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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 36 "Parametric Equations and \+
Arc Lengths" }}{PARA 0 "" 0 "" {TEXT -1 42 "Exercises: page 471, #5,7,
9,11,13, 17, 19." }}{PARA 0 "" 0 "" {TEXT -1 10 "Example 1." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "x:=ln(t); y:=exp(-t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot([x,y,t=1..2]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d1:=diff(x,t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d2:=diff(y,t);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "h:=t->((d1)^2+(d2)^2)^(1/2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "L:=Int(((d1)^2+(d2)^2)^(1/2),t=0..2*Pi);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "evalf(simpson(h(t), t=1..2, 10));" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "?simpson;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "evalf(middlesum(h(t),t=1..2,10));" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "evalf(trapezoid(h(t),t=1..2,10));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "x:=cos(t); y:=sin(t);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 22 "plot([x,y,t=0..2*Pi]);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 73 "The following command is to find the arc length of
 a parametric equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "
L:=Int((diff(x,t)^2+(diff(y,t)^2))^(1/2),t=0..2*Pi);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "plot([x,y,t=0..2*Pi],scaling=constrained);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 256 50 "The following example is from pag
e 470, example 4." }}{PARA 0 "" 0 "" {TEXT -1 9 "Example 2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "x:=2*(t-sin(t)); y:=2*(1-cos(t));" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot([x,y,t=0..2*Pi]);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d1:=diff(x,t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d2:=diff(y,t);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 40 "L:=Int(((d1)^2+(d2)^2)^(1/2),t=0..2*Pi);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 42 "plot([x,y,t=0..4*Pi],scaling=constrained)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 11 "Exercise 1." }}{PARA 0 "" 
0 "" {TEXT -1 81 "Estimate the length of the arc of the hyperbola xy =
 1 from (1, 1) to (2, 1/2).  " }}{PARA 0 "" 0 "" {TEXT 258 12 "Exercis
e 2. " }}{PARA 0 "" 0 "" {TEXT -1 108 "If  x=exp(t)*cos(t) and  y=exp(
t)*sin(t), where  t  is in [0,2*Pi],  plot the graph and find its arc \+
length." }}{PARA 0 "" 0 "" {TEXT 259 11 "Exercise 3." }}{PARA 0 "" 0 "
" {TEXT -1 280 "The the function  f  below represents a telephone wire
 hanging between two poles x = -5 and x = 5. It takes the shape of a c
atenary. Note that the value 'a' represents the lowest height of the w
ire above the ground. Find the arc lengths for the wire with a=5 and a
=6 respectively." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 44 "f:=proc(a,x) (a/2)*(exp(x/a)+exp(-x/a)) end;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(5,x);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 43 "Some more interesting parametric equation
s." }}{PARA 0 "" 0 "" {TEXT -1 11 "Example 3. " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "x := cos(7*t)*sin(t);" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "y:= cos(-3*t)*cos(t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot([x,y,t=0..Pi]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Example 4." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "x := cos(t)*sin(2*t);     " }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 20 "y:= cos(4*t)*sin(t);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "plot([x,y,t=-Pi..Pi]);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 10 "Example 5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "x := t*sin(t);    " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 
"y:= t^2 - t^(1/3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot
([x,y,t=0..16*Pi]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Example 6.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x := 2*t;   " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "y:= exp(-t)*cos(t) + t*cos(t
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot([x,y,t=0..16*Pi]
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Example 7." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x := sin(t)*cos(t) + sin(2*t)*cos(2
*t) + sin(3*t)*cos(3*t) + sin(4*t)*cos(4*t);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 81 "y := sin(t)*cos(2*t) + sin(3*t)*cos(4*t) + sin(5
*t)*cos(6*t) + sin(7*t)*cos(8*t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "plot([x,y,t=0..16*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "plot([37*sin(15*t), 89*cos(20*t), t=0..100]);" }}}
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