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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "                          \+
                                  " }{TEXT 256 23 "      SOLVING EQUAT
IONS" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "
We first introduce in this section the function notation as distinct f
rom the expression notation.  If" }}{PARA 0 "" 0 "" {TEXT -1 103 "you \+
are given a function and wish to operate on it (evaluate, graph, diffe
rentiate, etc.) it is usually" }}{PARA 0 "" 0 "" {TEXT -1 99 "advisabl
e to use the function notatation which assigns to an independent varia
ble x, or t, or u, or" }}{PARA 0 "" 0 "" {TEXT -1 95 "whatever, a  val
ue f(x), or g(t), or Q(u), etc.  Consider the polynomial P(x)=x^2-5*x-
14 and we" }}{PARA 0 "" 0 "" {TEXT -1 23 "write it as a function:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "P:=x->x^2-5*x-14;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGR6#%
\"xG6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"#\"\"\"\"\"\"F/!\"&!#9F2F(F(F
(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Now evaluate it at x=1.75." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "P(1.75);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$!'vo>!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "N
ow find its roots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve
(P(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"#\"\"(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 91 "The expression =0 may be omitted from the
 solve command in which case it is understood that" }}{PARA 0 "" 0 "" 
{TEXT -1 96 "you are solving P(x)=0.  If the right hand side is not ze
ro then make the necessary adjustments." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(P(x)=4,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"\"&\"\"#\"\"\"*$-%%sqrtG6#\"#(*\"
\"\"#F'F&,&F$F'F(#!\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The \+
solve( ) command will only find exact roots.  If it can't find any it \+
returns no answer or an" }}{PARA 0 "" 0 "" {TEXT -1 27 "unusable one. \+
 For instance" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(P(x)
=4*sin(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,*-%$sinG
6#%#_ZG\"\"%*$)-F*F)\"\"#\"\"\"!\"\"F.\"\"(\"#6\"\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 43 "Now we write a polynomial as an expressio
n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "P:=x^2-7*x-11;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,(*$)%\"xG\"\"#\"\"\"\"\"\"F(!\"
(!#6F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "P(1.75);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,(*$)-%\"xG6#$\"$v\"!\"#\"\"#\"\"\"\"\"\"F&!
\"(!#6F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The evaluation must b
e done differently and requires more effort." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "subs(x=1.75,P);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$!'v=?!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The solve comman
d can be used as before." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 
"solve(P,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"\"(\"\"#\"\"\"*$-
%%sqrtG6#\"#$*\"\"\"#F'F&,&F$F'F(#!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+\"Q
D=K)!\"*$!+\"QD=K\"F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The comm
and evalf( ) uses the very accurate floating point arithmetic.  It mus
t be used whenever " }}{PARA 0 "" 0 "" {TEXT -1 95 "irrational numbers
 occur and we recommend it be used  at all times.  To go directly to d
ecimal " }}{PARA 0 "" 0 "" {TEXT -1 50 "approximations of roots use th
e fsolve( ) command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fso
lve(P,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+!QD=K\"!\"*$\"+!QD=K)F
%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "However, fsolve( ) will only
 find real roots, and it produces approximate values." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Q:=x->x^3+5*x^2+6*x+8;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"QGR6#%\"xG6\"6$%)operatorG%&arrowGF(,**$)9$
\"\"$\"\"\"\"\"\"*$)F/\"\"#F1\"\"&F/\"\"'\"\")F2F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fsolve(Q(x),x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$!\"%\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "solve(Q(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"%,&#!\"\"\"\"
#\"\"\"*&%\"IGF(-%%sqrtG6#\"\"(\"\"\"#F(F',&F%F(F)F%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6%$!\"%\"\"!,&$!+++++]!#5\"\"\"%\"IG$\"+cc(GK\"!\"*,&F'F*F+$!+cc(GK\"
F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "If solving an equation is \+
a one shot deal we can conveniently type in the function in the solve(
 ) command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(x^4-x^
3-x^2+3*x-6,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&,&#\"\"\"\"\"#F%*&%
\"IGF%-%%sqrtG6#\"\"(\"\"\"F$,&F$F%F'#!\"\"F&*$-F*6#\"\"$F-,$F1F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6&,&$\"+++++]!#5\"\"\"%\"IG$\"+cc(GK\"!\"*,&F$F'F($!+cc
(GK\"F+$\"+330K<F+$!+330K<F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "
However, if the equation is not a polynomial then the fsolve( ) comman
d, which initiates a numerical " }}{PARA 0 "" 0 "" {TEXT -1 103 "zero \+
finding scheme,  must be used.  For instance,  to find the intersectio
n point of the graphs of 2*x" }}{PARA 0 "" 0 "" {TEXT -1 66 "and cos(x
) we must solve the transcendental equation cos(x)-2*x=0." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(cos(x)-2*x,x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,&%#_ZG\"\"#-%$cosG6#F'!\"\"" }}}
{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 22 "We must use fsolve( )." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "fsolve(cos(x)-2*x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+8h$
=]%!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "                                  \+
                               SOLVING AND GRAPHING" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "When using the fsolve(
 ) command to find real roots some graphing may be needed since the fs
olve( ) " }}{PARA 0 "" 0 "" {TEXT -1 28 "command is very nearsighted.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "PROBL
EM: Find the zeros of the following function." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "g:=x->x-x^2+x^4/(10*x+4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(,(9$\"\"\"*$)F-
\"\"#\"\"\"!\"\"*&*$)F-\"\"%F2F2,&F-\"#5F7F.!\"\"F.F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fsolve(g(x),x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "The fs
olve( ) command has only found the obvious zero.  A little analysis wi
ll convince us there are " }}{PARA 0 "" 0 "" {TEXT -1 49 "two more pos
itive zeros and we use some graphing." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "plot(g(x),x=0..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 475 
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wn we can give ranges for fsolve( ) to find the other zeros." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve(g(x),x=0.8..1.2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z]B'3\"!\"*" }}}{EXCHG {PARA 0 "> \+
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{XPPMATH 20 "6#$\"+B)H$4$*!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
101 "If we only need rough estimates of the zeros we can box the graph
s then put the cursor on the x-axis " }}{PARA 0 "" 0 "" {TEXT -1 43 "c
rossing and read off an approximate value." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "PROBLEM: Find the eighth positi
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{PARA 0 "" 0 "" {TEXT -1 73 "simultaneously and the easy way is the co
mmand plot(\{f(x),g(x)\}, x=a..b)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "plot(\{sin(x),x/40\},x=0..50);" }}{PARA 13 "" 1 "" 
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0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "f:=x->sin(x)-x/40=0;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(/,&-%$sinG6#9$\"\"\"F1#!
\"\"\"#S\"\"!F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Note that \+
we inserted the =0 term - this is good practice." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "solve(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'RootOfG6#,&%#_ZG\"\"\"-%$sinG6#F'!#S" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "fsolve(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 100 "As expected, nothing happened except the obvious.
  Box the graph and use the cursor to estimate the " }}{PARA 0 "" 0 "
" {TEXT -1 82 "x-coordinate of the intersection point of the two graph
s, then give a range for x." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "fsolve(f(x),x,x=24..26);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/2
\\$e#!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "PROBLEMS" }}{PARA 0 "
" 0 "" {TEXT -1 47 "1.  Find all roots of the following polynomials" }
}{PARA 0 "" 0 "" {TEXT -1 64 "       a) x^3-7*x^2+14*x-6    b) x^6-1  \+
  c) 5*x^4-12*x^2+8*x-1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 68 "2.  Find all zeros of the function 18*x^5-8*x^3+18
*x^2-2-2*sin(x)^2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 71 "3.  Find all intersection points of the graphs of y=x^2 a
nd y=sin(6*x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 100 "4.  A fixed point of a function f(x) is a solution of th
e equation x=f(x).  Find the fixed points of" }}{PARA 0 "" 0 "" {TEXT 
-1 32 "       a) 2^(-x)    b) cos(x^2)." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 101 "5.  Find a value k>0 for which exp(
x^2/k) will have a fixed point.  Estimate the smallest value of k." }}
}}{MARK "0 3 0" 41 }{VIEWOPTS 1 1 0 1 1 1803 }