\sqrt{25-x^2}$, ~~~

\noindent
(g) $(-x^2+2x+2)/(1-x)^2$, 
(h) $-25/\left(x^2\sqrt{25-x^2}\right)$, ~~~
(i)-(j) $-\sqrt{x}/\sqrt{169-x^2}-\frac{1}{ 2}\sqrt{169-x^2}/x^{3/2}$, ~~~

\noindent
(k) $\frac{1}{ 2}\left(3x^2-2x+(1/x^2)\right)/\sqrt{x^3-x^2-(1/x)}$, ~~~
(l) $300x/(100-x^2)^{5/2}$, ~~~

\noindent
(m) $(\frac{1}{ 3}+x^2)(x+x^3)^{-2/3}$.~~~
(n) $\left((x^2+1)^2+\sqrt{1+(x^2+1)^2}\right)^{-1/2}x(x^2+1)\Bigl(
2+\left(1+(x^2+1)^2\right)^{-1/2}\Bigr)$.


\noindent
2:
(a) $-9.8t+20$, $-9.8$; 
(b) $a$, 0; 
(c) $\frac{1}{ 2}a(at+b)^{-1/2}$,$-\frac{1}{ 4}a^2(at+b)^{-3/2}$; 
(d) $\frac{1}{ 2}(2at+b)(at^2+bt+c)^{-1/2}$,
$a(at^2+bt+c)^{-1/2}-\frac{1}{ 4}(2at+b)^2(at^2+bt+c)^{-3/2}$, which
simplifies to $\frac{1}{ 4}(4ac-b^2)(at^2+bt+c)^{-3/2}$.

\noindent
3:
$v(t)=-5+2t$, $a(t)=2$; the bicycle is moving leftward initially
at 5 m/sec, but its speed is decreasing; after 2.5 sec it has slowed to
0 m/sec, at which point it reverses direction and starts moving to the right.

\noindent
4:
$2+\frac{n}{ 4}\Delta x$. ~~~
%
5:
$\pm0.96\Delta x$. ~~~
%
6:
$\frac{500}{ 3}\pi\pm 40\pi(\Delta x)$.~~~
%
7:
(a) $f(t)=\sqrt{100+(78-4.9t^2)^2}$;
(b) $-9.8t(78-4.9t^2)/\sqrt{100+(78-4.9t^2)^2}$.


%\end{answers}


%\newlabel{homework:7}{{7}{5}}

%\begin{answers}{homework:7}
\subsection{Answers to Selected Problems in Chapter 7}

\noindent
1:
(a) 1.992, (b) 5.0129, (c) 4.00133. ~~~
%
2:
(a) $y'=x/y$, (b) $y'=-(2x+y)/(x+2y)$,
(c) $y'=(3x^2+y^2-2xy)/(3y^2+x^2-2xy)$. ~~~
%
3:
$y'=1$. ~~~
%
4:
$y=-\frac{1}{ 13}x+\frac{14}{ 13}$, 0.9923. ~~~
%
5:
$B$ is $4$ times as strong as $A$. ~~~
%  
6:
$y'=3/19$. ~~~  
%
7:
(a) $dT/dh=-38.64^\circ$F/hour, (b) $h=-0.0259 T+5.493$. ~~~
%
8:
(a) $\sqrt{100+(10+0.3t)^2}+\sqrt{(x-10)^2+(10-0.3t)^2}=$const, 
(b) 12.492 sec. 

%\end{answers}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newlabel{homework:8}{{8}{5}}


%\begin{answers}{homework:8}
\subsection{Answers to Selected Problems in Chapter 8}

\noindent
1:
(a) $\pi/12$, (b) $2\pi/3$, (c) $\pi/2$, (d) $20\pi$.  ~~~
%
2:
(a) 25 rad/sec (b) 6.944 rad/sec. ~~~ 
%
3:
(a) $30\sin\theta$, (b) $\tan\theta$, (c) $60(1-\cos\theta)$. ~~~
%
4:
$\sin(\alpha+\beta)=\overline{BC}=\overline{DF}=\overline{DE}+
\overline{EF}=\overline{AE}\sin\alpha+\overline{BE}\cos\alpha=
\cos\beta\sin\alpha+\sin\beta\cos\alpha$. ~~~
%
6:
warmest: 62${}^\circ$ on July 12; coolest: $-12{}^\circ$ on January 10.
~~~
%
7:
$F=50\sin\left(\frac{\pi}{ 2}t\right)+150$, 
$R=2500\sin\left(\frac{\pi}{ 2}(t-3)\right)+7500$. ~~~
%
8:
(a) $20\sin\left(\frac{2\pi}{ 360}(t-109)\right)+40$;
(b) $2\sin\left(\frac{2\pi}{ 360}(t-81)\right)+7$;
(c) $5\sin\left(\frac{2\pi}{ 24}(t-11)\right)+20$;
(d) $10\sin(5\pi t)$; 
(e) $0.4\sin\left(25(t-\frac{\pi}{ 50})\right)+0.4$ (or, equivalently, 
  $0.4(1-\cos(25t))$).

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newlabel{homework:9}{{9}{5}}


%\begin{answers}{homework:9}
\subsection{Answers to Selected Problems in Chapter 9}

\noindent
1:
0.96509 (exact value=0.96569).  ~~~
%
2:
(a) $-\text{csc}^2x$, (b) $-\text{cot}\,x\,\text{csc}\,x$. ~~~
%
3:
(a) 0.46977, (b) 0.484885, (c) 0.93018. ~~~
%
4:
(a) $\frac{\pi}{ 9}\cos\left(\frac{2\pi}{ 360}(t-109)\right)$;
(b) $\frac{\pi}{ 90}\cos\left(\frac{2\pi}{ 360}(t-81)\right)$; 
(c) $\frac{5\pi}{ 12}\cos\left(\frac{2\pi}{ 24}(t-11)\right)$;
(d) $50\pi\cos(5\pi t)$; (e) $10\sin(25t)$. ~~~
%

\noindent
5:
(a) $\cos^2x-\sin^2x$, 
(b) $-\sin(x)\cos(\cos x)$,
(c) $\left(x\sec^2x+\tan x\right)/\left(2\sqrt{x\tan x}\right)$,

\noindent
(d) $\Bigl((1+\sin x)\sec^2x-\sin x\Bigr)/(1+\sin x)^2$. ~~~
%
6:
0.7632. ~~~
%
7:
1.241. ~~~
%
8:
(b) $0.5=2\sin(\pi t)\sin(\pi x/3)$; 
(c) $x=0.5$; 
(d) $dx/dt=-3$;
(e) $x=0.5+(0.2-\frac{1}{ 6})(-3)=0.4$.

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newlabel{homework:10}{{10}{5}}


%\begin{answers}{homework:10}
\subsection{Answers to Selected Problems in Chapter 10}

\noindent
1:
(a) $x=2+3t$, $y=3-2t$; 
(b) $x=1+6t$, $y=2+2.5t$; 
(c) $x=5\sqrt{3}\,t$, $y=5t$.  ~~~
%
2:
(a) 25 rad/sec; 
(b) $\frac{25}{ 2\pi}$ rev/sec; 
(c) $4\pi$ m/sec; 
(d) 21.6
km/hr. ~~~
%
3:
(a) 34.3 mph; 
(b) 55 rad/sec, 8.75 rev/sec. ~~~
%
4:
(a) $20t$; 
(b) $15\pi\,t$; 
(c) $7.5t^2$. ~~~
%
5:
(a) $x=3\cos 0.5t$, $y=3\sin 0.5t$; 
(b) $x=3\sin 4\pi t$, $y=3\cos 4\pi t$;
(c) $x=-3\sin 0.4\pi t$, $y=-3\cos 0.4\pi t$;
(d) $x=3\sin 20\pi t$, $y=-3\cos 20\pi t$. ~~~
%
6:
(a) $x=10t$, $y=-4.9t^2+24.5t+1$; 
(b) 2.5 sec; 
(c) 5.04 sec. ~~~
%
7:
(a) $x=49.5t$, $y=-16t^2-49.5t+163$; 
(b) $(-32t-49.5)/49.5$;
(c) $52.9^\circ$, $58.7^\circ$; 
(d) 124 ft/sec at $66.4^\circ$ below horizontal (hits ground at $t=2$). ~~~
%
8:
(a) $x=(v\cos\alpha)t$, $y=\frac{1}{ 2}gt^2+(v\sin\alpha)t$;
(b) $2v\sin\alpha/(-g)$; 
(c) $2v^2\sin\alpha\>\cos\alpha/(-g)$. 

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newlabel{homework:11}{{11}{5}}

%\begin{answers}{homework:11}
\subsection{Answers to Selected Problems in Chapter 11}

\noindent
1:
(b) $\sin\theta/(1-\cos\theta)$. ~~~
%
2:
(a) $x=2t$; 
(b) $y=3\sin(4\pi(t-\frac{3}{ 8}))$; 
(c) $y=3\sin(2\pi (x-\frac{3}{ 4}))$. ~~~
%
3:
$y=3\sin(4\pi(\sqrt{x}-\frac{3}{ 8}))$. ~~~
%
4:
$x=2\sin(\pi t)$, $y=100-16t^2+2\cos(\pi t)$. ~~~
%
5:
$x=6t-0.5\sin(8t)$, $y=0.75-0.5\cos(8t)$.
%
6:
$x=10t+0.25\sin(40t)$, $y=0.25-4.9t^2+0.25\cos(40t)$.

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newlabel{homework:12}{{12}{5}}

%\begin{answers}{homework:12}
\subsection{Answers to Selected Problems in Chapter 12}

\noindent
1:
0.0199 cm/sec.~~~
%
2:
0.25 m/sec.~~~
%
3:
$80\pi$ mi/min (=4.2 mi/sec).~~~
%
4:
6.71 ft/sec.~~~
%
5:
2.12 cm/sec. ~~~
%
6:
0.65 ft/sec~~~
%
7:
7.91 m/sec~~~
%
8:
1.172 m/min~~~
%
9:
0.5454 m/min~~~
%
10:
6 ft/sec, 2.5 ft/sec~~~
%
%
11:
69.4 mph~~~
%
12:
0.1234 ft${}^3$/sec.~~~
%
13:
(a) $(a\dot{a}+b\dot{b}+0.5 a\dot{b}+0.5b\dot{a})/\sqrt{a^2+b^2+ab}$; 
(b) $(a\dot{a}+b\dot{b}- a\dot{b}\cos\theta-
   b\dot{a}\cos\theta+ab\dot{\theta}\sin\theta)/\sqrt{a^2+b^2-2ab\cos\theta}$.
~~~
%
14:
81.6 m/sec~~~
%
15:
(a) $y=a\cos(\theta)-a\sin(\theta)\cot(\beta+\theta)$ (or else, using
the law of sines, an equivalent answer is: $y=a\sin\beta/\sin(\theta+\beta)$);
(b) $dy/dt=\Bigl(-a\sin(\theta)(1-\text{csc}^2(\beta+\theta))-a\cos(\theta)\cot
(\beta+\theta)\Bigr)d\theta/dt$ (or else: $dy/dt=-a\Bigl(\sin\beta\cos(\theta
+\beta)/\sin^2(\theta+\beta)\Bigr)d\theta/dt$); 
(c) 3.79 cm/sec.~~~
%
16:
(a) $a(d\theta/dt)$; 
(b) closing at a little over 15 rad/sec;
(c) no violation of relativity, because no part of the scissors is moving
faster than 300 km/sec.~~~
%
17:
(a) -50 m/sec; (b) 68 m/sec; (c)... ~~~
~~~
%
18:
(a) $(a\dot{a}+b\dot{b})/\sqrt{a^2+b^2+h^2}$;
(b) $(a\dot{a}+b\dot{b}+h\dot{h})/\sqrt{a^2+b^2+h^2}$.

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newlabel{homework:13}{{13}{5}}


%\begin{answers}{homework:13}
\subsection{Answers to Selected Problems in Chapter 13}

\noindent
1:
Both $f'(x)=0$ and $f''(x)=0$ when $x=0$.  If $n$ is even, the origin is a
minimum, and the curve is always concave up.  If $n$ is odd, the origin is
a point of inflection, and the curve is concave up for $x>0$ and concave
down for $x<0$.
~~~

%
\noindent
2:
(a) min at $(0.5,-0.25)$, always concave up, no inflection;

(b) max at (1,4), min at $(-1,0)$, inflection at $(0,2)$, conc up for
$x<0$, conc down for $x>0$;~~~~

(c) max at (2,20), min at (4,16), inflection at (3,18), conc up for $x>3$,
conc down for $x<3$;~~~

(d) max at (0,3), min at $(\pm 1,2)$, inflections at $(\pm 0.577,2.444)$,
conc down for $-0.577<x<0.577$, otherwise conc up, symmetric with respect
to $y$-axis;

(e) min at $(1,-1)$, inflections at (0,0) and $(2/3,-0.5926)$, conc down
for $0<x<2/3$, otherwise conc up;

(f) no max, min or inflections, $y$-axis is asymptote, conc up for $x<0$,
conc down for $x>0$, the line $y=x$ is a slanted asymptote;
 
(g) no max or min, inflections at $(\pm 1,2)$, $y$-axis is asymptote, conc
up for $x<-1$ and $x>1$, otherwise conc down, symmetric with respect to
$y$-axis;

(h) min when $x=7\pi/12$ (and $x=7\pi/12\pm$any multiple of $\pi$), max
when $x=11\pi/12$ (again can add or subtract any multiple of $\pi$),
inflections when $x=\pi/4$, $3\pi/4$ ($\pm$ any multiple of $\pi$), conc up
for $\pi/4<x<3\pi/4$, general appearance is oscillation around the slanted
line $y=-x$;

(i) max at all points $(\frac{\pi}{ 4}\pm 2k\pi,\sqrt{2})$ (where $k$ is any
integer), min at $(\frac{5\pi}{ 4}\pm 2k\pi,-\sqrt{2})$, inflections at
$(\frac{3\pi}{ 4}\pm k\pi,0)$, you get a sinusoidal function;

(j) asymptotes at odd multiples of $\pi$, max when $x=-\frac{\pi}{ 2}\pm
2k\pi$, min when $x=\frac{\pi}{ 2}\pm 2k\pi$, inflections at even multiples
of $\pi$, in each section between asymptotes the left half is conc down,
the right half is conc up.

\noindent
5: (a) max at (0,0), min at $(\pm\sqrt{10/3},-500/27)$; (b) $x>\sqrt{2}$ and
$x<-\sqrt{2}$; (c) $(\pm\sqrt{2},-12)$; (d) use 2nd derivative test.

%\end{answers}

%\newlabel{homework:14}{{14}{5}}

%\begin{answers}{homework:14}
\subsection{Answers to Selected Problems in Chapter 14}

\noindent
1:~(a) max value $f(2)=5$; (b) min value $f(0)=1$. 
%
2:~$P/4$ by $P/4$.~~~
%
3:~${\root 3 \of {2V}}$ by ${\root 3 \of {2V}}$ by
$\frac{1}{ 2}{\root 3 \of {2V}}$; ratio=$1/2$.~~~
%
4:~$l^2/8$.~~~
%
5:~$x=2000$, $P=\$6$, profit=\$5000.~~~
%
6:~$r\sqrt{2}$ by $r\sqrt{2}/2$, area=$r^2$.~~~
%
7:~$h/r=2$.~~~
%
8:~$h/r=8/\pi$.~~~
%
9:~4/27.~~~
%
10:~Take $x=a$, where $f(x)$ has a corner (if $w>v$, then $f'(x)$ is never
zero).~~~
%
11:~$h/r=2$; $h/r=3.5$.~~~
%
12:~(a) $a/6$ on a side; (b) $\Bigl(a+b-\sqrt{a^2+ b^2-ab}\Bigr)/6$ on a
side.~~~
%
13:~$0.5-\frac{\pi}{ 4}k$ provided that $k<2/\pi$; otherwise,
the optimal window consists of only the semicircular piece.~~~
%
14:~$\theta=30^\circ$, max area$=l^2\sqrt{3}/12 \approx 0.144\>l^2$, 
bigger than in Problem 3.~~~
%
15:~0.00139 hours ($\approx 5$ sec), misses by 0.5726 mi.~~~
\noindent
16:~ratio=$a/b$.~~~
\noindent
17:~$\frac{2}{ 3}r\sqrt{3}$ by $\frac{2}{ 3}r\sqrt{6}$.~~~
%
18:~$1/\sqrt{3}$ ($\approx$58\%).~~~
%
19:~18 in $\times$ 18 in $\times$ 36 in.~~~
%
20:~$r=2.71$, $h=10.84$.~~~
%
21:~$h/r=\sqrt{2}$.~~~
%
22:~$h=1.1798R$, $r=0.9837R$, 28.54\%
%
23:~$c/(1+{\root 3 \of {b/a}})$ from $A$.~~~
%
24:~1/2.~~~
%
25:~$x=2500$, $P=\$5$, profit=\$7000.
%
26:~Snell's law: $\frac{\sin\theta_1}{ v_1}=\frac{\sin\theta_2}{ v_2}.$

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newlabel{homework:15}{{15}{5}}


%\begin{answers}{homework:15}
\subsection{Answers to Selected Problems in Chapter 15}

\noindent
 1:
$x_1=1.4765432$, $x_2=1.4757740$, $x_3=1.4757732.$~~~
%
 2:
$x_1=2.25$ (exactly), $x_3=x_2=2.19$ to three significant figures
($x_2=2.19369\dots$, $x_3=2.19192\dots$). ~~~
%
 3:
$\theta_1=1.9359512$, $\theta_2=1.9345639$, $\theta_3=1.9345632 $.~~~
%
 4:
Hint: Set the slope of the line joining $(-\pi/2,-1)$ to $(x,\sin x)$ equal
to the derivative of $\sin x$, then multiply through by the denominator to
get the $f(x)$ in Newton's method: $f(x)=(\frac{\pi}{ 2}+x)\cos x-\sin x-1$;
then $x_0=0.7$, $x_1=0.7633$, $x_3=x_2=0.7603$.~~~
%
 5:
$t=0.5062$ sec, height=$6.9938$ ft.~~~6:~take $t_0=0.7$, then $t_1=
0.68674$, $t_2=0.68726$.~~~
%
 7:
$x=40\sin(\pi t/10)-10\sin(\pi t/3)$, $y=50-40\cos(\pi t/10)-10\cos(
\pi t/3)$, 9.2655 sec, 98.56 ft.~~~
%
 8:
$l=\Bigl(9/\sin(\frac{\pi}{ 3}-\theta)\Bigr)+\Bigl(7/\sin\theta\Bigr)$,
0.49254 rad ($=28.2^\circ$), 31.89 ft

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newlabel{homework:16}{{16}{5}}


%\begin{answers}{homework:16}
\subsection{Answers to Selected Problems in Chapter 16}

\noindent
 1:
(a) $-\frac{1}{ 3x^3}+\frac{2}{ 5}x^{5/2}+5x+C$,
(b) $-0.1(1-t)^{10}+C$, (c) $\frac{1}{ 5}x^5+\frac{2}{ 3}x^3+x+C$,
(d) $-0.3(1-5t)^{2/3}+C$,\break (e) $\frac{1}{ 4}\sin^4x+C$,
(f) $-\frac{1}{ 3}(100-x^2)^{3/2}+C$, (g) $-\frac{2}{ 3}\sqrt{1-x^3}+C$,
(h) $-\frac{B}{ 2\pi}\cos\left(\frac{2\pi}{ B}(t-C)\right)
+Dt+C'$, (i) $-\frac{20}{ 3}\left(1-\sin\frac{t}{ 10}\right)^{3/2}+C$,
(j) $\frac{1}{\pi}\sin\bigl(\sin(\pi t)\bigr)+C$. ~~~
%
 2:
$-\frac{1}{\pi}\text{cot}(\pi t)+C$.~~~
%
 3:
$-\frac{1}{ 2}g(t-t_0)^2+v_0(t-t_0)+s_0$.~~~
%
 4:
$\frac{1}{ 3}t-\frac{1}{ 2}+1/(6t^2)$.~~~
%
 5:
(a) $\frac{1}{ 6}kt^3+v_0t+s_0$,
(b) $\frac{1}{ 15}(2t+1)^{5/2}+(v_0-\frac{1}{ 3})t+s_0-\frac{1}{ 15}$,
(c) $-\frac{1}{ 25}\sin(5t)+(v_0+\frac{1}{ 5})t+s_0$,
(d) $50\tan(0.1t)+(v_0-5)t+s_0$.~~~

%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newlabel{homework:17}{{17}{5}}

%\begin{answers}{homework:17}
\subsection{Answers to Selected Problems in Chapter 17}

\noindent
 1:
(a) 6.25 vs.~16/3=5.333, (b) 5.8125 vs.~5.333,
(c) 4.146 vs.~5.333, (d) 1.239 vs.~1, (e) 1.125 vs.~1.~~~
%
 2:
(a) 55, (b) 25/12, (c) $\frac{3}{ 2}+\frac{\sqrt{3}}{ 2}$. ~~~
%
 3:
(a) 28.5, (b) 14/9, (c) 2/3, (d) 0.24, (e) $2/\pi$.~~~
%
 4:
(a) 2/3, (b) $2-\sqrt{2}$, (c) $\frac{4}{\pi}+\frac{4}{ 3}$, (d) $\pi-2$.~~~
%
 5:
(a) 13/3, (b) 56/15, (c) 0, (d) $1/4\pi$, (e) 3/4, (f) 9.~~~
%
 6:
(a) 14600 degree-days, (b) 4523 degree-days, (c) 277 degree-days.
%
 7:
(a) $a(t)=\begin{cases}2t,& 0\le t\le 2\\ 4,& 2\le t\le 4\\ 8-t,& 4\le t\le 
8\end{cases}$; $v(t)=\begin{cases}t^2,& 0\le t\le 2\\ 4t-4,& 2\le t\le 4\\
8t-\frac{1}{ 2}t^2-12,& 4\le t\le 8\end{cases}$; $s(8)-s(0)=88$.
(b) $a(t)=\begin{cases}\sin t,& 0\le t\le 0.5110\\ 1-t,& 0.5110\le t\le 1\end{cases}$;
$v(t)=\begin{cases}1-\cos t,& 0\le t\le 0.5110\\ t-\frac{1}{ 2}t^2-0.2527,&
0.5110\le t\le 1\end{cases}$; $s(1)-s(0)=0.1234$.


%\end{answers}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newlabel{homework:18}{{18}{5}}

%\begin{answers}{homework:18}
\subsection{Answers to Selected Problems in Chapter 18}

\noindent
 1:
(a) $\sin^2u$, (b) $\sqrt{1+u+u^2}$, (c) $-\sqrt{1+\sin^2u}$,
(d) $2\sin^2u$; (e) 0.~~~
%
2:
(a) $b/(a+bt)$, 
(b) $(-gt+v_0)\sin^2\left(-\frac{1}{ 2}gt^2+v_0t+s_0\right)$,
(c) $3t^2\cos(t^6)-2t\cos(t^4)$,
(d) $A\alpha\cos(\alpha(t-C))\sqrt{1+\bigl(A\sin(\alpha(t-C))\bigr)^2}-
b\sqrt{1+(a+bt)^2}$.~~~
%
3:
(a) $\int_0^{2t}\left(20+10\cos^2(0.1\pi x)\right)dx$,
(b) $2\left(20+10\cos^2(0.2\pi t)\right)$.~~~
%
4:
(a) $\int_0^{2t}\frac{40\,dx}{ 1+(0.01x-1)^2}$, (b) $\frac{80}{ 1+(0.02t-1)^2}$.~~~
%
5:
(a) $2\int_{48-16t^2}^{48}48\cos\frac{\pi y}{ 96}\>dy$,
(b) $3072t\,\cos\left(\frac{\pi}{ 6}(3-t^2)\right)$.~~~
%
6:
(a) $2\int_{-1}^u\sqrt{1-x^2}\,dx$,
(b) $u(t)=\sin\bigl(\pi(t-\frac{1}{ 2})\bigr)$,
(c) $2\pi\cos\bigl(\pi(t-\frac{1}{ 2})\bigr)\sqrt{1-\sin^2\bigl(\pi(t-
\frac{1}{ 2})\bigr)}$.

%\end{answers}

\end{small}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Answers to Practice Midterms}

\begin{small}

\centerline{\bf Answers to Practice First Midterm \#1}
\smallskip

\noindent
1. (a) $0\le x\le 2$; (b) $-2\le x\le 0$; (c) all $x$ except 2 and 4;
(d) $-1<x<1$.

\noindent
2.~(a) ABIJ; (b) DEFG;  (c) GHI; (d) FJ; (e) J; (f) F.

\noindent
3.~$t=f(s)=\frac{130}{ 4s}-\frac{1}{ 2}s$, $f(5\frac{1}{ 12})\approx f(5)+\frac{1}{
12}f'(5)=4+\frac{1}{ 12} (-1.8)=3.85$ ft.  4.~$9\frac{7}{ 81}=9.0864$ cm.

\smallskip

\centerline{\bf Answers to Practice First Midterm \#2}
\smallskip
\noindent
1.~(a) like the $x^2$-parabola, but shifted to have its vertex at $(-2,-4)$;
(b) bottom half of ellipse, passes through $\pm 1$ on the $x$-axis and
$-5$ on the $y$-axis. 2.~$-\pi/2<x<\pi/2$.

\noindent
3.~(a) CDJ; (b) AFGH; (c) ABHI; (d) CGJ; (e) C; (f) G.
4.~(a) $v_0=9.5$ m/sec; (b) $v_0=f(t)=4.9t-\frac{75}{ t}$, $f(5\pm 0.1)\approx
9.5\pm 0.1\cdot 7.9=9.5\pm 0.79$ m/sec.

\noindent
5.~$y=f(x)=1000/x$, $f(10.77)\approx f(10)+(0.77)f'(10)=100+(0.77)(-10)=92.3$
m${}^2$.

\smallskip

\centerline{\bf Answers to Practice First Midterm \#3}
\smallskip

\noindent
1.~$0.5<x\le 2$.~~~
2.~(a) AFG; (b) CDI; (c) DE; (d) CFI; (e) C.

\noindent
3.~$g=f(t)=\frac{22.5}{ t^2}$; (a)~$f(3)=2.50$;
(b)~$f(3.1)\approx f(3)+0.1\cdot f'(3)=2.33$, so $g$ is between
$2.33$ and $2.50$ m/sec${}^2$.
4.~$y=f(x)=\sqrt{36^2-x^2}$, $f(21.6\pm 0.2)\approx 28.8$ in $\pm$ 0.15 in.


\smallskip


\centerline{\bf Answers to Practice First Midterm \#4}

\smallskip
\noindent
1.~(a) $-35<x<15$; (b) $-7\le x\le 7$.~~~2.~(b) when $t=12$ (parachute opens);
when $t=0$ (drop from plane); right after parachute opens.
3.~12.8. 4.~(a) $A=\frac{\pi}{ 4}r^2+rs$; (b)~$100\pi+600$;
(c)~$s=f(r)=\frac{100\pi+600}{ r}-\frac{\pi}{ 4}r$, $f(21)\approx f(20)+1\cdot
f'(20)=26.93$ cm.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\smallskip

\centerline{\bf Answers to Practice Second Midterm \#1}

\smallskip
\noindent
1.~19.861. 2.~(a) $(-1,-1.5)$ is max; (b) $x<-1$ and $x>0$; (c) $y$-axis;
(d) none; (e) see bottom of page.  3.~(b) $y=20\sin\left(\frac{2\pi}{ 12}(
t-9.5)\right)+55$; (c) mid-March to mid-November.
4.~$b,~C,\cos C$ are all constants, $a$ and $c$ are variables, take
$d/dt$ of everything, you know $da/dt=-5.70$, you get $dc/dt=-5.55$ m/sec.

\smallskip

\centerline{\bf Answers to Practice Second Midterm \#2}

\smallskip

\noindent
1.~(a) $(0,0)$, $(\sqrt{3},1.5)$, $(-\sqrt{3},1.5)$; (b) $-\infty<x<
-\sqrt{3}$, and $0<x<\sqrt{3}$; (c) $(1,5/6)$, $(-1,5/6)$;
(d) $-1<x<1$; (e) see below.

\noindent
2.~(a) $5\pi$ rad/sec; (b) $\dot{x}=5(1-\cos\theta)$, $\dot{y}=5\sin\theta$;
(c) $\dot{x}=2.50$ m/sec, $\dot{y}=-4.33$ m/sec, $dy/dx=-1.73$.
3.~(a) $x=\sqrt{100-4y^2}$; (b) $A=\pi(100-4y^2)$; (c) 0.63 m${}^2$/sec.

\noindent
4.~(a) $\sqrt{(x+5)^2+y^2}+\sqrt{x^2+(y+5)^2}=20$; (b) $y'=1/5$.

\smallskip

\centerline{\bf Answers to Practice Second Midterm \#3}

\smallskip
\noindent
1.~(a) $(0,0)$; (b) $(1,2.75)$ and $(3,6.75)$; (c) see bottom of page; (d)
$1<x<3$.

\noindent
2.~$C=f(c)$ ($a,b$ are const), $\pm 0.6175^\circ$.
3.~$y=5\cot\theta$, $0.0436$ ft/min.

\noindent
4.~(a) $y=19.6-4.9t^2$; (b) $x=2\sin(\pi(t-\frac{1}{ 2}))+2$;
(c) $\dot{x}=-4.443$ m/sec, $\dot{y}=-12.25$ m/sec, $dy/dx=2.757$.

%\smallskip
\newpage
\centerline{\bf Answers to Practice Second Midterm \#4}

\smallskip
\noindent
1.~(a) max: $(0,0)$, min: $(\pm\sqrt{6},-4.5)$; (b) $-\sqrt{6}<x<0$ and
$\sqrt{6}<x<\infty$; (c) $(\pm\sqrt{3},-2.925)$;

\noindent
(d) $-\sqrt{3}<x<\sqrt{3}$; (e) see bottom of page. 
2.~$\pi R^2T=1$ m${}^3$; take $d/dt$ of both sides; $0.804$ m/hr.
3.~$B=f(A)$ ($a,b$ are const), $61.732^\circ$.
4.~(a) $V=\frac{1000\pi}{ 3}\tan^2(\frac{1}{ 2}\theta)$; (b)
$349$ ft${}^3~\pm~14$ ft${}^3$.


%%--- figure OI (answers to Practice Midterm #2) ----------------
\begin{center}
\begin{psfigure}{5in}{3.5in}{figOI.ps}
\end{psfigure}\\
\end{center}
%------------------------------

\end{small}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Answers to Practice Final Exams}

\begin{small}


\centerline{\bf Answers to Practice Final Exam \#1} 
\smallskip
\noindent
1:~(a) -15/4; (b) $\frac{5}{\pi}\sin^2\left(\frac{\pi}{ 10}(t-1)\right)+C$, or
else $-\frac{5}{\pi}\cos^2\left(\frac{\pi}{ 10}(t-1)\right)+C$.
2:~-78.125 m/sec.

\noindent
3:~(b) $v=\frac{1}{ 2}t^2,~~0\le t\le 1;~~~v=t-\frac{1}{ 2},~~1\le t\le 2;~~~
v=3/2,~~2\le t\le 3$; (c) 8/3 m.

\noindent
4:~area of hexagon $=2\sqrt{3}r^2$, optimal ratio $h/r$ is $4\sqrt{3}/\pi
=2.2053\cdots$.

\noindent
5:~(a) 4.60555 cm; (b) $x=2\cos(7\pi t)+\sqrt{16-4\sin^2(7\pi t)}$;

\noindent
(c) $dx/dt=-14\pi\sin(7\pi t)-28\pi\sin(7\pi t)\cos(7\pi t)\Bigl(16-
4\sin^2(7\pi t)\Bigr)^{-1/2}$;

\noindent
(d) $f'(1/21)=-48.654$, $f(1/20)\approx 4.4897$ cm.

\noindent
6:~(a) $x=5t-10\sin(t/4)$; (b) 8.18361 cm; (c) $t_1=3.57285$ sec,
$t_2=3.55148$ sec, $x(t_2)=10.00011$ cm. 

\smallskip
\centerline{\bf Answers to Practice Final Exam \#2}
\smallskip

\begin{rightindent}{5in}
\noindent
1:~(a) $\frac{1}{ 4}(1-2t)^{-2}+C$; (b) 2/9.
2:~(a) $20+\frac{10}{\pi}$ rev; (b) $40\pi+20$ rad;

\noindent
(c) 109.25 ft. 3.~16 ft/sec.  4:~2.033 ft.  5:~$s=1.59$ ft, $l=4.76$ ft.

\noindent
6:~(b) $2x+2\cos(2x)=0$; (c) $(\pi/12,\pi^2/144+1/2)$ and

\noindent
$(5\pi/12,25\pi^2/144+1/2)$; (d) $-\pi/4$; (e) $x_1=-0.5236$,

\noindent
$x_2=-0.5150$; (f) no other max/min; (g) graph at right: 

\noindent
7:~(a) $\pi/6$ rad/sec; (b)~$x=20\cos(\pi t/6)$, $y=20+20\sin(\pi t/6)$;

\noindent
(c) $v_{0,\rm horiz}=-30$ ft/sec, $v_{0,\rm vert}=52$ ft/sec;

\noindent
(d) $x=75-30t$, $y=-16t^2+52t$; (e) 5.6737 ft;

\noindent
(f) dist$=\sqrt{(75-30t-20\cos(\pi t/6))^2+(-16t^2+52t-20-20\sin(\pi t/6))^2}
$

\noindent
(g) let $f(t)=\frac{d}{ dt}($dist${}^2)=2(75-30t-20\cos(\pi t/6))(-30+
(20\pi/6)\sin(\pi t/6))+$

\noindent
$+2(-16t^2+52t-20-20\sin(\pi t/6))(-32t+52-(20\pi/6)\cos(\pi t/6))$.
\end{rightindent}
\hfill
%--- figure UA (practice final #2 answer to problem  6(g) ) ---------------
%
\begin{psfigure}{1.25in}{1.75in}{figUA.ps}
\puttext{-10,-47}{\small min}
\puttext{40,-25}{\small inflec. pts.}
\puttext{25,-13}{\vector(-1,2){7}}
\puttext{15,-25}{\vector(-1,0){10}}
\end{psfigure}
%------------------------------

\smallskip
\centerline{\bf Answers to Practice Final Exam \#3}
\smallskip
\noindent
1:~(a) $-\frac{2}{ 7}\sqrt{1-7z}+C$; (b) $(4-\sqrt{2})/3=0.8619$.

\noindent
2:~(b) $2-t$ for $0\le t\le 1$, $0.5+0.5(2-t)^2$ for $1\le t\le 2$,
$0.5$ for $2\le t\le 3$, $2-0.5t$ for $3\le t\le 4$; (c) 2.92 m.
3:~$-9.823$ ft/sec.
4:~differentiate $x+y+\sqrt{x^2+y^2}=$const, get $y'=-32/49$, answer is
24.013.

\noindent
5:~(a) vol$=\frac{1}{ 3}\pi r^3x$, area$=\pi r^2\sqrt{1+x^2}$;
(b) $r=\sqrt{C/\pi}\Bigl(1+x^2\Bigr)^{-1/4}$. (c) $x=\sqrt{2}$.

\noindent
6:~Choose $f(x)=x^{1/2}+x^{1/3}-x$ in Newton's method, so that
$f'(x)=\frac{1}{ 2}x^{-1/2}+\frac{1}{ 3}x^{-2/3}-1$; then
$x_0=3$, $x_1=3.3163$, $x_2=3.3094$.

\noindent
7:~(a) $x=\frac{1}{ 2}\sin(2\pi(t-\frac{1}{ 4}))+\frac{1}{ 2}$;
(b) $(4-4.9t^2)^2+(\frac{1}{ 2}\sin(2\pi(t-\frac{1}{ 4}))+\frac{1}{ 2})^2$;
(c) $f(t)=-19.6t(4-4.9t^2)+\bigl(\sin(2\pi(t-\frac{1}{ 4}))+1\bigr)\pi
\cos(2\pi(t-\frac{1}{ 4}))$.

\smallskip
\centerline{\bf Answers to Practice Final Exam \#4}
\smallskip
\noindent
1:~(a) 28/15; (b) $3B/16\pi$.~~~
%
2:~(a) $3+3t-\frac{3}{ 2}t^2$, $0\le t\le 2$; $9-3t$, $2\le t\le 3$; 
(b) 9.5 m.

\noindent
3:~(a) Vol$=50\cdot 4\cos\theta(14+4\sin\theta)$; (b) d(Vol)$/d\theta=
-400(4X^2+7X-2)$ (where $X=\sin\theta$); (c) 2904 in${}^3$.

\noindent
4:~A good initial guess is $s_0=10$ (since vol=1000, surface area=600).  Choose
$f(s)=s^3+6s^2-1500$; then $s_1=9.762$, $s_3=s_2=9.757$.

\noindent
5:~(a) $x_A=\cos(2\pi t/9)$, $y_A=\sin(2\pi t/9)$, $x_B=2-t$, $y_B=t$;

\noindent
(b) $2\bigl(2-t-\cos(2\pi t/9))(-1+(2\pi/9)\sin(2\pi t/9)\bigr)+
2\bigl(t-\sin(2\pi t/9))(1-(2\pi/9)\cos(2\pi t/9)\bigr)$.

\noindent
6:~(a) $x=(10\cos\alpha)t$, $y=-4.9t^2+(10\sin\alpha)t+50$; (b)
$T=1.0204\bigl(\sin\alpha+\sqrt{\sin^2\alpha+9.8}\bigr)$; (c) $X=(10\cos
\alpha)T$ (write $T$ in terms of $\alpha$ using (b));  (d) find
$\frac{\pi}{ 180}\cdot\frac{dX}{ d\alpha}$ and then take $\alpha=60^\circ$:
$-0.5781$ (the minus sign means that a positive error in $\alpha$ leads
to a negative error in $X$). 



\end{small}

