{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "File: s1-4.mws Date: 3-s ep-2004 By: bob jantzen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 34 "Graphing Calculators and Computers" }} {PARA 18 "" 0 "" {TEXT -1 81 "[being an intelligent user in plotting f unctions and choosing the viewing window]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "finding the graph? [plus \+ example 3]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x->sqrt(x-2 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(sqrt(x-20),x); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The default horizontal window " }{XPPEDIT 18 0 "-10 .. 10;" "6#;,$\"#5!\"\"F%" }{TEXT -1 143 " is n ot in the domain! nothing to plot! hence the error message. But 20 is \+ in the domain where the function is zero, but this is the right edge" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(sqrt(x-20),x=0..20); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "By looking at the guts of the plot, you see that single point " }{XPPEDIT 18 0 "[20, 0];" "6#7$\"#? \"\"!" }{TEXT -1 104 " but apparently Maple wants at least two points \+ to connect together to make a CURVE, so you see nothing." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plt:=plot(sqrt(x-20),x=0..20);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(sqrt(x-20),x=19..30,-3. .3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 267 "The beginning is not con nected to the horizontal axis as it should (make the plot live by clic king on it and then click on the points option from the toolbar to see the sampled points that are then connected smoothly); we increase the sampling from the default of 49:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(sqrt(x-20),x=19..30,-3..3,numpoints=100);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(x^3-49*x,x=-5..5,y=-5.. 5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(x^3-49*x,x=-10. .10,y=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(x^3 -49*x,x=-10..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Try this on e. Is anything going on near the origin? you have to zoom in to find o ut by changing the horizontal window:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot((100*x)^3-49*(100*x),x=-10..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "This is just the previous problem with t he input multiplied by 100, so it contracts the graph horizontally by \+ a factor of 100!" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "finding the right scale [example 4]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " plot(sin(50*x),x); # try \"points\" option" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(sin(50*x),x=-1.5..1.5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(sin(50*x),x=-2*Pi/50..2*Pi/50);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The period is just " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 90 " divided by 50, since the graph is contracted horizontally by multiplying the input by 50. " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 44 "function with 2 different s cales [example 5]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=x->s in(x)+1/100*cos(100*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " plot(f(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(f(x), x=-0.1..0.1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "You cannot see \+ both of these curving properties in the same plot because they occur a t such different scales." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 53 "fun ction with one scale but zeros on different scales" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "f:=x->0.01*x^3-x^2+5;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "plot(f(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(f(x),x=-100..100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(f(x),x=-400..400);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(x^3/100,x=-infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(x^3/100-x^2,x=-infinity..infin ity);\n# of course this is wrong: a bug, no doubt?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(f(x)=0,x);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 211 "In this case one zero is about 2 orders of magnitude b igger than the other two zeros, so one plot cannot show them all. Mapl e can solve third degree polynomial equations exactly, but it is not v ery enlightening:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve( x^3/100-x^2+5,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(% );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(fnormal,[%]);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "That was just a trick to remove \+ the small complex numerical error. But it still did not remove the zer o imaginary part, so we could simply take the real part explicitly:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "map(Re,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(f(x)=0,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 290 "Notice that these all differ by a change in th e final digit (803 versus 799 is a difference of 4 in the final digit) because different procedures were used to evalulate these numbers and hence the last digit is unreliable because of truncation error that o ccurred differently along the way." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 58 "families of functions depending on a parameter [example 8]" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "If an expression has any unevaluat ed constants left in it, MAPLE cannot plot it:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "f:=x->x^3+c*x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-2..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Explore a parameter range by a discrete family of representive functions, namely a sequence of functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "seq(f(x),c=[-2,-1.5,-1,-0.5,0,0.5,1,1.5,2]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot([%],x=-2..2,-2..2);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Click on the graph and Play from the VCR \+ button:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "animate(x^3+t*x, x=-2..2,t=-2..2,view=[-2..2,-2..2]);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "intersecting graphs [example 9]" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 143 "When looking for numerical solutions of equations, one must always look first graphically to decide which interval to look f or a specific root." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot ([x,cos(x)],x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot([x, cos(x)],x=0.5..0.9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plo t([x,cos(x)],x=0.73..0.74);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fsolve(x-cos(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " plot([x,tan(x)],x,-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fsolve(x-tan(x)=0,x=2..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 257 66 "The Failures of App roximating by Sampling (Student Calc 1 package)" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "with(Student[Calculus1]):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 222 "One reason for studying derivatives is to get qua litative information about a function. The easiest way to sketch a fu nction is to sample it at a number of points and connect the dots. Fo r example, sampling the function " }{XPPEDIT 18 0 "sin(12*x)" "6#-%$si nG6#*&\"#7\"\"\"%\"xGF(" }{TEXT -1 15 " at the points " }{TEXT 256 1 " x" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "0, 1, 2, 3, 4" "6'\"\"!\"\"\"\"\" #\"\"$\"\"%" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "5" "6#\"\"&" }{TEXT -1 178 " suggests the following approximation (shown in blue). Knowing that the sine function oscillates, you may be satisfied with this res ult. The actual expression is plotted in red." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "PointInterpolation(sin(12*x), x=0..5 );" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "In the following example, the glo bal cubic behavior is very well approximated by the sampling, but the \+ asymptote at " }{XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 11 " is missed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Point Interpolation( (x^4 - 2*x^3 - 3*x^2 + 3*x + 1)/(x + 1), x=-6..6 ); " } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "In other cases, some of the beh avior of the expression occurs outside the sampling region. The follow ing misses that the expression goes to " }{XPPEDIT 18 0 "infinity" "6# %)infinityG" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 19 " increases and not " }{XPPEDIT 18 0 "-infinity" "6#,$%)infinity G!\"\"" }{TEXT -1 22 " as the plot suggests." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "PointInterpolation(x^4-3*x^3-x+3, x=-2..2);" }}} }}{MARK "0 0 0" 50 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }