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-1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 331 26 "Three-Dimensional Graph ics" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "There are several package s of programs in Maple that we will find useful. For many calculus op erations we will need \223" }{TEXT 320 7 "student" }{TEXT -1 42 "\224. For vector operations the package is \223" }{TEXT 321 6 "linalg" } {TEXT -1 146 "\224. One of the real strengths of Maple is its ability to graph curves and surfaces in a three-dimensional coordinate system . We need the package \223" }{TEXT 322 5 "plots" }{TEXT -1 69 "\224 in order to do this. There are two basic ways to use the command \223" }{TEXT 323 6 "plot3d" }{TEXT -1 21 "\224. The first plots " } {XPPEDIT 18 0 "z =f (x,y)" "6#/%\"zG-%\"fG6$%\"xG%\"yG" }{TEXT -1 45 " , while the second and most versatile plots " }{XPPEDIT 18 0 "[x,y,z] " "6#7%%\"xG%\"yG%\"zG" }{TEXT -1 22 " parametrically with " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y" "6#% \"yG" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 " z " "6#%\"zG" }{TEXT -1 461 " as functions of two variables. The reader is encouraged to exe cute the commands being discussed and to try the suggestions to see th e effects that they have. It may be to your advantage to save the Map le work that you type in to test because you may be able to cut, paste , and edit them when you need similar entries later. Remember, always start at the top of a worksheet and hit all the way down if y ou have edited the worksheet. The command \223" }{TEXT 328 8 "resta rt:" }{TEXT -1 64 " \224 clears the Maple kernel of all internal memo ry. Some put \223" }{TEXT 324 8 "restart:" }{TEXT -1 65 "\224 on th e first line of a worksheet before any packages such as \223" }{TEXT 325 7 "student" }{TEXT -1 6 "\224 or \223" }{TEXT 327 5 "plots" } {TEXT -1 103 "\224 so that confusion is avoided if one \222s fr om the first line all the way through. Do NOT put \223" }{TEXT 326 8 "restart:" }{TEXT -1 115 "\224 on a line AFTER you have listed packa ges because that will erase the packages that you think have been incl uded.\n" }{MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot3d(x^2+y^2,x=-5..5,y=-5..5,axes=BOXED);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "implicitplot3d(x^2+y^2+z^2=1,x=-2..2,y=-2 ..2,z=-2..2,axes=boxed);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "implicitplot3d(x^2+y^2=z,x=- 2..2,y=-2..2,z=-2..2,axes=BOXED);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "implicitplot3d(\{x^2+y^2=z,y=1\},x=-2..2,y=-2..2,z=-2 ..2,axes=BOXED);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "implici tplot3d(\{x^2+y^2=z,x=1\},x=-2..2,y=-2..2,z=-2..2,axes=BOXED);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "The figure above shows the the 3d graph of z=x^2+y^2. (Do you know what this is?) We can see how the 'c ontour lines' are obained by slicing the graph at certain heights:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot3d(\{x^2+y^2,0,2,4\},x= -5..5,y=-5..5,axes=BOXED);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "Ca n you tell the what the graphs of the intersections among (z=0, z=x^2 +y^2), (z=2, z=x^2+y^2) and (z=4, z=x^2+y^2). These curves are calle d the level curves for z=f(x,y)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "implicitplot(\{x^2+y^2=0,x^2+y^2=2,x^2+y^2=4\},x=-2..2,y=-2..2); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Example: For f(x,y)=x^2-y^2. We will investigate the level curves for f as follows: x^2-y^2=0, x ^2-y^2=1, and x^2-y^2=-1 as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot3d(\{x^2-y^2,0,1,-1\},x=-2..2,y=-2..2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Now, we plot the level curves as f ollows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "implicitplot(\{x ^2-y^2=0,x^2-y^2=1,x^2-y^2=-1\},x=-2..2,y=-2..2);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 77 "The following examples are graphs, which have rest rictions on their domains. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 332 10 "Example 1:" }{TEXT -1 54 " Draw three faces o f the rectangular box defined by " }{TEXT 333 60 "[0 ,2]\327[1 ,3] \327[1 ,2]= \{(x,y,z ): 0<=x<=2 ,1<=y<=3, 1<=z<=2 \}" }{TEXT -1 53 " \+ and include coordinate axes. Let's start with the " }{TEXT 339 1 "x" }{TEXT -1 21 "-axis. The command \221" }{TEXT 340 10 "spacecurve" } {TEXT -1 117 "\222 has the parametric form of a curve as its argument, along with the range and choice of color. All we need here is " } {TEXT 334 7 "[t,0,0]" }{TEXT -1 73 " to generate a portion of the axis . We must suppress the output with a \221" }{MPLTEXT 1 0 1 ":" } {TEXT -1 84 "\222so that all the pieces may be displayed at one time. \+ In the face labelled \221A\222 the " }{TEXT 336 1 "y" }{TEXT -1 10 " \+ value is " }{TEXT 335 1 "1" }{TEXT -1 6 ", and " }{TEXT 337 1 "x" } {TEXT -1 5 " and " }{TEXT 338 1 "z" }{TEXT -1 67 " may vary. Analyze \+ the lines below and predict the output of each." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 325 "with(plots):\nxaxis:=spacecurve([t,0,0],t=0.. 3,color=black):\nyaxis:=spacecurve([0,t,0],t=0..3,color=black):\nzaxis :=spacecurve([0,0,t],t=0..3,color=black):\nA:=plot3d([x,1,z],x=0..2,z= 1..2,color=red):\nB:=plot3d([x,y,1],x=0..2,y=1..3,color=green):\nC:=pl ot3d([2,y,z],y=1..3,z=1..2,color=magenta):\ndisplay(xaxis,yaxis,zaxis, A,B,C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 10 "Example 2:" }{TEXT -1 39 " Draw the portion of the para boloid " }{XPPEDIT 18 0 "z = 4-x^2-y^2;" "6#/%\"zG,(\"\"%\"\"\"*$%\" xG\"\"#!\"\"*$%\"yGF*F+" }{TEXT -1 42 " that is over the quarter-disk of radius " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 24 " in the first quadrant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot3d(4-x^2- y^2,y=0..sqrt(4-x^2),x=0..2,color=blue);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Note how we kept " }{TEXT 257 1 "x" }{TEXT -1 22 " betwee n 0 and 2, but " }{TEXT 258 1 "y" }{TEXT -1 19 " was between 0 and " } {XPPEDIT 18 0 "sqrt(4-x^2)" "6#-%%sqrtG6#,&\"\"%\"\"\"*$%\"xG\"\"#!\" \"" }{TEXT -1 951 " . At this point you should have a three-cornered \+ sheet of blue lines appearing. Move the pointer onto the figure and c lick once. A rectangle should appear around the figure and a new set \+ of menu options are seen above. The button 1:1 adjusts the ratios of \+ the axes. There are four red spheres next to 1:1, click on each of th em and note how you get different ways of showing the axes on the figu re, with one option being no axes. There are 7 black spheres to the le ft of the four red ones. One at a time, click on each of the spheres. You may wish to end this sequence with the middle sphere. Now, clic k on the figure and hold down the left button of the mouse. Move the \+ mouse so as to move the pointer and note how the figure rotates. On t he left end of the line above with the spheres you will find two angle s displayed. As you rotate the figure the values of those angles chan ge and are displayed accordingly. The angle on the left is " } {XPPEDIT 18 0 "Theta" "6#%&ThetaG" }{TEXT -1 19 ", or in lower case " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 35 " , and measures rota tion about the " }{TEXT 259 1 "z" }{TEXT -1 14 "-axis. When " } {XPPEDIT 18 0 "Theta" "6#%&ThetaG" }{TEXT 260 3 "=0 " }{TEXT -1 25 "yo u are looking down the " }{TEXT 261 1 "x" }{TEXT -1 28 "-axis. The se cond angle is " }{XPPEDIT 18 0 "Phi" "6#%$PhiG" }{TEXT -1 19 ", or in \+ lower case " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 4 " or " } {TEXT 262 1 "j" }{TEXT -1 29 ", and measures how much the " }{TEXT 329 1 "z" }{TEXT -1 20 "-axis has deflected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "Phi" "6#% $PhiG" }{TEXT -1 28 "=0 you are looking down the " }{TEXT 263 1 "z" } {TEXT -1 22 "-axis from above.When " }{XPPEDIT 18 0 "Theta" "6#%&Theta G" }{TEXT -1 8 "=45 and " }{TEXT 285 1 "j" }{TEXT -1 18 "=75, you have the " }{TEXT 264 1 "x" }{TEXT -1 27 "-axis to your left and the " } {TEXT 265 1 "y" }{TEXT -1 39 "-axis to your right equally, while the \+ " }{TEXT 266 1 "z" }{TEXT -1 214 "-axis is tipped forward so as to giv e you the usual perspective one gets when sketching in 3-d. This will all make more sense to you after you have been introduced to cylindri cal and spherical coordinate systems.\n" }}{PARA 0 "" 0 "" {TEXT -1 362 "Before you move on, click carefully on the lower (right-hand) cor ner of the rectangle and drag it towards the opposite corner until you have a small square of about two inches, release and the figure will \+ redraw within the box. You are expected to reduce the size of your pl ots in the assignments so as to save paper. We would have gotten the \+ same result from\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot3 d([x,y,4-x^2-y^2],y=0..sqrt(4-x^2),x=0..2,color=blue);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "which is the parametric form of the same surface. The value of the parametric form is that vertical surfaces are easily handled, but of course they cannot occur as " }{TEXT 267 10 "z =f (x,y)" }{TEXT -1 416 ". Our fo cus here is on the quadric surfaces such as paraboloids, hyperboloids, ellipsoids, spheres, and cones. But in order to display them it is b est to learn how to show the result of cutting these \221solids\222wit h planes. When one variable is held constant we have simply intersect ed the figure with a plane that is parallel to the coordinate plane of the remaining two variables. The result is called a \221trace\222.\n " }}{PARA 0 "" 0 "" {TEXT 268 11 "Example 3: " }{TEXT -1 47 " Draw the solid figure bounded on the sides by " }{TEXT 269 4 " y=x" }{TEXT -1 2 ", " }{TEXT 270 9 " y=2 - x" }{TEXT -1 8 " , and " }{TEXT 271 3 "x =0" }{TEXT -1 11 ", below by " }{TEXT 272 3 "z=0" }{TEXT -1 16 ", and above by " }{XPPEDIT 18 0 "z=4-x^2-y^2" "6#/%\"zG,(\"\"%\"\"\"*$%\"xG \"\"#!\"\"*$%\"yGF*F+" }{TEXT -1 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 130 "We will use this same example to introduce double integrals later ,so a little effort here will be helpful. If you draw the lines " } {TEXT 273 3 "y=x" }{TEXT -1 5 " and " }{TEXT 274 7 "y = 2-x" }{TEXT -1 46 " in the first quadrant and then draw the line " }{TEXT 275 3 "x =0" }{TEXT -1 48 ", you will see that a triangle has been formed.\n" } }{PARA 0 "" 0 "" {TEXT 276 7 "BEWARE!" }{TEXT -1 9 " In the " }{TEXT 330 2 "xy" }{TEXT -1 16 "-plane the line " }{TEXT 277 3 "x=0" }{TEXT -1 20 " is vertical and is " }{TEXT 278 3 "NOT" }{TEXT -1 5 " the " } {TEXT 279 1 "x" }{TEXT -1 28 "-axis. Note that using the " }{TEXT 280 1 "x" }{TEXT -1 146 "-axis as a boundary defines a different trian gle. This error frequently occurs. Please observe the result of the \+ following plot on your screen.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot([x,2-x],x=0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 252 "The top line should be green and the bottom should be re d. If the triangle was blue then you would have a good view looking do wn on the solid. But the red and green lines will be edges of the ver tical surfaces. The top of our solid is the paraboloid " }{XPPEDIT 18 0 "z=4-x^2-y^2" "6#/%\"zG,(\"\"%\"\"\"*$%\"xG\"\"#!\"\"*$%\"yGF*F+ " }{TEXT -1 61 " . Keep this in mind when you consider the upper boun ds for " }{TEXT 281 1 "z" }{TEXT -1 82 " when plotting the sides. How do we know when a \221side\222is vertical? The variable " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 84 " will be missing from that equation. \+ We begin with a surface in the vertical plane " }{TEXT 282 3 "y=x" } {TEXT -1 15 " so that every " }{TEXT 283 1 "y" }{TEXT -1 16 " is repla ced by " }{TEXT 284 1 "x" }{TEXT -1 3 " .\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot3d([x,x,z],z=0..4-2*x^2,x=0..1,color=red);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 244 "You are probably a little puzzle d by the result - a vertical red line. Using what you learned up abov e, rotate the figure to the right 30 degrees or so. Now you should se e a surface. Maple orients the initial plot so that the vertical plan e " }{TEXT 286 3 "y=x" }{TEXT -1 39 " is directly towards the viewer, \+ i.e., " }{XPPEDIT 18 0 "Theta" "6#%&ThetaG" }{TEXT -1 46 "=45. If you examine our command you will see " }{TEXT 287 7 "[x,x,z]" }{TEXT -1 40 " which means our plot lies in the plane " }{TEXT 288 3 "y=x" } {TEXT -1 160 " since the first and second coordinates are the same. W e wish to add another plane to the situation and we will draw it separ ately before combining our plots.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot3d([x,2-x,z],z=0..4-x^2-(2-x)^2,x=0..1,color=gree n);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Because this surface res ides in the plane " }{TEXT 289 5 "y=2-x" }{TEXT -1 12 " , wherever " } {TEXT 290 2 "y " }{TEXT -1 35 "would occur we have replaced it by " } {TEXT 291 3 "2-x" }{TEXT -1 43 " . In particular, note the upper bound for " }{TEXT 292 1 "z" }{TEXT -1 140 " . To display these plots join tly, we must give the plots names and suppress their outputs with colo ns at the ends of their command lines.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "A1:=plot3d([x,x,z], z=0..4-2*x^2,x=0..1,color=red):\nA2:=plot3d([x,2-x,z],z=0..4-x^2-(2-x) ^2,x=0..1,color=green):\ndisplay(A1,A2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "After rotating and including the axes you should see the figure above. We add our top, suppressing its output,and display all three together.When drawing the top, observe that for any fixed " } {TEXT 293 1 "x" }{TEXT -1 3 ", " }{TEXT 294 1 "y" }{TEXT -1 30 " will vary between the values " }{TEXT 295 1 "x" }{TEXT -1 5 " and " } {TEXT 296 4 "2-x," }{TEXT -1 24 " i.e. \223curve-to-curve\224. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A3:=plot3d([x,y,4-x^2-y^2],y =x..2-x,x=0..1,color=blue):\ndisplay(A1,A2,A3);\n" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 178 "We see two sides and a top of the paraboloid. Th e third vertical side, if it were needed, would result from the plot l abelled \223A4\224. This side is viewed by rotating to the left." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A4:=plot3d([0,y,z],z=0..4-y^2,y=0..2,color=magenta):\ndisplay(A1 ,A2,A3,A4);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 302 10 "Example 4:" } {TEXT -1 55 " Display the portion of the hyperboloid of one sheet " }{XPPEDIT 18 0 "x^2+y^2-z^2=1" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$% \"zGF'!\"\"F(" }{TEXT -1 15 " for which " }{XPPEDIT 18 0 "x<=0" "6 #1%\"xG\"\"!" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "y>=0" "6#1\"\"!%\"yG " }{TEXT -1 8 " , and " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 23 " \+ is between -1 and 1.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Looking at this solid from a point out on the " }{TEXT 303 2 "x-" }{TEXT -1 78 "axis we would see a flat surface bounded on the left by a vertical line, (the " }{TEXT 304 2 "z-" }{TEXT -1 69 "axis), on the top and bo ttom by horizontal lines (edges of planes) " }{XPPEDIT 18 0 "z=-1" " 6#/%\"zG,$\"\"\"!\"\"" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "z=1" "6#/% \"zG\"\"\"" }{TEXT -1 105 " , and on the right by a hyperbola that be nds to the left in the middle. Rotate this surface about the " } {TEXT 305 1 "z" }{TEXT -1 216 "-axis 90 degrees and you have a surface that will be hidden from our view, but it serves as the domain of our parametrization of the curved surface. We realize that we cannot draw the curved surface as a function of " }{TEXT 306 1 "x" }{TEXT -1 5 " \+ and " }{TEXT 307 1 "y" }{TEXT -1 24 " . So, let\222s solve for " } {TEXT 308 1 "y" }{TEXT -1 13 " in terms of " }{TEXT 309 1 "x" }{TEXT -1 5 " and " }{TEXT 310 1 "z" }{TEXT -1 75 " . We are being careful t o go \221curve-to-curve and point-to-point\222 here so " }{TEXT 311 1 "x" }{TEXT -1 35 " must vary from 0 to a function of " }{TEXT 312 1 "z " }{TEXT -1 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2+y^2-z^2=1" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'!\"\"F(" } {TEXT -1 13 " implies " }{XPPEDIT 18 0 "y^2=1+z^2-x^2" "6#/*$%\"yG \"\"#,(\"\"\"F(*$%\"zGF&F(*$%\"xGF&!\"\"" }{TEXT -1 18 " which impli es " }{XPPEDIT 18 0 "y=sqrt(1+z^2-x^2)" "6#/%\"yG-%%sqrtG6#,(\"\"\"F) *$%\"zG\"\"#F)*$%\"xGF,!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Let " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" } {TEXT -1 18 " and solve for " }{TEXT 313 1 "x" }{TEXT -1 15 " in \+ terms of " }{TEXT 314 1 "z" }{TEXT -1 5 " : " }{XPPEDIT 18 0 "x^2 = \+ 1+z^2;" "6#/*$%\"xG\"\"#,&\"\"\"F(*$%\"zGF&F(" }{TEXT -1 19 " which \+ implies " }{XPPEDIT 18 0 "x = -sqrt(1+z^2);" "6#/%\"xG,$-%%sqrtG6#,& \"\"\"F**$%\"zG\"\"#F*!\"\"" }{TEXT -1 22 " . In Maple,we have\n" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "H1:=plot3d([x,sqrt(1+z^2-x^ 2),z],x=-sqrt(1+z^2)..0,z=-1..1,color=blue):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The top is a quarter of a disk. The radius is determined by putting " }{TEXT 315 2 "z " } {TEXT -1 51 "= \2611 in the equation of the hyperboloid. We get " } {XPPEDIT 18 0 "x^2+y^2-1=1" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(!\"\" F(" }{TEXT -1 17 " which implies " }{XPPEDIT 18 0 "x^2+y^2=2" "6#/,& *$%\"xG\"\"#\"\"\"*$%\"yGF'F(F'" }{TEXT -1 6 " and\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "H2:=plot3d([x,y,1],y=0..sqrt(2-x^2) ,x=-sqrt(2)..0,color=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "The remaining surface we need lives in th e plane " }{TEXT 316 2 "x " }{TEXT -1 25 "= 0. As in the domain of " } {TEXT 317 3 "H1 " }{TEXT -1 2 ", " }{TEXT 318 1 "z" }{TEXT -1 69 " can not be the dependent variable. The display can be created below.\n" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "H3:=plot3d([0,y,z],y=0..sqr t(1+z^2),z=-1..1,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(H1,H2,H3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT 319 10 "Example 5:" }{TEXT -1 40 " Display the hyperboloid of two sheets " }{XPPEDIT 18 0 "x^2-y^2-z^2=1 " "6#/,(*$% \"xG\"\"#\"\"\"*$%\"yGF'!\"\"*$%\"zGF'F+F(" }{TEXT -1 23 " between th e planes " }{XPPEDIT 18 0 "x = sqrt(5);" "6#/%\"xG-%%sqrtG6#\"\"&" } {TEXT -1 8 " and\n " }{XPPEDIT 18 0 "x = -sqrt(5);" "6#/%\"xG,$-%%sq rtG6#\"\"&!\"\"" }{TEXT -1 348 " . The result can be created below. \+ We list the answer without much comment. The use of 1.999 where it i s clearly meant to be 2 is to avoid losing a portion of the graph when the square root of a negative number (which is meant to be 0) occurs. (A better, more natural way, to handle this involves parameters desc ribed in the later sections.)\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "H5:=plot3d([-sqrt(1+y^2+z^2),y,z],z=-sqrt(4-y^2)..sqrt(4-y^2), y=-(1.999)..(1.999),color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "H6:=plot3d([sqrt(1+y^2+z^2),y,z],z=-sqrt(4-y^2)..sqrt (4-y^2),y=-(1.999)..(1.999),color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "H7:=plot3d([sqrt(5),y,z],z=-sqrt(4-y^2)..sqrt(4-y^2), y=-(1.999)..(1.999),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(H5,H6,H7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 21 "Challenging Problems\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 2 "1." }{TEXT -1 158 " Use Maple to plot the coordinat e axes and the remaining three faces of the box in Example 1. Rotate \+ the output so that a portion of each face may be seen.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Use Maple to plot the regions defined in \+ problems 2, 3, and 4.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 2 "2." } {TEXT -1 48 " R is the region in the first octant between " } {XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 40 " and y =2 x for x <=2. Also, 0<=z<=3." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 1 "3" } {TEXT -1 26 ". S is the region inside " }{XPPEDIT 18 0 "x^2+y^2=9" "6 #/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(\"\"*" }{TEXT -1 18 " that is below " }{XPPEDIT 18 0 "z=5" "6#/%\"zG\"\"&" }{TEXT -1 3 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 300 1 "4" }{TEXT -1 39 ". T is the region inside the cylinder " }{XPPEDIT 18 0 "x^2+y^2=4" "6#/,&*$%\"xG\"\"#\"\"\"*$ %\"yGF'F(\"\"%" }{TEXT -1 34 " that is above z = 0 and below " } {XPPEDIT 18 0 "z=9-x^2-y^2" "6#/%\"zG,(\"\"*\"\"\"*$%\"xG\"\"#!\"\"*$% \"yGF*F+" }{TEXT -1 3 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 301 2 "5." }{TEXT -1 5 " For " }{XPPEDIT 18 0 "k " "6#%\"kG" }{TEXT -1 152 " a constant assigned by your instructor (s ee the preface), use Maple to plot the \"bowl with lid\" described as \+ follows. The bowl is blue with equation " }{XPPEDIT 18 0 "z=x^2+y^2" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "z<=k+1" "6#1%\"zG,&%\"kG\"\"\"F'F'" }{TEXT -1 83 " and the lid is a gold disk that fits the bowl (lying in the plane with eq uation " }{XPPEDIT 18 0 "z=k+1" "6#/%\"zG,&%\"kG\"\"\"F'F'" }{TEXT -1 20 " ). (See Example 2.)" }}}}{MARK "57" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }