{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 21 "Limits and Continuity" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Maple Worksheet for Section 2.2 of Marsden and Tromba" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 28 "Prepared by Douglas B. Meade" }}{PARA 0 "" 0 "" {TEXT -1 15 "17 January 1997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "URL: http://www.math.sc.edu/~meade/math550-S97/maple /sec2-2.mws" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Objectives" }} {EXCHG {PARA 15 "" 0 "" {TEXT -1 87 "provide plots of functions that a ppear in selected Examples and Problems in section 2.2" }}{PARA 15 "" 0 "" {TEXT -1 68 "provide a graphical understanding of plots in more t han one variable" }}{PARA 15 "" 0 "" {TEXT -1 77 "demonstrate how Mapl e can be used to examine functions along specified curves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( p lots );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "setoptions3d( axes=BOXED );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Example 13" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := sin( x^2 + y^2 ) / (x^2+y^2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "plot3d( f, x=-3..3, y=-3..3, title=`Figure 2.2.17` \+ );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Example 15" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "f(x,y) = x^2/(x^2+y^2)" "/-%\"fG6$%\"xG%\"yG*&F &\"\"#,&*$F&\"\"#\"\"\"*$F'\"\"#F-!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := x^2 / ( x^2 + y^2 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Graphical" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plo t3d( f, x = -2 .. 2, y = -2 .. 2, title=`Figure 2.2.18a` );" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "To see some of the different beha viors that this funciton exhibits as x and y approach 0, rotate t he plot to different perspectives." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Algebraic" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "Another source of information ab out limits is to examine different paths towards the limit point (here , the origin). The simplest two curves approaching the origin are the \+ " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 6 "- and " }{XPPEDIT 18 0 "y " "I\"yG6\"" }{TEXT -1 13 "-axes, i.e. " }{XPPEDIT 18 0 "y=0" "/%\"yG \"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "x=0" "/%\"xG\"\"!" } {TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "xaxis := y = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs( xaxis, f );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "yaxis := x = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs( yaxis, f );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "These show that the function is constant along each axis . Since the constants are not equal, the limit cannot exist." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "But, we can go \+ further. The next few commands define additional paths through the ori gin:" }}{PARA 0 "" 0 "" {TEXT -1 10 "the line " }{XPPEDIT 18 0 "y=x" "/%\"yG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "line1 := y \+ = x;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "the line with slope " } {XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 7 " (and " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 39 "-intercept 0 -- why is this necessary?)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "linem := y = m * x;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "the parabolas with vertex at the o rigin" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "parab := y = a * x ^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "the parabolas with one roo t at the origin" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "parab2 : = y = a * x^2 + b * x;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "invest igate the behavior of the function along each of these classes of curv es by modifying the first argument to the " }{TEXT 19 4 "subs" }{TEXT -1 8 " command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "F := simp lify( subs( parab, f ) );" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Not e about " }{TEXT 19 8 "simplify" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "the use of " }{TEXT 19 8 "simplify" }{TEXT -1 138 " \+ ensures that all possible cancellations are made; it's not always need ed, but it doesn't hurt either (try some of the curves without the " } {TEXT 19 8 "simplify" }{TEXT -1 9 " command)" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "b) " }{XPPEDIT 18 0 "f(x,y) = 2*x^2*y/(x^2+y^2)" "/-%\"fG6$%\"xG%\"yG**\"\"#\"\"\"*$F&\"\"#F*F'F*,& *$F&\"\"#F**$F'\"\"#F*!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "f := 2 * x^2 * y / ( x^2 + y^2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "G raphical" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot3d( f, x = - 2 .. 2, y = -2 .. 2, title=`Figure 2.2.18b` );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Do you think the limit exists at the origin?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 9 "Algebraic" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 301 "We can investigate the limits along simple paths as before. Note however, th at in many cases the function does not simplify to a constant. Thus, i t will be necessary to actually compute some limits. While most of the se limits are easily computed by hand, I'll also demonstrate how Maple could be used." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "F := sim plify( subs( parab, f ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "lim := Limit( F, x=0 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "lim = value( lim );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "Do no t make the mistake of saying that the limit exists because the same li mit is obtained for each of the paths considered here. This technique \+ can " }{TEXT 270 5 "never" }{TEXT -1 71 " be used to show the limit ex ists. It can be used to show a limit does " }{TEXT 271 3 "not" }{TEXT -1 68 " exist (if two different paths lead to two different values) an d to " }{TEXT 272 7 "suggest" }{TEXT -1 24 " the value of the limit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Probl em 15" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "sin ( x+y ) / (x+y)" "*&-%$sinG6#,&%\"xG\"\"\"%\"yGF(F(,&F'F(F)F(!\"\"" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := sin( x + y ) / ( x + y );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot 3d( f, x = -2..2, y = -2 .. 2, title=`Problem 15a` );" }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 45 "Is this an accurate picture of this func tion?" }}{PARA 15 "" 0 "" {TEXT -1 74 "Is there a curve through the or igin on which this function does not exist?" }}{PARA 15 "" 0 "" {TEXT -1 66 "What does this say about the existence of the limit at the orig in?" }}{PARA 15 "" 0 "" {TEXT -1 78 "(What does this say about the acc uracy of the answer in the back of the book?)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "b) " } {XPPEDIT 18 0 "x*y/(x^2+y^2)" "*(%\"xG\"\"\"%\"yGF$,&*$F#\"\"#F$*$F%\" \"#F$!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f := x * y / ( x^2 + y^2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot3d( f, x = -2 .. 2, y = -2 .. 2, title=`Problem 15b` );" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Proble m 16" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 18 0 "( sin (2*x) - 2*x ) / x^3" "*&,&-%$sinG6#*&\"\"#\"\"\"%\"xGF)F)*&\"\"#F)F*F) !\"\"F)*$F*\"\"$F-" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "This is a problem in one-variable calculus. Here's the function an d a reasonable plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f : = ( sin( 2*x ) - 2*x ) / x^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot( f, x = -2 .. 2, view = [ -2 .. 2, -2 .. 0 ], title=`Proble m 16a` );" }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 45 "What value does thi s suggest is the limit as " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 15 " approaches 0? " }}{PARA 15 "" 0 "" {TEXT -1 40 "Can you verify th is result analytically?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "b) " }{XPPEDIT 18 0 "( sin( \+ 2*x ) - 2*x + y ) / ( x^3 + y )" "*&,(-%$sinG6#*&\"\"#\"\"\"%\"xGF)F)* &\"\"#F)F*F)!\"\"%\"yGF)F),&*$F*\"\"$F)F.F)F-" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f := ( sin( 2*x ) - 2*x + y \+ ) / ( x^3 + y );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot3d( f, x=-2..2, y=-2..2, view=[-2..2,-2..2,-10..10], title=`Problem 16b` \+ );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "This plot is not so nice. T he worst behavior occurs along a single path." }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 18 "What is this path?" }}{PARA 15 "" 0 "" {TEXT -1 75 " How does this function behave along lines through the origin? on parab olas?" }}{PARA 15 "" 0 "" {TEXT -1 35 "Does the limit exist at the ori gin?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 }