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1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 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 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 2 "C3" }{TEXT 341 2 "M1" } {TEXT 343 1 "\n" }{TEXT 342 26 "Three-Dimensional Graphics" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "There are several packages of programs i n Maple that we will find useful. For many calculus operations we wil l need \223" }{TEXT 328 7 "student" }{TEXT -1 42 "\224. For vector op erations the package is \223" }{TEXT 329 6 "linalg" }{TEXT -1 146 " \224. One of the real strengths of Maple is its ability to graph curv es and surfaces in a three-dimensional coordinate system. We need the \+ package \223" }{TEXT 330 5 "plots" }{TEXT -1 69 "\224 in order to do t his. There are two basic ways to use the command \223" }{TEXT 331 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 a nd most versatile plots " }{XPPEDIT 18 0 "[x,y,z]" "6#7%%\"xG%\"yG%\"z G" }{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 var iables. The reader is encouraged to execute the commands being discus sed and to try the suggestions to see the effects that they have. It \+ may be to your advantage to save the Maple work that you type in to te st 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 workshe et and hit all the way down if you have edited the worksheet. \+ The command \223" }{TEXT 336 8 "restart:" }{TEXT -1 64 " \224 clea rs the Maple kernel of all internal memory. Some put \223" }{TEXT 332 8 "restart:" }{TEXT -1 65 "\224 on the first line of a worksheet \+ before any packages such as \223" }{TEXT 333 7 "student" }{TEXT -1 6 " \224 or \223" }{TEXT 335 5 "plots" }{TEXT -1 103 "\224 so that confusi on is avoided if one \222s from the first line all the way thro ugh. Do NOT put \223" }{TEXT 334 8 "restart:" }{TEXT -1 115 "\224 on a line AFTER you have listed packages because that will erase the pac kages that you think have been included.\n" }}{PARA 0 "" 0 "" {TEXT 257 10 "Example 1:" }{TEXT -1 54 " Draw three faces of the rectangula r box defined by " }{TEXT 258 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 coordin ate axes. Let's start with the " }{TEXT 337 1 "x" }{TEXT -1 21 "-axis . The command \221" }{TEXT 338 10 "spacecurve" }{TEXT -1 117 "\222 ha s the parametric form of a curve as its argument, along with the range and choice of color. All we need here is " }{TEXT 259 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 261 1 "y" }{TEXT -1 10 " value is " }{TEXT 260 1 "1 " }{TEXT -1 6 ", and " }{TEXT 263 1 "x" }{TEXT -1 5 " and " }{TEXT 264 1 "z" }{TEXT -1 67 " may vary. Analyze the lines below and predic t the output of each." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 325 "w ith(plots):\nxaxis:=spacecurve([t,0,0],t=0..3,color=black):\nyaxis:=sp acecurve([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:=plot3 d([x,y,1],x=0..2,y=1..3,color=green):\nC:=plot3d([2,y,z],y=1..3,z=1..2 ,color=magenta):\ndisplay(xaxis,yaxis,zaxis,A,B,C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 10 "Example 2:" }{TEXT -1 39 " Draw the portion o f the paraboloid " }{XPPEDIT 18 0 "z = 4-x^2-y^2;" "6#/%\"zG,(\"\"% \"\"\"*$%\"xG\"\"#!\"\"*$%\"yGF*F+" }{TEXT -1 42 " that is over the q uarter-disk of radius " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 24 " i n the first quadrant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "pl ot3d(4-x^2-y^2,y=0..sqrt(4-x^2),x=0..2,color=blue);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Note how we kept " }{TEXT 265 1 "x" }{TEXT -1 22 " between 0 and 2, but " }{TEXT 266 1 "y" }{TEXT -1 19 " was betwee n 0 and " }{XPPEDIT 18 0 "sqrt(4-x^2)" "6#-%%sqrtG6#,&\"\"%\"\"\"*$%\" xG\"\"#!\"\"" }{TEXT -1 951 " . At this point you should have a three -cornered sheet of blue lines appearing. Move the pointer onto the fi gure and click 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 them and note how you get different ways of showing the axes o n the figure, with one option being no axes. There are 7 black spheres to the left of the four red ones. One at a time, click on each of th e spheres. You may wish to end this sequence with the middle sphere. \+ Now, click 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 rota tes. On the left end of the line above with the spheres you will find two angles displayed. As you rotate the figure the values of those a ngles change 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 r otation about the " }{TEXT 267 1 "z" }{TEXT -1 14 "-axis. When " } {XPPEDIT 18 0 "Theta" "6#%&ThetaG" }{TEXT 268 3 "=0 " }{TEXT -1 25 "yo u are looking down the " }{TEXT 269 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 270 1 "j" }{TEXT -1 29 ", and measures how much the " }{TEXT 339 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 271 1 "z" } {TEXT -1 22 "-axis from above.When " }{XPPEDIT 18 0 "Theta" "6#%&Theta G" }{TEXT -1 8 "=45 and " }{TEXT 293 1 "j" }{TEXT -1 18 "=75, you have the " }{TEXT 272 1 "x" }{TEXT -1 27 "-axis to your left and the " } {TEXT 273 1 "y" }{TEXT -1 39 "-axis to your right equally, while the \+ " }{TEXT 274 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 275 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 276 11 "Example 3: " }{TEXT -1 47 " Draw the solid figure bounded on the sides by " }{TEXT 277 4 " y=x" }{TEXT -1 2 ", " }{TEXT 278 9 " y=2 - x" }{TEXT -1 8 " , and " }{TEXT 279 3 "x =0" }{TEXT -1 11 ", below by " }{TEXT 280 3 "z=0" }{TEXT -1 16 ", and above by " }{XPPEDIT 18 0 "z=4-x^2-y^2" "6#/%\"zG,(\"\"%\"\"\"*$%\"xG \"\"#!\"\"*$%\"yGF*F+" }{TEXT -1 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 130 "We will use this same example to introduce double integrals later ,so a little effort here will be helpful. If you draw the lines " } {TEXT 281 3 "y=x" }{TEXT -1 5 " and " }{TEXT 282 7 "y = 2-x" }{TEXT -1 46 " in the first quadrant and then draw the line " }{TEXT 283 3 "x =0" }{TEXT -1 48 ", you will see that a triangle has been formed.\n" } }{PARA 0 "" 0 "" {TEXT 284 7 "BEWARE!" }{TEXT -1 9 " In the " }{TEXT 340 2 "xy" }{TEXT -1 16 "-plane the line " }{TEXT 285 3 "x=0" }{TEXT -1 20 " is vertical and is " }{TEXT 286 3 "NOT" }{TEXT -1 5 " the " } {TEXT 287 1 "x" }{TEXT -1 28 "-axis. Note that using the " }{TEXT 288 1 "x" }{TEXT -1 146 "-axis as a boundary defines a different trian gle. This error frequently occurs. Please observe the result of the \+ following plot on your screen.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot([x,2-x],x=0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 252 "The top line should be green and the bottom should be re d. If the triangle was blue then you would have a good view looking do wn on the solid. But the red and green lines will be edges of the ver tical surfaces. The top of our solid is the paraboloid " }{XPPEDIT 18 0 "z=4-x^2-y^2" "6#/%\"zG,(\"\"%\"\"\"*$%\"xG\"\"#!\"\"*$%\"yGF*F+ " }{TEXT -1 61 " . Keep this in mind when you consider the upper boun ds for " }{TEXT 289 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 290 3 "y=x" } {TEXT -1 15 " so that every " }{TEXT 291 1 "y" }{TEXT -1 16 " is repla ced by " }{TEXT 292 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 294 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 295 7 "[x,x,z]" }{TEXT -1 40 " which means our plot lies in the plane " }{TEXT 296 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 297 5 "y=2-x" }{TEXT -1 12 " , wherever " } {TEXT 298 2 "y " }{TEXT -1 35 "would occur we have replaced it by " } {TEXT 299 3 "2-x" }{TEXT -1 43 " . In particular, note the upper bound for " }{TEXT 300 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 301 1 "x" }{TEXT -1 3 ", " }{TEXT 302 1 "y" }{TEXT -1 30 " will vary between the values " }{TEXT 303 1 "x" }{TEXT -1 5 " and " } {TEXT 304 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 310 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 311 2 "x-" }{TEXT -1 78 "axis we would see a flat surface bounded on the left by a vertical line, (the " }{TEXT 312 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,$\"\"\"!\"\"" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "z=1" "6#/% \"zG\"\"\"" }{TEXT -1 105 " , and on the right by a hyperbola that be nds to the left in the middle. Rotate this surface about the " } {TEXT 313 1 "z" }{TEXT -1 216 "-axis 90 degrees and you have a surface that will be hidden from our view, but it serves as the domain of our parametrization of the curved surface. We realize that we cannot draw the curved surface as a function of " }{TEXT 314 1 "x" }{TEXT -1 5 " \+ and " }{TEXT 315 1 "y" }{TEXT -1 24 " . So, let\222s solve for " } {TEXT 316 1 "y" }{TEXT -1 13 " in terms of " }{TEXT 317 1 "x" }{TEXT -1 5 " and " }{TEXT 318 1 "z" }{TEXT -1 75 " . We are being careful t o go \221curve-to-curve and point-to-point\222 here so " }{TEXT 319 1 "x" }{TEXT -1 35 " must vary from 0 to a function of " }{TEXT 320 1 "z " }{TEXT -1 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2+y^2-z^2=1" "6#/,(*$%\"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)*$%\"xGF,!