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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Exam 2 MCS 320 Friday 2 Ap
ril 2004" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "1. Give the Maple \+
commands for the following tasks:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 201 "(a) Use stats[random,uniform[0,2*Pi]] to gener
ate 10 random angles.  Use these angles to generate 10 random points o
n the unit circle x^2 + y^2 - 1 = 0, using x = cos(a), y = sin(a), for
 every angle a." }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Sto
re the points as a list of lists of coordinates [x,y], i.e., as [[x1,y
1],[x2,y2],[x3,y3],...]. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 84 "(b) Make a lists of plots of disks of radius 0.01 cente
red at the ten random points." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "
ANSWER to (a):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "a := stats[random,uniform[0,2*Pi]](10);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "xp := map(cos,[a]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "yp := map(sin,[a]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 14 "ANSWER to (b):" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "l := zip((x,y)->[x,y],xp,yp)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "lp := map(x-> plottool
s[disk](x,0.01),l):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot
s[display](lp); # this is extra" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "2. What is the
 assume facility in Maple.  Why is it needed?" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "    Give an example of a good use \+
of the assume command." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 7 "ANSWER:" }{MPLTEXT 1 0 0 "" }}}{EXCHG }{EXCHG {PARA 0 "" 
0 "" {TEXT -1 71 "The assume facility allows the user to impose assump
tions on variables." }}{PARA 0 "" 0 "" {TEXT -1 72 "This is needed to \+
simplify expression or to evaluate improper integrals." }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "For example:" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "assume(x > 0); sqr
t(x^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "assume(a < 0); i
nt(exp(a*t),t=0..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 227 "3. The Cantor set
 for any interval [a,b] is constructed by removing the middle third fr
om the interval [a,b], leaving the two intervals [a,a+(b-a)/3] and [b-
(b-a)/3,b], and repeating this removal on the intervals that are left.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "Write a procedure cantor wit
h as index a natural number n, which takes on input the endpoints a an
d b of the interval [a,b] (thus, a < b). Calling cantor[n](a,b) return
s a sequence of 2^n intervals." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
129 "For example: cantor[0](0,1) = [0,1], cantor[1](0,1) = [0,1/3],[2/
3,1], cantor[2](0,1) = [0,1/9],[2/9,1/3],[2/3,7/9],[8/9,1], etc." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "cantor \+
:= proc(a,b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "  local n:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "  n := op(procname);" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "  if n = 0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "     return [a,b]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "  else
 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "     return cantor[n-1](a,a+(b
-a)/3),cantor[n-1](b-(b-a)/3,b):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "
  end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "cantor[0](0,1);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 15 "cantor[1](0,1);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "cantor[2](0,1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "cantor[3](0,1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "4. Co
nsider the integral of e^(sin(x)) for x in [0,t].  Give the Maple comm
ands" }}{PARA 0 "" 0 "" {TEXT -1 89 "  (a) to define this integral as \+
a formula (call it f), without asking Maple to evaluate;" }}{PARA 0 "
" 0 "" {TEXT -1 82 "  (b) to compute a 10-th order Taylor approximatio
n (call it T) for it around t=0;" }}{PARA 0 "" 0 "" {TEXT -1 115 "  (c
) to use the Taylor approximation to create a function F which returns
 a number, e.g.: F(2) returns 12496/2835." }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := Int(exp(sin(x)),x=0..t)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "T := taylor(f,t,10);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "F := unapply(convert(T,po
lynom),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 84 "5. Consider the system 1 + y - x y + y^2 = 0, x + \+
y^2 - 2 z^2 = 0, 1 - z + y z = 0. " }}{PARA 0 "" 0 "" {TEXT -1 66 " Gi
ve all Maple commands needed to answer the following questions." }}
{PARA 0 "" 0 "" {TEXT -1 28 " Also, justify your answers." }}{PARA 0 "
" 0 "" {TEXT -1 13 "(a) How many " }{TEXT 258 7 "complex" }{TEXT -1 
33 " solutions does this system have?" }}{PARA 0 "" 0 "" {TEXT -1 13 "
(b) How many " }{TEXT 259 4 "real" }{TEXT -1 33 " solutions does this \+
system have?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
14 "ANSWER to (a):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "p1 := 1 + y - x*y + y^2;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "p2 := x + y^2 - 2*z^2;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "p3 := 1 - z + y*z;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "gb := grobner[gbasis]([p1,p2,p3],[x,y,z],plex);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "From the triangular structure, we
 can simply make the product of the degrees to count the number of com
plex roots:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 48 "degree(gb[1],x)*degree(gb[2],y)*degree(gb[3],z);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 57 "To find the solutions, we solve the last \+
equation for z :" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "fsolve(gb[nops(gb)],z,complex);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 161 "We see there are five solutions.  Since the first
 two polynomials in gb are linear in x and y, for every value of z, we
 find one corresponding value for x and y." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 14 "ANSWER to (b):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "fsolve(gb[nops(gb)],z);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 121 "We see three real solutions.  Substitution in the f
irst two polynomials will yield real corresponding values for x and y.
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta
rt;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 258 "6. The lemniscate of Bern
oulli is defined by (x^2+y^2)^2 = x^2 - y^2.  Give all Maple commands \+
to convert this expression into polar coordinates.  Give also the defi
nition for the lemniscate you obtain in polar coordinates and the comm
and to plot this curve." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "ANSWER \+
:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "lem
niscate := (x^2 + y^2)^2 = x^2 - y^2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "polar_lmn := subs(\{x = r*cos(t),y=r*sin(t)\},lemnisc
ate);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "nf := normal(polar
_lmn/r^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "snf := simpli
fy(nf);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "s := solve(snf,r
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plots[polarplot](s[1]
,numpoints=600);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "7. Consider the initial value
 problem dy/dx = sin(2x-y), y(0) = 0.5." }}{PARA 0 "" 0 "" {TEXT -1 
11 "  (a) Give " }{TEXT 256 3 "all" }{TEXT -1 87 " Maple commands (not
 the output) to define this initial value problem and to obtain an " }
{TEXT 257 5 "exact" }{TEXT -1 10 " solution." }}{PARA 0 "" 0 "" {TEXT 
-1 54 "  (b) Give the first three significant digits of y(1):" }}
{PARA 0 "" 0 "" {TEXT -1 152 "  (c) Use the exact solution to create a
 function which gives a hardware float numerical approximation for eve
ry x.  Give the Maple command(s) you used." }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "ANSWER to (a):" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ivp := diff(y(x),x) = s
in(2*x-y(x)),y(0)=0.5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s
ol := dsolve([ivp],y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "ANSW
ER to (c):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "yx := subs(body = rhs(sol),x->evalhf(body));" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 14 "ANSWER to (b):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "yx(1);" }}}{EXCHG }}{MARK "71" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }