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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 9 "Functions" }}{PARA 257 "
" 0 "" {TEXT -1 12 "March 2005\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 252 "In the worksheet on Fundamentals of Maple we already say a lis
itng of elementary functions, and how to invoke them in maple.  In the
 sequel we shall revisit this list, but before we do so let's look at \+
the fundamentals of the definition of a function. " }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 113 "Remember to use the sequence -> to defin
e a function, which is a dash followed by an inequality symbol.  Examp
le:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := x -> x + 1/(1+x^2);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Once the function is defined, you
 may perform all kinds of operations with it.  Examples:" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "f(x);  #  the output from f" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "f(2*x);  # horizontal compression" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f(t);  # the name of the ind
ependent variable is immaterial  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "(f(x+h) - f(x))/h;  # Difference Quotient" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 57 "Polyn
omials, Rational Functions, Roots and Absolute Value" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 134 "For polynomials it is important to remember to include the sta
r (*) for multiplication; carrets (^) are used for exponents.  Example
s:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p := x -> 3*x^3 - 4*x^2 + 6*x
 - 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "factor(p(x));  # j
ust because it can be factored" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "q := x -> (x-1)*(x+3)*x^2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "expand(q(x));  # 'expand' is the opposite of 'factor'
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Rational functions are equall
y simple to deal with." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r := x ->
 (3*x -5)/(4- 25*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r(x)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
30 "The square root is entered as " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "f:= x -> sqrt(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For othe
r order roots use" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := x -> root
[4](x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f(x);  # check\n
f(625);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 15 "Absolute value:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "f := x -> abs(x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "abs(-4);  # check" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Exponential
s and Logarithms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The generic function  " }
{XPPEDIT 18 0 "a^x;" "6#)%\"aG%\"xG" }{TEXT -1 20 "  can be entered as
 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f := x ->  a^x;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 56 "and you can actually do some math with it
.  For example " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f(1/2);\nf(t)*f(
s); \nsimplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "If you want
 to temporarily use a specific value for a, say a=5/2, you may substit
ute this value:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(a=5/2,f(x))
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "You can also define " }
{XPPEDIT 18 0 "f(x) = (5/2)^x;" "6#/-%\"fG6#%\"xG)*&\"\"&\"\"\"\"\"#!
\"\"F'" }{TEXT -1 13 "  directly.  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "f := x -> (5/2)^x;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "The nat
ural exponential function using the Euler constant e = 2.71828....  is
 given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x -> exp(x);" }}}
{EXCHG {PARA 258 "" 0 "" {TEXT -1 13 "This is wrong" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 35 "e^x; #  WARNING, don't to this !!!!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 123 "This does not work because maple conside
rs e just as any other constant, and it does not associate e = 2.71828
...  Compare:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "e^2.5;\nexp(2.5);
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 48 "The natural logarithm can be entered in two ways" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ln(x);\nlog(x);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 47 "A logarithm with a different base is entered as" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> log[4](x);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "f(32);  " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "4^%;  # Check\nsimplify(%);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 176 "Logs are inverses of exponential functions.  For th
is demo we have to make the assumption that we are dealing with real n
umbers (since maple also works for complex arithmetic):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "assume(x,real);\nln(exp(x));\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "assume(a,positive);\na^x;\nlog[a](%
);\nsimplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Trigonometric Functions" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 72 "When dealing with trigonometric functions it is ve
ry important to enter " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 23 " \+
 with an upper case P." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Pi;\npi; \+
# WRONG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "There doesn't appear \+
to be any difference among the two, but evaluation as floating point n
umbers will reveal the distinction:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "evalf(Pi);\nevalf(pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "We
 conclude that 'pi' is just some constant, while 'Pi' associates the v
alue 3.14159....  with that constant. " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 42 "The trig functions are entered as follows:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 47 "sin(x);\ncos(x);\ntan(x);\ncot(x);\nsec(x);\nc
sc(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Never enter the sine as
 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sin*x;  #  WRONG, don't do thi
s" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 272 "While it almost looks like \+
the sine function, 'sin*x' represents a constant, called sin, multipli
ed by x.  Keep in mind that sin(x) is a function, just like f(x), exce
pt that we use three letters (s-i-n) to represent it, rather than a si
ngle letter f.  Some more examples:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "f := x -> 2*cos(x) + sin(2*x);\nf(0);\nf(Pi/3);\nf(Pi);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "tan(x)/sec(x);\nsimplify(%);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Here is a way to enter  power
s of trig functions.   Example:  We all know that   1 - " }{XPPEDIT 
18 0 "sin^2;" "6#*$%$sinG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x;" "6
#%\"xG" }{TEXT -1 4 "  = " }{XPPEDIT 18 0 "cos^2;" "6#*$%$cosG\"\"#" }
{TEXT -1 2 "  " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 24 " .  To che
ck this, enter" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "1-sin(x)^2;\nsimp
lify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Or, for example  " }
{XPPEDIT 18 0 "sin^2;" "6#*$%$sinG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "Pi/3;" "6#*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 42 "   means to take t
he square of the number " }{XPPEDIT 18 0 "sin(Pi/3) = sqrt(3)/2;" "6#/
-%$sinG6#*&%#PiG\"\"\"\"\"$!\"\"*&-%%sqrtG6#F*F)\"\"#F+" }{TEXT -1 20 
"    .  Thus we enter" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sin(Pi/3)^
2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Inverse functions are denot
ed with the prefix 'arc'.  Examples:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "arcsin(x);\narctan(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "arcsin(1/2);  # angle with sine = 1/2" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 48 "arctan(sqrt(3));  # angle with tangent = sqrt(3)" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sin(Pi/6);  # Checks\ntan
(Pi/3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "tan(arcsin(x)); \+
 #  Draw a triangle to check this" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Piecewise F
unctions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "It's a little tricky to define pi
ecewise function.  The syntax is piecewise( condition1, expression1, c
ondition2, expressions2, ....., otherwise).  Let's look at an example:
   " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f := x -> piecewise( x<1, x^
2, x<=4, x-3, sqrt(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
f(x);  # display f(x)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "pl
ot(f, -2..8,discont=true);  # plot the function" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "f(1), f(4); # Test at the transition points" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The absolute value function as a
 piecewise function would be entered as " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "f := x -> piecewise(x<0,-x,x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "f(x) - abs(x);  # Check\nsimplify(%);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 1 0" 11 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }