{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "" 1 14 62 1 204 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 19 1 248 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 33 "Contour Plots and Critica l Points" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Load the plotting package." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w ith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 40 "Part 1. Exploration of a Sample Surface" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Enter the definition of t he function f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f := (x,y) -> cos(x-y)*(1-x^2+y)*exp(x*y^2/(1+x^2*y^2 ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Construct a contou r plot of f on the domain [-3,3] x [-5,5], and enter your answers f or Step 1.." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "contourplot(f(x,y),x=-3..3,y=-5..5, contours=15, colo r=blue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 192 "Use the following commands to zoom in smaller portions of the contour plot. \+ The plotted rectangle is [a,b] x [c,d]. Change the values of a, b, c , and d to get the rectangles of your choice." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "a:=0; b:=1; c:=0; d :=1;\ncontourplot(f(x,y),x=a..b, y=c..d, contours=15, color=green);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Enter your answers for \+ Steps 2-4 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "The following commands define the partial derivatives " }{XPPEDIT 18 0 "f[x];" "6#&%\"fG6#%\"xG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "f[y];" "6#&%\"fG6#%\"yG" }{TEXT -1 15 " as f unctions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "fx:=D[1](f);\nfy:=D[2](f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "Answer Steps 5 and 6 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 55 "P art 2. First- and Second-Degree Taylor Approximations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 152 "Define the second p artial derivatives. (We use colons here rather than semicolons, so tha t we do not waste time and space displaying the long formulas.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "fxx :=D[1,1](f):\nfxy:=D[1,2](f):\nfyy:=D[2,2](f):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "Enter a point (" }{XPPEDIT 18 0 "x[0];" " 6#&%\"xG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6#\" \"!" }{TEXT -1 13 ") with -3 < " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\" \"!" }{TEXT -1 16 " < 3 and -5 < " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6 #\"\"!" }{TEXT -1 5 " < 5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x0:=???;\ny0:=???;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Finish the following expression for the f irst-degree Taylor approximation to f at (" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6# \"\"!" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "L:=(x,y)-> ???;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 47 "Compare the contour plots of f and L near (" } {XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6#\"\"!" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "delta:=0.5;\nplot1:=conto urplot(f(x,y),x=x0-delta..x0+delta,y=y0-delta..y0+delta,color=red):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "plot2:=contourplot(L(x,y),x=x0-del ta..x0+delta,y=y0-delta..y0+delta,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "display(plot1,plot2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 50 "Extend L to the quadratic approximation Q at ( " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6#\"\"!" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Q:=(x,y)->L (x,y)+1/2*(fxx(x0,y0)*(x-x0)^2+2*fxy(x0,y0)*(x-x0)*(y-y0)+fyy(x0,y0)*( y-y0)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Make a co ntour plot of the quadratic approximation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot3:=contourplot(Q(x,y) ,x=x0-delta..x0+delta,y=y0-delta..y0+delta,color=green):%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Show all three contour plots to gether." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display(plot1,plot2,plot3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 46 "Finish answering the questions in Part 2 here." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 37 "Part 3. Locating the Cr itical Points" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Your functions " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "f[x];" "6#&%\"fG6#%\"xG" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "f[y];" "6#&%\"fG6#%\"yG" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "f[xx];" "6#&%\"fG6#%#xxG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "f[yy];" "6#&%\"fG6#%#yyG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT -1 8 ", and " }{XPPEDIT 18 0 "Q;" "6#%\"QG" }{TEXT -1 18 " are all defined " }{TEXT 260 12 "as functions" }{TEXT -1 39 " , using values of the derivatives at (" }{XPPEDIT 18 0 "x[0];" "6#&% \"xG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6#\"\"! " }{TEXT -1 42 ") as coefficients in the definitions of " }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "Q;" "6#%\"QG " }{TEXT -1 40 ". Thus, when you change the values of " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y [0];" "6#&%\"yG6#\"\"!" }{TEXT -1 61 ", all of the function definitio ns get updated automatically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "Choose one of your approximate critical v alues, and enter its coordinates as " }{XPPEDIT 18 0 "x[1];" "6#&%\"x G6#\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y[1];" "6#&%\"yG6#\" \"\"" }{TEXT -1 143 ". The reason for this choice of names will appear very soon. In particular, the next thing that happens is that these \+ values get assigned to " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x1:=???; y1:=???;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "(*) This step is where you return when you start each new iteratio n. Transfer the values of " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\" " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y[1];" "6#&%\"yG6#\"\"\"" } {TEXT -1 6 " to " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6#\"\"!" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x0:=x1; y0:=y1;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "So lve for the critical point of Q." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solns:=solve(\{diff(Q(x,y),x)=0.,di ff(Q(x,y),y)=0\},\{x,y\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "x1:=e valf(subs(solns,x));\ny1:=evalf(subs(solns,y));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 89 "Check the function value and the values o f the two partial derivatives for the new point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalf(f(x1,y1));\ne valf(fx(x1,y1));\nevalf(fy(x1,y1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 40 "Plot the contours of f and Q near (" } {XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6#\"\"!" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "plot4:=contourplot(f(x,y ),x=x0-0.1..x0+0.1, y=y1-0.2..y1+0.2, color=red):\nplot5:=contourplot( Q(x,y),x=x0-0.1..x0+0.1, y=y1-0.2..y1+0.2, color=blue):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "display(plot4,plot5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 141 "Go back to (*), and iterate these steps \+ until you are sure you have the coordinates of the critical point corr ect to five significant digits." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "Check to see what so rt of critical point you have." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "evalf(fxx(x1,y1));\nevalf(fxx(x1,y1 )*fyy(x1,y1)-fxy(x1,y1)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Now return to the start of Part 3, change the starting point, a nd locate a saddle point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "Go back to Part 1 to find an a pproximate location for another max/min point. Use the coordinates to start the interation process again, and find the location of the extr eme point to 5SD." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 16 "Part 4. Summary" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Answer the summary questions here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 3 0" 12 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }