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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 110 "10. Consider the function f(t) defined b
y the integral of (1-e^t), for t >= 0.  Define this function in Maple.
" }}{PARA 0 "" 0 "" {TEXT -1 20 "     What is f'(0) ?" }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := t -> int(1-exp(
x),x=0..t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "df := D(f);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "df(0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "15. T
he logarithmic spiral is defined by r = a*exp(b*t) in polar coordinate
s." }}{PARA 0 "" 0 "" {TEXT -1 86 "  (a) Give the Maple commands to ma
ke a plot for a = 0.5 and b = 0.07 for t = 0..6*pi." }}{PARA 0 "" 0 "
" {TEXT -1 103 "  (b) Create an animation of 10 frames,for a = 0.5 and
 b going from 0.01 to 0.1 (also for t = 0..6*pi)." }{MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Answer to (a):" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 44 "plots[polarplot](0.5*exp(0.07*t),t=0..6*Pi);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Answer to (b):" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 70 "frames := [seq(plots[polarplot](0.5*exp(k*t/100),t
=0..6*Pi),k=1..10)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plo
ts[display](frames,insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "16. C
onsider the curve x^4 - 3*x*y + y^4.  Give all Maple commands" }}
{PARA 0 "" 0 "" {TEXT -1 65 "   (a) to make a plot for x and y both ra
nging between -2 and +2;" }}{PARA 0 "" 0 "" {TEXT -1 55 "   (b) to con
vert the curve into polar coordinates; and" }}{PARA 0 "" 0 "" {TEXT 
-1 46 "   (c) to plot the curve in polar coordinates." }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := x^4 + 3*x*y + y
^4;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Answer to (a):" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plots[implicitpl
ot](f,x=-2..2,y=-2..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Answer
 to (b):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "pf := subs(\{x = r*cos(
t),y = r*sin(t)\},f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "nf
 := normal(pf/r^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "s :=
 solve(nf,r);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Answer to (c):" 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plots[
polarplot](s[1]);  # same curve with s[2]" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "19.
 Consider the initial value problem" }}{PARA 0 "" 0 "" {TEXT -1 55 "  \+
     x''(t) + 4*x(t) = sin(t),  x(0) = 1,  x'(0) = 0." }}{PARA 0 "" 0 
"" {TEXT -1 82 "   (a) Give all Maple commands to define this problem \+
and to solve it numerically." }}{PARA 0 "" 0 "" {TEXT -1 69 "   (b) De
fine a function which returns for every t the value of x(t)." }}{PARA 
0 "" 0 "" {TEXT -1 50 "   (c) Plot the solution for t going from 0 to \+
10." }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Answer
 to (a):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "ode := diff(x(t),t$2) + 4*x(t) = sin(t);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "ini := x(0) = 1, D[x](0) = 0;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "ivp := [ode,ini];" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "s := dsolve(ivp,x(t),numeric);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 14 "Answer to (b):" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := t -> rhs(s(t)[2]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Answer to (c):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "plot(f,0..10);" }}}}{MARK "21 0 0" 8 }{VIEWOPTS 1 
1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }