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0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 
0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 
11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 
-1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "
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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 5 "Lab 5" }}{PARA 19 "" 0 "" {TEXT 
-1 35 "Math 181: Computational Mathematics" }}{PARA 19 "" 0 "" {TEXT 
-1 11 "R. Robinson" }}{PARA 0 "" 0 "" {TEXT -1 10 "9/99, 2/01" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "M
aple Number Formats" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "If you type
 a number without a decimal point, Maple interprets it as an " }{TEXT 
257 7 "integer" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 5 "1024;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 47 "You type evalf to convert it to floating
 point:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "evalf(1024);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 42 "Maple also allows numbers to be store
d as " }{TEXT 258 8 "rational" }{TEXT -1 9 " numbers " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "22/7;" }{TEXT -1 0 "" }}{PARA 11 "" 
1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "But Maple au
tomatically " }{TEXT 259 7 "reduces" }{TEXT -1 17 " to lowest terms." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "440/(-140);" }{TEXT -1 0 
"" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 85 "A rational number in Maple, then, is a pair of integers (that a
re relatively prime). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 262 13 "Theoretically" }{TEXT -1 179 " any rational number \+
can represented but since there is a practical limit to how large an i
nteger can be in Maple (the machine integer size: 32 bits for a PC), t
his restricts the " }{TEXT 263 8 "accuracy" }{TEXT -1 49 " of the rati
onal numbers that can be represented." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 264 11 "Exercise 1:" }{TEXT -1 64 " See if y
ou can discover (about) how big the largest integre is." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "You can convert a rational numbe
r to a floating point number using \"evalf\". " }}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 82 "If you type a number wi
th a decimal point, Maple automatically interprets it as a " }{TEXT 
256 4 "real" }{TEXT -1 23 " floating point number:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "3.14159;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 66 "Maple's floating point numbers are stored with a b
ase 10 mantissa." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 149 "Floating point numbers can also be input in (base 10) sc
ientific notation. Maple has its own way to normalize floating point n
umbers that it outputs." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "6
02e21;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 195 "However, Maple represents floating point
 numbers internally as a pair of integrs (so rationals and floating po
int numbers are similar in this respect). Here is the actual data stru
cture involved:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "n:=Float(602,23);" }}{PARA 11 "" 1 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 262 "\nThe following is som
e general Maple technique that is useful in this context. The \"op\" c
ommand is used to pick off a part of a Maple object. So \"op(1,num);\"
 should pick off part 1 of num and \"op(2,num)\" should pick off part \+
2. \"op(0,num)\" reads the object type." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 265 11 "Exercise 2:" }{TEXT -1 152 " Try th
is (i.e., op(0,num), op(1,num), op(2,num)) on a few real numbers, inte
gers and rationals. This will give you a clue about how Maple really w
orks." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 291 "Even though Maple can s
tore vary large integers, and floating point constructed out of intege
rs, floating point numbers in Maple retain only a fixed number of digi
ts in the mantissa. The number of such digits is controlled by an envi
ronment variable called \"Digits\". The default value is 10." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "You should rea
d the \"Help\" page on \"Float\". In particular, you will note that a \+
floating point number can be a lot bigger than an integer." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 11 "Exercise 3:" }
{TEXT -1 82 " What is the largest possible floating point number in Ma
ple? (Assuming Digits=10)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Maple has a third kind of nu
mber: " }{TEXT 267 8 "hardware" }{TEXT -1 25 " floating point numbers.
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalhf(22/7);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "You can get the approximate number
 of decimal digits of accuracy using:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "evalhf(Digits);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
241 "This shows that these are pretty much the same as Matlab numbers.
 Computations with these machine numbers are faster (by a factor of 40
 or so) than Maple's ordinary floating point numbers, which are really
 \"software\" floating point numbers. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 39 "Matlab is probably a bit faster still
. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "On
 the other hand a Fortran or C program is probably 40 times faster sti
ll (see Essential Maple by Robert M. Corless, Springer Verlag, 1995." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 5 "Note:" 
}{TEXT -1 314 " There are complicated rules and restriction about usin
g Maple's hardware floating point numbers. In particular, it is easy t
o \"contaminate\" a hardware number calculation so that it actually ev
eluates in software floating point. This will rob your program of the \+
hardware speed advantage. See \"Help\" on this topic." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 8 "Comment:
" }{TEXT -1 94 " This lab is on Maple numbers, but similar principles \+
apply to other Computer Algebra Systems." }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 28 "Experimenting with accuracy " }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 36 "Let's reduce accuracy to Digits:=2. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 278 "The goal here is to use \+
Maple's control (as a CAS) to do experiments in numerical analysis! In
 particular, we can force Maple to use a rediculously innaccurate numb
er system to illustrate features of all number systems which may be mo
re difficult to observe in a better systems." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "As we did in Lecture 3 and Lab \+
3, we will use quadrature as an example." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "Digits:=2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 " A
n integral (which we discussed in Lab 3) is:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "Int(1/x,x=1/10..1);" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 116 "Note the upper case \"I\" in \"Int\". Th
is shows how to get Maple to just print an integral, but not try to ev
aluate it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 55 "Now of course this integral is easy and Maple can just " }
{TEXT 260 3 "do " }{TEXT -1 38 "in an exact way. This is a CAS answer.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int(1/x,x=1/10..1);" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 
"However our task is to experiment with round off error. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "To this end we imp
lement the trapezoid rule as a function of n. " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 70 "trap1:=n->(10+sum(2*(1/(1/10+(9*i)/(10*n))),i=
1..n-1)+1)*(9/10)/(2*n);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Before going on, let me ex
plain the syntax. The n->... notation defines a " }{TEXT 270 0 "" }
{TEXT 271 8 "function" }{TEXT 272 0 "" }{TEXT -1 53 " of n. This is es
sentially a simple Maple \"program\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 119 "Unlike MATLAB, we can enter a progra
m right inside the Maple GUI. This is convenient for short programs li
ke this one. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 182 "More complicated programs can also be edited inside sepa
rate files and read in. As we will see below Maple has all the usual p
rogramming control structures (loops, conditions, etc.)." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Note that the inte
grand (the unction f(x)=1/x) is built in to this expression. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "You might
 want to verify this is a correct implemtation of the trapezoid rule.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Now t
ry evaluating this. Replace the 12's with some other numbers. Look at \+
the plot" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "trap1(12);" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "err1:=n->ab
s(evalf(trap1(n))-evalf(log(10)));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 9 "err1(12);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "list1:=[seq
([n,err1(n)],n=1..12)];" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "plot(list1);" }{TEXT -1 0 "" }}
{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 11 
"Exercise 3:" }{TEXT -1 216 " The error here seems less than theory mi
ght predict. In particular, we see the truncation error but not the ro
und-off error. Note the \"rational\" outputs from trap1. Can you expla
in what Maple is actually doing here?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 182 "Hint: Plug \"n\" into 'err1\" instea
d of a number. Look up the Greek letter you see in \"Help\". You might
 also look up \"sum\". It is less straightforward than might appear on
 the surface." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 155 "The answer to this question explains why a more sophisti
cated approach is necessary if we want to do \"numerical analysis\" ex
periments with a CAS (Maple).  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 193 "Here is a different approach, which more
 closely resembles traditional programming. It also illustrates some M
aple features, including control structures and how to pass functions \+
to functions." }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 111 "trapc:=proc(k,n) # Makes the coeffs 1,2,2,2,.
..,1 \n  if k=1 or k=n then \n     1; \n  else \n     2; \n  fi \n  en
d;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "trap2
:=proc(n,a,b,fun) # Implements the trapezoid rule.\n   local h,s,i;\n \+
  s:=0;\n   h:=(b-a)/n;\n   for i from 0 to n do\n     s:=s+trapc(k,n)
*fun(a+i*h);\n   od;\n   (s*h)/2;\nend;" }{TEXT -1 0 "" }}{PARA 12 "" 
1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "rec:=x
->1/x; # The function" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "trap2(12.3,.1,1,rec);" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "err2:=n->abs(trap2(n,.1,1,rec)-eval
f(log(10)));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "err2(12);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "list2:=[seq
([n,err2(n)],n=1..100)];" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "plot(list2);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 274 11 "Exercise 4." }{TEXT -1 220 " Try several
 different values of n and different numbers of digits. Try some diffe
rent functions. What sort of functions would likely lead to round off \+
errors? Write a short text explaining your results and observations." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 21 "Example from robotics" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 79 "The main point of a CAS is that it can solve big systems \+
of equations exactly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 51 "We will now illustrate this point with an example. \+
" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
24 "This example comes from " }{TEXT 261 28 "Computer Algebra in Indus
try" }{TEXT -1 74 " by Arjeh M. Cohen, Wiley (1993). The discussion th
ere is more extensive. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 97 "This problem concerns a robotic arm with two links a
nd three segments (see the picture in class)." }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 78 "We want to find values for a1, a2 and a3 (i.e., design
 an arm) so that if the " }{TEXT 275 5 "crank" }{TEXT -1 55 " has a sp
ecific angular velocity and acceleration, the " }{TEXT 276 8 "follower
" }{TEXT -1 66 " will also have a specified velocity and acceleration \+
(see below)." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Algebra" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 131 "siderels:=\{x1=-a1*cos(phi), y1=a1*sin(phi)
, x2=-a4-a3*cos(psi), \ny2=a3*sin(psi), cos(phi)^2+sin(phi)^2=1, cos(p
si)^2+sin(psi)^2=1\};" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "vars:=[x1,x2,y1,y2,a1,a
2,a3,a4,cos(phi),sin(phi),cos(psi),sin(psi)];" }{TEXT -1 0 "" }}{PARA 
12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "
equn1:=(x2-x1)^2+(y2-y1)^2=a2^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 36 "exp1:=simplify(equn1,siderels,vars);" }{TEXT 
-1 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 85 "See \"Help\" to understand this useful command. Note that
 the siderels were input as a " }{TEXT 277 3 "set" }{TEXT -1 17 " (cur
ly bracket: " }{TEXT 278 9 "unordered" }{TEXT -1 1 " " }{TEXT 280 4 "d
ata" }{TEXT -1 35 "), wheras the vars were input as a " }{TEXT 279 4 "
list" }{TEXT -1 18 " (square bracket: " }{TEXT 281 7 "ordered" }{TEXT 
-1 1 " " }{TEXT 282 4 "data" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "exp2:=combine(exp1,'trig');" }{TEXT -1 0 "" }}
{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 
"Here we have applied trig identities. Next we use a library function \+
called \"isolate\" that we need to read in before we use it." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "readlib(isolate);" }{TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "exp3:=isolate(exp
2,cos(phi-psi));" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "collect(exp3,\{cos(psi),cos(
phi)\});" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "Note the
 form of the RHS: \"-K1*cos(psi)+K2*Cos(phi)+K3. Now we solve explicit
ly for psi (with dummy variables):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "exp4:=solve(cos(phi-psi)=-K
1*cos(phi)+K2*cos(psi)+K3,psi);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 171 "A careful examination of this (massive) output will
 reveal two solutions. These correspond to two positions of the linkag
e. We concentrate on the first, psi+ (or psiplus)." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "exp5:=exp4[1];" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "exp6:=subs(sin(phi)^2=1-cos(phi)^2,
exp5);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ps
iplus:=simplify(exp6);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 64 "This shows how psi depends on theta, exactly. Pretty impr
essive!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 8 "Calculus" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 94 "Now we define some \"aliases\" for psi and phi as functions of \+
t, as well as their derivatives. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 63 "(Note the I. Maple comes with one built i
n alias I=(-1)^(1/2).)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "
alias(\nphi=phi(t),psi=psi(t), \nw_phi=diff(phi(t),t), w_psi=diff(psi(
t),t),\na_phi=diff(phi(t),t$2),a_psi=diff(psi(t),t$2));" }{TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "eqn1:=cos(phi-psi)=-K1*
cos(phi)+K2*cos(psi)+K3;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "eqn2:=diff(eqn1,t);" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "eqn3:=diff(eqn1,t$2);" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eqns:=\{eqn.(1..3)\}; #Note \+
some interesting syntax here" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 65 "Here we see the purpose of the aliases: implicit differ
entiation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=solve(eq
ns,\{K1,K2,K3\});" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 285 11 "Exercise 5:" }{TEXT 286 1 " " }
{TEXT -1 91 "The next line is a bit of a cheat! Can you figure out a w
ay to do it using Maple commands? " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "sys:=\{K1=a4/a3, K2=a4/a1, K3=(a1^2+a3^2+a4^2-a2^2)/(
2*a1*a3)\};" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "solutions:=solve(sys,\{a1,a2,a3\});" }{TEXT -1 0 "" }}{PARA 12 "" 
1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "The RootOf i
n the output indicates a root of a polynomial equation with no algebra
ic solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 63 "Now we input the design specifications from the Cohen article. \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "Name
ly, when the crank is upright and moving forward with a uniform veloci
ty, the follower is also upright and moving forward, but accelerating.
" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "values:=\{phi=Pi/2, w_phi=3, a_phi=0, psi=Pi/2, w_psi
=6/5, a_psi=81/50\};" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "kvals:=eval(subs(values,sol));" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "assign(kvals);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "subs(\{RootOf(_Z^2-4)=2,a4=4\},solu
tions);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Comment
: There must be a better way to do this! Can you find it?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assign(%); a4:=4; " }{TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "psiplus;" }{TEXT -1 0 "" }
}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 47 "alias(phi=phi); #Turn phi back into a variable!" }{TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(psiplus,phi=0..2*P
i);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 11 "Exercise 6:" }{TEXT -1 
52 " Can you fix this plot so it does not have the jump?" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 27 "We can also draw the crank:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "phi:
=Pi/3; " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "x
1:=evalf(-a1*cos(phi)); y1:=evalf(a1*sin(phi));" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "x2:=evalf(-a4-a3*cos(psiplus
)); y2:=evalf(a3*sin(psiplus));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 49 "plot([[0,0],[x1,y1],[x2,y2],[-a4,0]],style=LIN
E);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 11 "
Exercise 7:" }{TEXT -1 91 " Show the arm in a few other positions. (Fo
r a real challenge consider doing an animation!)" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 277 "Comment: For some
 reason Maple seems to choose different orders for the two solutions p
si+ and psi- different times (at least for me). If you get psi- by mis
take you can go back to the algebra section and make the opposite choi
ce. This should not happen, but sometimes it does!" }}}}}}{MARK "7" 0 
}{VIEWOPTS 1 1 0 3 2 1804 }