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Let " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" } {TEXT -1 18 " and solve for " }{TEXT 321 1 "x" }{TEXT -1 15 " in \+ terms of " }{TEXT 322 1 "z" }{TEXT -1 5 " : " }{XPPEDIT 18 0 "x^2 = \+ 1+z^2;" "6#/*$%\"xG\"\"#,&\"\"\"F(*$%\"zGF&F(" }{TEXT -1 19 " which \+ implies " }{XPPEDIT 18 0 "x = -sqrt(1+z^2);" "6#/%\"xG,$-%%sqrtG6#,& \"\"\"F**$%\"zG\"\"#F*!\"\"" }{TEXT -1 22 " . In Maple,we have\n" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "H1:=plot3d([x,sqrt(1+z^2-x^ 2),z],x=-sqrt(1+z^2)..0,z=-1..1,color=blue):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The top is a quarter of a disk. The radius is determined by putting " }{TEXT 323 2 "z " } {TEXT -1 51 "= \2611 in the equation of the hyperboloid. We get " } {XPPEDIT 18 0 "x^2+y^2-1=1" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(!\"\" F(" }{TEXT -1 17 " which implies " }{XPPEDIT 18 0 "x^2+y^2=2" "6#/,& *$%\"xG\"\"#\"\"\"*$%\"yGF'F(F'" }{TEXT -1 6 " and\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "H2:=plot3d([x,y,1],y=0..sqrt(2-x^2) ,x=-sqrt(2)..0,color=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "The remaining surface we need lives in th e plane " }{TEXT 324 2 "x " }{TEXT -1 25 "= 0. As in the domain of " } {TEXT 325 3 "H1 " }{TEXT -1 2 ", " }{TEXT 326 1 "z" }{TEXT -1 69 " can not be the dependent variable. The display can be created below.\n" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "H3:=plot3d([0,y,z],y=0..sqr t(1+z^2),z=-1..1,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(H1,H2,H3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT 327 10 "Example 5:" }{TEXT -1 40 " Display the hyperboloid of two sheets " }{XPPEDIT 18 0 "x^2-y^2-z^2=1 " "6#/,(*$% \"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,$-%%sq rtG6#\"\"&!\"\"" }{TEXT -1 348 " . The result can be created below. \+ We list the answer without much comment. The use of 1.999 where it i s clearly meant to be 2 is to avoid losing a portion of the graph when the square root of a negative number (which is meant to be 0) occurs. (A better, more natural way, to handle this involves parameters desc ribed in the later sections.)\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "H5:=plot3d([-sqrt(1+y^2+z^2),y,z],z=-sqrt(4-y^2)..sqrt(4-y^2), y=-(1.999)..(1.999),color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "H6:=plot3d([sqrt(1+y^2+z^2),y,z],z=-sqrt(4-y^2)..sqrt (4-y^2),y=-(1.999)..(1.999),color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "H7:=plot3d([sqrt(5),y,z],z=-sqrt(4-y^2)..sqrt(4-y^2), y=-(1.999)..(1.999),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(H5,H6,H7);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 14 "C 3M1 Problems\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 305 2 "1." }{TEXT -1 158 " Use Maple to plot the coordinate axes and the remaining three f aces of the box in Example 1. Rotate the output so that a portion of \+ each face may be seen.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Use M aple to plot the regions defined in problems 2, 3, and 4.\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 306 2 "2." }{TEXT -1 48 " R is the regio n in the first octant between " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG " }{TEXT -1 40 " and y =2 x for x <=2. Also, 0<=z<=3." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 307 1 "3" }{TEXT -1 26 ". S is the region inside " }{XPPEDIT 18 0 "x^2+y^2=9" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(\"\" *" }{TEXT -1 18 " that is below " }{XPPEDIT 18 0 "z=5" "6#/%\"zG\" \"&" }{TEXT -1 3 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 308 1 "4" } {TEXT -1 39 ". T is the region inside the cylinder " }{XPPEDIT 18 0 " x^2+y^2=4" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(\"\"%" }{TEXT -1 34 " \+ that is above z = 0 and below " }{XPPEDIT 18 0 "z=9-x^2-y^2" "6#/%\"z G,(\"\"*\"\"\"*$%\"xG\"\"#!\"\"*$%\"yGF*F+" }{TEXT -1 3 " ." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 2 "5. " }{TEXT -1 5 " For " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 152 " a \+ constant assigned by your instructor (see the preface), use Maple to p lot the \"bowl with lid\" described as follows. The bowl is blue with equation " }{XPPEDIT 18 0 "z=x^2+y^2" "6#/%\"zG,&*$%\"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 t he bowl (lying in the plane with equation " }{XPPEDIT 18 0 "z=k+1" "6 #/%\"zG,&%\"kG\"\"\"F'F'" }{TEXT -1 20 " ). (See Example 2.)" }}}} {MARK "35 0 3" 22 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }