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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 15 "Maple Project 2" }}
{PARA 0 "" 0 "" {TEXT -1 209 "This lab we will fool around with expone
ntials and logs.  One thing to keep in mind: it might be a good idea t
o save your worksheet from time to time, in case your computer locks u
p or won't print or whatever." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 95 "First let's introduce a few functions.  H
ere is how we write the exponential function in Maple:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "exp(x);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 21 "Here is the number e:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "exp(1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Maple
 doesn't feel like giving us a decimal.  Whenever we want to convert s
ome numerical expression to a decimal, we can use the \"evalf\" comman
d." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(exp(1));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "If we want to see e evaluated to 2
0 decimal places, we simply write:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "evalf(exp(1),20);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 69 "Let's just assign this value to the letter \"e\" so we can refe
r to it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "e:=exp(1);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "The inverse of the exponential, th
e natural logarithm, is written as we would expect:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 6 "ln(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 68 "It's a good idea to remember these 2 simple values of the logar
ithm:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ln(1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ln(e);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 90 "So now let's try a few things.  First, we can get Maple t
o do limits.  We can verify that " }{XPPEDIT 18 0 "limit((1+x)^(1/x),x
 = 0);" "6#-%&limitG6$),&\"\"\"\"\"\"%\"xGF)*&\"\"\"F)F*!\"\"/F*\"\"!
" }{TEXT -1 16 " = e  by typing:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "limit((1+x)^(1/x),x=0);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 183 "The first thing in the parentheses is the function we wi
sh to take the limit of; the next is the number we want to approach.  \+
Simple, no?   We can easily verify our homework problem: " }{XPPEDIT 
18 0 "limit((exp(3*x)-exp(-3*x))/(exp(3*x)+exp(-3*x)),x = -infinity)" 
"6#-%&limitG6$*&,&-%$expG6#*&\"\"$\"\"\"%\"xGF-F--F)6#,$*&\"\"$F-F.F-!
\"\"F4F-,&-F)6#*&\"\"$F-F.F-F--F)6#,$*&\"\"$F-F.F-F4F-F4/F.,$%)infinit
yGF4" }{TEXT -1 73 "  =  -1.  First we will use the \"limit\" command \+
with a capital \"L\"; this " }}{PARA 0 "" 0 "" {TEXT -1 196 "will spit
 the limit back out at us so we can verify if we typed in the correct \+
expression or not.  Note how the parentheses are used in the command s
o that fraction is in the form we want it.     " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 61 "Limit((exp(3*x)-exp(-3*x))/(exp(3*x)+exp(-3*x)
),x=-infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 329 "We can see this, of course, \+
in the behavior of the graph.  (Save yourself some time with this next
 command: highlight the \"Limit\" command above with your cursor and u
se the \"Copy\" command in the file menu to copy it, then use the \"Pa
ste\" command in the same menu to paste it where you want it, then jus
t make some minor changes.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "plot((exp(3*x)-exp(-3*x))/(exp(3*x)+exp(-3*x)),x=-20..20);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "From this plot we can also see th
at the limit as x goes to infinity is 1 (which can also easily be chec
ked with the \"limit\" command)..  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 183 "It is instructive to see what happens i
f we don't put in the parentheses correctly in our limit command; the \+
whole thing will be screwed up.  Watch out for that stuff!  Whoooaaa!!
!!  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Limit(exp(3*x)-exp(
-3*x)/exp(3*x)+exp(-3*x),x=-infinity);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 194 "That certainly isn't what we wanted.  Anyway, just remem
ber: proper placement of parentheses is positively paramount to the pe
rfection of your project.  Next: we know Maple can take derivatives." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(x^3*exp(x),x);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Another example:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "f:=ln(x-3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "diff(f,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 180 "B
ut what if I asked you what its 100th derivative was?  What would you \+
do then, huh?  You think you're so smart, eh?  One way to do it is jus
t use the \"$\" notation in our command. " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "diff(f,x$100);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
311 "That's the hundreth derivative.  To get any other derivative, say
 the third derivative, you would just use \"x$3\" in the above command
.  THere is also another way to do this.  First define ln(x-3) as a FU
NCTION.  (What we defined above as \"f\" is just an expression.  A fun
ction we can actually plop values into.)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "f:=x->ln(x-3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
77 "Now we type the following expression, to get the derivative of thi
s function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The second derivative is:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(D@@2)(f)(x);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 24 "Its 100th derivative is:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "(D@@100)(f)(x);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 260 "The neat thing about this way of doing it is we \+
can just plop values right into the derivative if we want, which we co
uld not do with the \"diff\" command.  To EVALUATE the derivative at a
 particular value, say at x=2, instead of typing \"D(f)(x)\" we simply
 type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The same goes for any other deriva
tive:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(D@@2)(f)(2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "(D@@100)(f)(2);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 84 "If you're not flabbergasted at the sheer \+
power of Maple, you should leave right now." }}{PARA 0 "" 0 "" {TEXT 
-1 81 "Start a new worksheet, put your name at the top.  Do these prob
lems, why dontcha." }}{PARA 0 "" 0 "" {TEXT -1 29 "1)  The question is
, what is " }{XPPEDIT 18 0 "limit(ln(ln(x))/ln(x),x = infinity);" "6#-
%&limitG6$*&-%#lnG6#-F(6#%\"xG\"\"\"-%#lnG6#%\"xG!\"\"/%\"xG%)infinity
G" }{TEXT -1 94 "  ?  If wer're lazy, we could just plot it and try to
 eyeball the limit.  Plot the curve y =  " }{XPPEDIT 18 0 "ln(ln(x))/l
n(x);" "6#*&-%#lnG6#-F%6#%\"xG\"\"\"-%#lnG6#%\"xG!\"\"" }{TEXT -1 348 
" on the interval [2,30].  What's your guess on what the limit might b
e?  Now plot it on [2,100].  What's your guess now?  Plot it on [2,100
0].  How 'bout now?  Now actually calculate the limit using Maple.  Th
e moral of the story is, you can't really tell what the function is do
ing as x gets large by looking at small values of x; it can fool you.
" }}{PARA 0 "" 0 "" {TEXT -1 586 "2) Find the equation of the tangent \+
line to the curve y = ln (2x+1) at x=20, and plot this curve and the l
ine on the same graph over the interval [1,50].  DO EVERYTHING WITH MA
PLE COMMANDS, AND MAPLE COMMANDS ONLY.  (Hint: take this one step by s
tep.  First define ln(2x+1) as a function.  Then think about what you \+
need to get the equation of a line.  You need: 1)  a slope, which can \+
be found by taking a derivative and plugging in your x-value, and  2) \+
your point, of which you already have the x-value, so just find the y-
value.  Then write your equation of a line and do the plot.)" }}}}
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