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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "Basics on Functions and Plotting \+
in Maple" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Introduction" }}
{PARA 0 "" 0 "" {TEXT -1 220 "In this worksheet we will take a look at
 functions: see how a shift in the dependent or independent variable a
ffects the graph, and caution about using such sophisticated plotting \+
tools as Maple (or graphing calculators)" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 18 "Defining Functions" }}{PARA 0 "" 0 "" {TEXT -1 99 "In Map
le a function is defined analogously to the definition given in class,
 i.e., a function is a " }{TEXT 256 4 "rule" }{TEXT -1 47 " (or mappin
g) which assigns a specified value (" }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6
#%\"xG" }{TEXT -1 31 ") to the independent variable (" }{XPPEDIT 18 0 
"x" "6#%\"xG" }{TEXT -1 47 "). Maple's notation for defining the funct
ion  " }}{PARA 0 "" 0 "" {TEXT -1 20 "                    " }{XPPEDIT 
18 0 "f(x) = sin(x)/x-sqrt(x)/(x^2+1);" "6#/-%\"fG6#%\"xG,&*&-%$sinG6#
F'\"\"\"F'!\"\"F-*&-%%sqrtG6#F'F-,&*$)F'\"\"#F-F-\"\"\"F-F.F." }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "is" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "f:= x -> sin(x)/x - sqrt(x)/(x^2+1);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 27 "We can now try to evaluate " }{XPPEDIT 
18 0 "f" "6#%\"fG" }{TEXT -1 22 " at any point we wish:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 39 "f(1); f(2); f(3.5); f(Pi); f(0); f(-1);" }}}
{PARA 0 "" 0 "" {TEXT -1 41 "We see that Maple attempts to give us an \+
" }{TEXT 256 12 "exact answer" }{TEXT -1 20 " whenever possible. " }}
{PARA 0 "" 0 "" {TEXT -1 9 "Clearly, " }{XPPEDIT 18 0 "f(0)" "6#-%\"fG
6#\"\"!" }{TEXT -1 36 " is not defined, and in the case of " }
{XPPEDIT 18 0 "f(-1)" "6#-%\"fG6#,$\"\"\"!\"\"" }{TEXT -1 121 " Maple \+
treats the function as a complex-valued function, i.e., Maple views th
e square root of -1 as the imaginary number " }{XPPEDIT 18 0 "i" "6#%
\"iG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 179 "With some more \+
sophisticated programming (which we will not worry about) we could mak
e sure that Maple only returns real solutions, and complains if the fu
nction value is complex." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "We can
 also plot the graph of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 6 ", \+
e.g." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(f, 0..10);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 62 "or, if we pick a different range for the \+
independent variable," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f, 0.
.infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The command" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(f(x), x=0..10);" }}}{PARA 0 "
" 0 "" {TEXT -1 74 "does the same as the first plot command used - but
 since Maple knows that " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 90 " \+
is a function, it is not really necessary to specify the independent v
ariable explicitly." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Shifting \+
Graphs" }}{PARA 0 "" 0 "" {TEXT -1 109 "We can nicely visualize how a \+
shift in the dependent or independent variable affects the graph of a \+
function." }}{PARA 0 "" 0 "" {TEXT -1 48 "To this end we define a coup
le of new functions:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "h :=
 x -> f(x+a);\nv := x -> f(x) + a;" }}}{PARA 0 "" 0 "" {TEXT -1 33 "As
 you will remember from class, " }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT 
-1 18 " should result in " }{TEXT 256 10 "horizontal" }{TEXT -1 12 " s
hifts and " }{XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 4 " in " }{TEXT 
256 9 "vertical " }{TEXT 18 5 "ones." }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 43 "To illustrate the effects of the parameter " }{XPPEDIT 18 0 "a
" "6#%\"aG" }{TEXT -1 13 " we create a " }{TEXT 256 8 "sequence" }
{TEXT -1 46 " of functions. This is done in Maple with the " }{TEXT 
256 3 "seq" }{TEXT -1 9 " command." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "H := seq(h(x), a=-3..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "
Now we can graph all of these functions together in one plot. In order
 to do this the sequence has to be enclosed in curly brackets to turn \+
it into a set." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(\{H\}, x=0..
10);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "A Tough Function to Gra
ph" }}{PARA 0 "" 0 "" {TEXT -1 227 "The following is an example of a p
roblem where a straightforward use of Maple as a plotting tool is not \+
very helpful. In order to get a meaningful graph some Calculus knowled
ge (our luck and a lot of trial and error) is needed." }}{PARA 0 "" 0 
"" {TEXT -1 264 "The hydrogen atom is composed of one proton in the nu
cleus and one electron, which moves about the nucleus. In the quantum \+
theory of atomic structures, it is assumed that the electron does not \+
move in a well-defined orbit. Instead, it occupies a state known as an
 " }{TEXT 257 7 "orbital" }{TEXT -1 125 ", which may be thought of as \+
a \"cloud\" of negative charge surrounding the nucleus. At the state o
f lowest energy, called the " }{TEXT 258 12 "ground state" }{TEXT -1 
6 ", or 1" }{TEXT 260 9 "s-orbital" }{TEXT -1 182 ", the shape of this
 cloud is assumed to be a sphere centered at the nucleus. This sphere \+
is described in terms of the probability density function \n          \+
                        " }{XPPEDIT 18 0 "p(r) = 4/a[0]^3" "6#/-%\"pG6
#%\"rG*&\"\"%\"\"\"*$&%\"aG6#\"\"!\"\"$!\"\"" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 " r^2*exp(-2*r/a[0]))" "6#*&%\"rG\"\"#-%$expG6#,$*(\"\"#
\"\"\"F$F,&%\"aG6#\"\"!!\"\"F1F," }{TEXT -1 6 ",     " }{XPPEDIT 18 0 
"r>=0" "6#1\"\"!%\"rG" }{TEXT -1 8 ",\nwhere " }{XPPEDIT 18 0 "a[0]" "
6#&%\"aG6#\"\"!" }{TEXT -1 15 " is called the " }{TEXT 259 17 "first B
ohr radius" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "a[0] =5.29*10^(-11)" "6#/
&%\"aG6#\"\"!*&$\"$H&!\"#\"\"\")\"#5,$\"#6!\"\"F," }{TEXT -1 74 " m). \+
\nTry to create a meaningful plot of the probability density function \+
" }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 48 " which shows its local ma
ximum and behavior for " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 19 " a
pproaching 0 and " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 23 " approac
hing infinity.\n" }}{PARA 0 "" 0 "" {TEXT -1 31 "Here is how you would
 define p:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "a[0] := 5.29*1
0^(-11);\np := r -> (4/a[0]^3)*r^2*exp(-2*r/a[0]);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 74 "The following command is a valid Maple command - \+
but it doesn't work here." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(p
(r), r=0..infinity);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Suggest
ed Exercises" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.1:" }}{PARA 0 "
" 0 "" {TEXT -1 29 "a) What does the behavior of " }{XPPEDIT 18 0 "f" 
"6#%\"fG" }{TEXT -1 15 " seem to be as " }{XPPEDIT 18 0 "x" "6#%\"xG" 
}{TEXT -1 23 " tends toward infinity?" }}{PARA 0 "" 0 "" {TEXT -1 41 "
b) Try to create a graph of the function " }{XPPEDIT 18 0 "f" "6#%\"fG
" }{TEXT -1 35 " which illustrates the behavior of " }{XPPEDIT 18 0 "f
" "6#%\"fG" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 
22 " tends to zero better." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.
2:" }}{PARA 0 "" 0 "" {TEXT -1 15 "In the plot of " }{XPPEDIT 18 0 "H
" "6#%\"HG" }{TEXT -1 29 ", which curve corresponds to " }{XPPEDIT 18 
0 "a=0" "6#/%\"aG\"\"!" }{TEXT -1 44 ", i.e., the unshifted version, w
hich one to " }{XPPEDIT 18 0 "a=3" "6#/%\"aG\"\"$" }{TEXT -1 19 ", and
 which one to " }{XPPEDIT 18 0 "a=-3" "6#/%\"aG,$\"\"$!\"\"" }{TEXT 
-1 1 "?" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.3:" }}{PARA 0 "" 0 
"" {TEXT -1 65 "Repeat the visualization of the shifted graphs with th
e function " }{XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 1 "." }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 5 "Ex.4:" }}{PARA 0 "" 0 "" {TEXT -1 32 "Crea
te the \"meaningful plot\" of " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT 
-1 17 " mentioned above." }}}}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 }