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{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 500 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT -1 34 " Math 181Computational Mathematics" }}{PARA 337 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT -1 19 "Lab 4: Basic Maple" }}{PARA 336 "" 0 "" {TEXT -1 0 "" }}{PARA 335 "" 0 "" {TEXT -1 10 "9/99, 2/01 " }}{PARA 334 "" 0 "" {TEXT -1 11 "R. Robinson" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 498 "" 0 "" {TEXT -1 61 "This is the revised ver sion 2/21/01. Sorry about the mix-up! " }}{PARA 500 "" 0 "" {TEXT -1 0 "" }}{PARA 499 "" 0 "" {TEXT -1 240 "This version has Problems 1-3 & 5. Problem 4, on the incorrectly posted worksheet, was to evaluate th e Onsager integral using Maple. This is harder than it might seem, and to accomplish it you may need to write your own quadrature routine. \+ " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "0. Welcome to Maple." } {TEXT 259 0 "" }}{EXCHG {PARA 338 "" 0 "" {TEXT -1 94 " The goal in th is first part of the lab is to introduce you to a few basic features o f Maple. " }}{PARA 327 "" 0 "" {TEXT -1 0 "" }}{PARA 326 "" 0 "" {TEXT -1 126 "This file is a maple \"worksheet\". Worksheets combine t ext with executable code. The extension of a Maple worksheet is \".mws \". " }}{PARA 457 "" 0 "" {TEXT -1 0 "" }}{PARA 456 "" 0 "" {TEXT -1 536 "I will use worksheets as interactive lecture notes. You should al so use worksheets to hand in your Maple homework. When you do that, pl ease make sure you upload them in \"binary\" mode. Also, to save space , please delete all output (I will run your input) and all graphics. T his will save a lot of memory in your web site. Finally, please note t hat we are using Maple V, Release 5. The newest version of Maple is Ma ple 6. If you use that to do your assignments, please test to make sur e your worksheets are compatible with this version. " }}{PARA 340 "" 0 "" {TEXT -1 0 "" }}{PARA 339 "" 0 "" {TEXT -1 169 "The worksheet wil l change as you work with it. You can save the changes if you want, bu t, you can always start over by download a fresh copy from the cours e web site. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 111 "Our initial tour of Maple will be very brief. Next week we will use Maple to experiment with round-off errors." }}{PARA 342 " " 0 "" {TEXT -1 0 "" }}{PARA 341 "" 0 "" {TEXT -1 114 "You can find ou t more about Maple using the excellent built-in help system (click on \+ \"Help\" above) or in the book " }{TEXT 261 22 "Maple V Learning Guide " }{TEXT -1 106 ". In particular, the latter is a good place to find \+ out how to structure your own worksheets (pp 17-21). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "1. Getting S tarted" }}{EXCHG {PARA 0 "" 0 "" {TEXT 260 102 "Most Maple commands lo ok similar to their equivalents in mathematical notation (this is like MATLAB). " }}{PARA 329 "" 0 "" {TEXT -1 0 "" }}{PARA 328 "" 0 "" {TEXT -1 26 "An important rule is that " }{TEXT 256 3 "all" }{TEXT -1 43 " Maple commands require a \";\" at the end. " }}{PARA 464 "" 0 " " {TEXT -1 0 "" }}{PARA 463 "" 0 "" {TEXT -1 71 "This is different fro m Matlab, where the \";\" is used to supress output." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 330 "" 0 "" {TEXT 270 8 "Example:" }{TEXT -1 47 " Type \"2+2;\" after the \">\", then press \"Enter\"." }{TEXT 257 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 269 108 "Notice that Maple's input is printed in red and its out put is printed in blue (it is a little to the right)." }}{PARA 460 "" 0 "" {TEXT -1 0 "" }}{PARA 459 "" 0 "" {TEXT -1 92 "To save Lab time I will sometimes type executables. You can just press \"Enter\" to run \+ them. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 458 "" 0 "" {TEXT 271 5 "Note:" }{TEXT -1 80 " you can get an extra \">\" by clicking the \" [>\" button at the top of the screen. " }}{PARA 462 "" 0 "" {TEXT -1 0 "" }}{PARA 461 "" 0 "" {TEXT -1 110 "It will come out at the end of \+ the current \"execution group\" (the text enclosed in the bracket to t he right). " }}{PARA 466 "" 0 "" {TEXT -1 0 "" }}{PARA 465 "" 0 "" {TEXT -1 95 "You can merge different execution groups using the \"Spli t or Join\" command in the \"Edit\" menu. " }}{PARA 468 "" 0 "" {TEXT -1 0 "" }}{PARA 467 "" 0 "" {TEXT -1 133 "You can also convert an exec ution group from \"Maple Input\" to \"Text Input\" using the \"Insert \" menu. Go ahead and experiment a little." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 262 19 "2. Ba sic arithmetic" }}{EXCHG {PARA 402 "" 0 "" {TEXT 258 144 "All basic ar ithmetic works as you might expect. Note that multiplication in Maple \+ requires a \"*\" (which makes Maple different from Mathematica)." } {TEXT -1 0 "" }}}{EXCHG {PARA 403 "> " 0 "" {MPLTEXT 1 0 8 "345*271;" }}{PARA 404 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 405 "> " 0 "" {MPLTEXT 1 0 4 "2^6;" }}{PARA 406 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 407 "> " 0 "" {MPLTEXT 1 0 8 "2^(2^6);" }}{PARA 408 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 409 "" 0 "" {TEXT -1 98 "This is a large integer. It would be pretty hard to factor it by hand, but with Maple it's a snap:" }}}{EXCHG {PARA 410 "> " 0 "" {MPLTEXT 1 0 19 "ifactor( 2^(2^6)-1);" }}{PARA 411 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 412 "" 0 "" {TEXT -1 18 "Pretty impressive!" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "3. Exact Aritmetic" }}{EXCHG {PARA 331 "" 0 "" {TEXT -1 124 "Unlike conventional programming languages (Fortran and C) and MAT LAB, (but like Mathematica) Maple uses \"exact\" arithmetic. " }} {PARA 333 "" 0 "" {TEXT -1 0 "" }}{PARA 332 "" 0 "" {TEXT -1 133 "By t his I mean that Maple carries along \"symbolic\" calculations until yo u explicitly tell it to evaluate to a floating point number. " }} {PARA 360 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}} {EXCHG {PARA 416 "> " 0 "" {MPLTEXT 1 0 8 "2^(1/2);" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 356 "" 0 "" {TEXT -1 107 "The evaluation is carried out using the \"evalf\" command. The percent \+ \"%\" refers to the most recent output." }}}{EXCHG {PARA 361 "> " 0 " " {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 362 "" 0 "" {TEXT -1 58 "You can ask for more digits if y ou need them (lots more!)." }}}{EXCHG {PARA 363 "> " 0 "" {MPLTEXT 1 0 19 "evalf(sqrt(2),300);" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 364 "" 0 "" {TEXT -1 27 "Which is more e^pi or pi^e?" }}}{EXCHG {PARA 365 "> " 0 "" {MPLTEXT 1 0 3 "Pi;" }{TEXT -1 0 "" }}}{EXCHG {PARA 366 "> " 0 "" {MPLTEXT 1 0 10 "e:=exp(1);" }{TEXT -1 0 "" }}} {EXCHG {PARA 367 "> " 0 "" {MPLTEXT 1 0 13 "evalf(Pi,30);" }{TEXT -1 0 "" }}}{EXCHG {PARA 368 "> " 0 "" {MPLTEXT 1 0 12 "evalf(e,30);" } {TEXT -1 0 "" }}}{EXCHG {PARA 369 "> " 0 "" {MPLTEXT 1 0 5 "E^Pi;" }}} {EXCHG {PARA 370 "> " 0 "" {MPLTEXT 1 0 12 "evalf(e^Pi);" }}}{EXCHG {PARA 371 "> " 0 "" {MPLTEXT 1 0 12 "evalf(Pi^e);" }}}{EXCHG {PARA 372 "" 0 "" {TEXT -1 24 "How close is Pi to 22/7?" }}}{EXCHG {PARA 373 "> " 0 "" {MPLTEXT 1 0 5 "22/7;" }{TEXT -1 0 "" }}}{EXCHG {PARA 374 "> " 0 "" {MPLTEXT 1 0 12 "evalf(22/7);" }}}{EXCHG {PARA 375 "> " 0 "" {MPLTEXT 1 0 20 "evalf(abs(22/7)-Pi);" }{TEXT -1 0 "" }}}{EXCHG {PARA 376 "" 0 "" {TEXT -1 8 "Not bad!" }}{PARA 377 "" 0 "" {TEXT -1 0 "" }}{PARA 357 "" 0 "" {TEXT -1 45 "Here's a better rational approxi mation of Pi." }}{PARA 359 "" 0 "" {TEXT -1 0 "" }}{PARA 358 "" 0 "" {TEXT -1 158 "Keep taking reciprocals and \"peeling off\" the integer \+ part. Then put the integers you get back together. The rational number you get is called a \"convergent\"." }}{PARA 470 "" 0 "" {TEXT -1 0 " " }}{PARA 469 "" 0 "" {TEXT -1 164 "This scheme is called the \"contin ued fraction\" approximation of a number. It gives the \"best\" possib le rational approximation of an irrational number by rationals." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 378 "> " 0 "" {MPLTEXT 1 0 19 "f1:=evalf(Pi-3,30);" }}}{EXCHG {PARA 379 "> " 0 "" {MPLTEXT 1 0 19 "g1:=evalf(1/f1,30);" }{TEXT -1 0 "" }}}{EXCHG {PARA 380 "> " 0 " " {MPLTEXT 1 0 19 "f2:=evalf(g1-7,30);" }{TEXT -1 0 "" }}}{EXCHG {PARA 413 "> " 0 "" {MPLTEXT 1 0 11 "P0:=3+(1/7)" }{TEXT -1 0 "" } {MPLTEXT 1 0 1 ";" }{TEXT -1 0 "" }}}{EXCHG {PARA 414 "> " 0 "" {MPLTEXT 1 0 16 "g2:=evalf(1/f2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 381 "> " 0 "" {MPLTEXT 1 0 18 "P:=3+(1/(7+1/15));" }}}{EXCHG {PARA 382 "> \+ " 0 "" {MPLTEXT 1 0 9 "evalf(P);" }{TEXT -1 0 "" }}}{EXCHG {PARA 415 " " 0 "" {TEXT -1 0 "" }}{PARA 383 "" 0 "" {TEXT 288 10 "Problem 1:" } {TEXT -1 75 " What do you get if you carry this one step further? How \+ much better is it?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "4. Algebra" }}{EXCHG {PARA 343 "" 0 "" {TEXT -1 0 "" }}{PARA 344 "" 0 "" {TEXT -1 176 "So far Maple may not seem so very different from Matlab, but now we come to the big difference. It can do Algebra (and Calculus) the way you would do it with pencil and paper. " }}{PARA 472 "" 0 "" {TEXT -1 0 "" }}{PARA 471 "" 0 "" {TEXT -1 104 "This is characteristic of \"Computer Algebra Systems\" (CAS) \+ like Maple, Mathematica, Mu-Pad and Macsyma." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 384 "" 0 "" {TEXT -1 94 "As we saw above, it is useful to be able to give names to expressions. For this we use \":=\". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 345 "> " 0 "" {MPLTEXT 1 0 13 "p:=x^2-3*x+2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 346 "> " 0 "" {MPLTEXT 1 0 10 "factor(p);" }{TEXT -1 0 "" }}}{EXCHG {PARA 389 "" 0 " " {TEXT -1 51 "Note the difference between \"factor\" and \"ifactor\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 386 "" 0 "" {TEXT -1 55 "The normal equal sign \"=\" is used in writing equations:" }}}{EXCHG {PARA 347 "> " 0 "" {MPLTEXT 1 0 11 "solve(p=0);" }{TEXT -1 0 "" }}} {EXCHG {PARA 390 "" 0 "" {TEXT -1 0 "" }}{PARA 348 "" 0 "" {TEXT -1 25 "Here is a harder example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 349 "> " 0 "" {MPLTEXT 1 0 11 "q:=x^2-x-1;" }{TEXT -1 0 " " }}}{EXCHG {PARA 350 "> " 0 "" {MPLTEXT 1 0 10 "factor(q);" }{TEXT -1 0 "" }}}{EXCHG {PARA 391 "" 0 "" {TEXT -1 0 "" }}{PARA 351 "" 0 "" {TEXT -1 73 "This polynomial is \"irreducible\" over the integers. Bu t all is not lost." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 352 "> " 0 "" {MPLTEXT 1 0 11 "solve(q=0);" }{TEXT -1 0 "" }}}{EXCHG {PARA 392 "" 0 "" {TEXT -1 0 "" }}{PARA 353 "" 0 "" {TEXT -1 148 "This (the largest root) is an important number called the \"golden mean\". It is the irrational number which is most poorly approximated by rati onals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 393 "" 0 "" {TEXT -1 0 "" }}{PARA 354 "" 0 "" {TEXT 272 10 "Problem 2:" }{TEXT -1 311 " Find the continued fraction approximation of the golden mean. To do this, first proceed as above for several steps. Then formulate a c onjecture (and prove your conjecture if you can!). Finally, compute th e first 10 convergents of the continued fraction approximation. Estima te the relative and absolute errors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 355 "> " 0 "" {MPLTEXT 1 0 14 "gms:=evalf(%);" }{TEXT -1 0 "" }}}{EXCHG {PARA 394 "" 0 "" {TEXT -1 0 "" }}{PARA 385 "" 0 "" {TEXT -1 130 "An important technical skill with Maple (or any online m athematical environment) is being able to extract just part of an answ er. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "> " 0 "" {MPLTEXT 1 0 11 "gm:=gms[1];" }{TEXT -1 0 "" }}}{EXCHG {PARA 395 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 39 "Here is a word probl em from Calculus I:" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 192 "The post office will deliver a package only if leng th plus girth is less than 108 inches (girth is the distance around th e package). Find the largest volume tube (cylinder) that can be mailed ." }}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 152 "Here we will use x for length (the letter l is confusing in type) and r for radius. The girth is the circumference 2*Pi*r, so length + girt h=2*Pi*r + x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 387 "> \+ " 0 "" {MPLTEXT 1 0 18 "eq1:=2*Pi*r+x=108;" }{TEXT -1 0 "" }}}{EXCHG {PARA 388 "> " 0 "" {MPLTEXT 1 0 12 "V:=Pi*r^2*x;" }{TEXT -1 0 "" }}} {EXCHG {PARA 396 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 73 "We solve Eq1 for x, then substitute into V (replacing V with a new value)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 264 "> " 0 " " {MPLTEXT 1 0 17 "xi:=solve(eq1,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 265 "> " 0 "" {MPLTEXT 1 0 16 "V:=subs(x=xi,V);" }{TEXT -1 0 "" }}} {EXCHG {PARA 266 "> " 0 "" {MPLTEXT 1 0 13 "V:=expand(V);" }}}{EXCHG {PARA 397 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 29 "We ca n see a picture of this:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 268 "> " 0 "" {MPLTEXT 1 0 16 "plot(V,r=0..20);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" }{TEXT 264 40 "We will finish this in the next section." }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "4. Calculus" }}{EXCHG {PARA 269 " " 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 66 "Here's how to co mpute a derivative (we use the example from above)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 14 "DV:=diff( V,r);" }{TEXT -1 0 "" }}}{EXCHG {PARA 272 "> " 0 "" {MPLTEXT 1 0 21 "p lot(\{V,DV\},r=0..20);" }{TEXT -1 0 "" }}}{EXCHG {PARA 273 "> " 0 "" {MPLTEXT 1 0 18 "soln:=solve(DV=0);" }}}{EXCHG {PARA 274 "> " 0 "" {MPLTEXT 1 0 15 "r_ans:=soln[2];" }{TEXT -1 0 "" }}}{EXCHG {PARA 275 " > " 0 "" {MPLTEXT 1 0 13 "evalf(r_ans);" }{TEXT -1 0 "" }}}{EXCHG {PARA 276 "> " 0 "" {MPLTEXT 1 0 23 "V_ans:=subs(r=r_ans,V);" }}} {EXCHG {PARA 277 "> " 0 "" {MPLTEXT 1 0 13 "evalf(V_ans);" }}}{EXCHG {PARA 399 "" 0 "" {TEXT -1 0 "" }}{PARA 278 "" 0 "" {TEXT -1 70 "This \+ is a pretty big tube. How big would a cube of the same volume be?" }} {PARA 400 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 0 14 "(V_ans)^(1/3);" }}}{EXCHG {PARA 280 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 398 "" 0 "" {TEXT -1 0 "" } }{PARA 281 "" 0 "" {TEXT -1 29 "Now let's try some integrals:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 0 11 "int(x^2,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 401 "" 0 "" {TEXT -1 37 "MATLAB can't do this! At best it can " }{TEXT 273 11 "approxima te" }{TEXT -1 11 " integrals." }}}{EXCHG {PARA 283 "> " 0 "" {MPLTEXT 1 0 17 "int(x^2,x=-2..4);" }}}{EXCHG {PARA 284 "> " 0 "" {MPLTEXT 1 0 14 "int(sec(x),x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 285 "> " 0 "" {MPLTEXT 1 0 22 "int(5*x/(7*x^2-10),x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 420 "" 0 "" {TEXT -1 0 "" }}{PARA 421 "" 0 "" {TEXT 265 33 "Here is a problem from last week." }}{PARA 422 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 417 "> " 0 "" {MPLTEXT 1 0 20 "int(1/t,t=1/10..1);\n" }}} {EXCHG {PARA 418 "" 0 "" {TEXT -1 12 "This is the " }{TEXT 266 6 "exac t " }{TEXT -1 7 "answer." }}{PARA 423 "" 0 "" {TEXT -1 0 "" }}{PARA 286 "" 0 "" {TEXT -1 74 "We can force Maple to evaluate the integral n umerically by applying evalf:" }}{PARA 424 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(int(1/t,t=1/10..1));" }}}{EXCHG {PARA 426 "" 0 "" {TEXT -1 99 "Can you see why Maple does th e next integral as a numerical inegral without the need to type evalf? " }}}{EXCHG {PARA 427 "> " 0 "" {MPLTEXT 1 0 17 "int(1/t,t=.1..1);" }} }{EXCHG {PARA 428 "" 0 "" {TEXT -1 0 "" }}{PARA 429 "" 0 "" {TEXT -1 42 "Here is an integral without a closed form:" }}{PARA 430 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 431 "> " 0 "" {MPLTEXT 1 0 10 "e:=exp(1) ;" }}}{EXCHG {PARA 432 "> " 0 "" {MPLTEXT 1 0 21 "int(e^(-x^3),x=0..1) ;" }{TEXT -1 0 "" }}}{EXCHG {PARA 433 "" 0 "" {TEXT -1 89 "Maple just \+ gives back the integral itself. But, we can always force numerical eva luation:" }}}{EXCHG {PARA 434 "> " 0 "" {MPLTEXT 1 0 28 "evalf(int(e^( -x^3),x=0..1));" }{TEXT -1 0 "" }}}{EXCHG {PARA 435 "" 0 "" {TEXT -1 84 "Hers is a similar (but more famous) integral. It also has no \"clo sed form\" but look:" }}}{EXCHG {PARA 436 "> " 0 "" {MPLTEXT 1 0 21 "i nt(e^(-x^2),x=0..1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 419 "" 0 "" {TEXT -1 0 "" }}{PARA 287 "" 0 "" {TEXT -1 151 "Here are two really ha rd integrals. Maybe you should try them by hand first to see how hard \+ they are! Can you remember from Calculus II how to do them?" }}{PARA 425 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 288 "> " 0 "" {MPLTEXT 1 0 22 "int(x^5*(sin(x))^3,x);" }}}{EXCHG {PARA 289 "> " 0 "" {MPLTEXT 1 0 23 "int((x^5-5)/(x^6-1),x);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "5. Graphics" }}{EXCHG {PARA 290 "" 0 "" {TEXT -1 0 "" }}{PARA 291 "" 0 "" {TEXT -1 54 "Maple is great at graphing! Here are a few exampl es. \n" }}}{EXCHG {PARA 292 "" 0 "" {TEXT -1 0 "" }}{PARA 293 "> " 0 " " {MPLTEXT 1 0 25 "plot(sin(x)/x,x=-10..10);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 294 "> " 0 "" {MPLTEXT 1 0 27 "plot(sin( x)/x,x=-100..100);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 295 "" 0 "" {TEXT -1 0 "" }}{PARA 296 "" 0 "" {TEXT -1 156 "For best r esults you need to \"work smart,\" especially in terms of choosing goo d plot ranges. Remenber, graphics is an exercise in art as much as in \+ science!" }}{PARA 474 "" 0 "" {TEXT -1 0 "" }}{PARA 473 "" 0 "" {TEXT -1 24 "Here is a \"bad\" example:" }}}{EXCHG {PARA 297 "" 0 "" {TEXT -1 0 "" }}{PARA 298 "> " 0 "" {MPLTEXT 1 0 26 "plot(x/(x^2-1),x=-20..2 0);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 299 "" 0 "" {TEXT -1 65 "Try again with x=-2..2 (use the mouse to change the input line). " }}{PARA 437 "" 0 "" {TEXT -1 0 "" }}{PARA 300 "" 0 "" {TEXT -1 54 "Then try again adding \", y=-10..10\" after the x range." }} {PARA 301 "" 0 "" {TEXT -1 0 "" }}{PARA 302 "" 0 "" {TEXT -1 17 "See w hat I mean? " }}{PARA 303 "" 0 "" {TEXT -1 0 "" }}{PARA 438 "" 0 "" {TEXT 267 138 "You can also try Maple's smartplot. To do this, highlig ht the word \"smartplot\" as the text suggests or click on \"Help\" on the upper right." }}{PARA 475 "" 0 "" {TEXT -1 0 "" }}{PARA 439 "" 0 "" {TEXT -1 27 "How did smartplot work out?" }}{PARA 440 "" 0 "" {TEXT -1 0 "" }}{PARA 441 "" 0 "" {TEXT 268 68 "So (here at least) you need to do a little thinking before plotting." }}{PARA 442 "" 0 "" {TEXT -1 0 "" }}{PARA 443 "" 0 "" {TEXT -1 20 "A few more examples:" } }}{EXCHG {PARA 304 "" 0 "" {TEXT -1 0 "" }}{PARA 305 "> " 0 "" {MPLTEXT 1 0 23 "plot(e^(-x^2),x=-2..2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 306 "> " 0 "" {MPLTEXT 1 0 28 "ent:=-x*ln(x)-(1-x)*ln(1-x);" } {TEXT -1 0 "" }}}{EXCHG {PARA 307 "> " 0 "" {MPLTEXT 1 0 19 "plot(ent, x=0..1.1);" }{TEXT -1 1 "\n" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 308 "" 0 "" {TEXT -1 45 "This function arises in the stud y of entropy." }}{PARA 309 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {TEXT 274 10 "Problem 3:" }{TEXT -1 129 " Find the exact (symbolic) ta ngent line to the entropy function at x0=1/3, and plot the function to gether with its tangent line. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 444 "" 0 "" {TEXT 275 5 "Hint:" }{TEXT -1 125 " Compute the deri vative using \"diff\" and use \"subs\" to plug in and get the slope m. Then use subs to find the y0 for x0=1/3. " }}{PARA 445 "" 0 "" {TEXT -1 0 "" }}{PARA 446 "" 0 "" {TEXT -1 30 "Now recall that y-y0=m(x-x0). " }}{PARA 447 "" 0 "" {TEXT -1 0 "" }}{PARA 448 "" 0 "" {TEXT -1 86 " We want an expression for y in terms of x (a variable) and x0, y0, and m (constants). " }}{PARA 449 "" 0 "" {TEXT -1 0 "" }}{PARA 450 "" 0 " " {TEXT -1 87 "It would be easy to solve this on paper, but the challe nge is to do this strictly via \"" }{TEXT 276 6 "e-math" }{TEXT -1 9 " \", i.e., " }{TEXT 277 24 "completely on the screen" }{TEXT -1 48 ". S ee if you can do it with the \"solve\" command." }}{PARA 451 "" 0 "" {TEXT -1 0 "" }}{PARA 452 "" 0 "" {TEXT -1 143 " If you call the resul ting tangent line function (an expression in x) \"tline\" then you ca n plot command plot both functions with the command: " }}{PARA 311 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 312 "> " 0 "" {MPLTEXT 1 0 27 "plot (\{ent,tline\},x=0..1.1);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 313 "" 0 "" {TEXT -1 75 "Here are two (of many) other kinds of p lots Maple can do: Implicit and 3-D." }}{PARA 476 "" 0 "" {TEXT -1 1 " " }}{PARA 314 "" 0 "" {TEXT -1 50 "We will do a lot more with graphic s in later labs:" }}{PARA 315 "" 0 "" {TEXT -1 0 "" }}{PARA 316 "> " 0 "" {MPLTEXT 1 0 51 "plots[implicitplot](x^3+y^3-3*x*y,x=-3..3,y=-4.. 2);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 453 "" 0 "" {TEXT -1 0 "" }}{PARA 317 "" 0 "" {TEXT -1 157 "This is called the \"F olium of Descartes\". Add the option \"grid=[50,50]\" to improve the r esolution. To see a slightly different picture, subtract .1 from the \+ " }}{PARA 318 "" 0 "" {TEXT -1 10 "equation. " }}{PARA 478 "" 0 "" {TEXT -1 0 "" }}{PARA 477 "" 0 "" {TEXT -1 62 "Can you figure out how \+ to plot both pictures at the same time?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 454 "" 0 "" {TEXT 278 8 "Co mment:" }{TEXT -1 161 " The syntax here is telling Maple to look in on e of its libraries for the \"implicitplot\" command. Maple conserves m emory by only loading in the basics initially." }}}{EXCHG {PARA 479 " " 0 "" {TEXT -1 0 "" }}{PARA 319 "" 0 "" {TEXT -1 35 "This is a saddle , or \"hyperboloid\"." }}{PARA 320 "" 0 "" {TEXT -1 0 "" }}{PARA 321 " > " 0 "" {MPLTEXT 1 0 32 "plot3d(x^2-y^2,x=-2..2,y=-2..2);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 322 "" 0 "" {TEXT -1 29 "This is the \"monkey saddle\". " }}{PARA 323 "" 0 "" {TEXT -1 0 "" }} {PARA 324 "> " 0 "" {MPLTEXT 1 0 37 "plot3d(x^3-3*x*y^2, x=-2..3,y=-2. .2);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 455 "" 0 "" {TEXT -1 0 "" }}{PARA 325 "" 0 "" {TEXT -1 89 "Use the mose to move th e plots around. Then right click to redraw or to draw differently." } {MPLTEXT 1 0 1 "\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "6. Basic \+ Linear Algebra" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 68 "Start any session whe re you plan on using matrices with the command:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 138 "Maple has a very strang e notation for the identity matrix. It is usually recommended that it \+ be \"aliased\" to a more memorable expression: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "alias(Id=&* ());" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 106 "Maple's input for matrices is longer winded than MATLAB's (th ere are other ways to do this; consult help)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=matrix([ [0,1],[3,2]]);" }}}{EXCHG {PARA 483 "" 0 "" {TEXT -1 0 "" }}{PARA 482 "" 0 "" {TEXT -1 73 "Evaluation of a matrix is not automatic. You need to tell Maple to do it;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(A);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 80 "Let's co mpute the eigenvalues of A. We start with the characteristic polynomia l:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "p:=det(A-x*Id);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evs:=solve(p=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 23 "We want them separately" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ev1:=evs[1];" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "ev2:=evs[2];" }}}{EXCHG {PARA 481 "" 0 "" {TEXT -1 157 "Now to get the corresponding eigenvectors we do the Gaus s-Jordan method, using the convenient Maple command \"backsub\". Also \+ observe the notation for vectors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z:=vector([0,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A11:=augment(A-ev1*Id,z);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A12:=augment(A-ev 2*Id,z);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "v1t:=backsub(gausselim(A11),false,t);" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "v1:= subs(t[1]=1,evalm(v1t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "v2t:=backsub(gausselim(A12),false,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "v2:=subs(t[1]=1,evalm(v2t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 102 "Recall that the eige nvectors can be used to diagonalize a matrix. Make them the columns of a matrix P:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "P:=evalm((augment(v1,v2))^(-1));" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 480 "" 0 "" {TEXT -1 81 "Here is ou r diagonalization. Observe the notation for the matrix inverse and the " }{TEXT 281 7 "unusual" }{TEXT -1 36 " notation for matrix multiplic ation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Di:=evalm(P^(-1)& *A&*P);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 38 "One ca n also use the simpler commands:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nullspace(A-ev1*Id);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nullspace(A-ev2*Id);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 17 "or simpler still:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "jordan(A,'P1');" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 10 "evalm(P1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalm(P1^(-1)&*A&*P1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 490 "" 0 "" {TEXT -1 0 "" }}{PARA 484 "" 0 "" {TEXT -1 71 "The m ain difference is that Maple matrices can have variables in them. " }} {PARA 492 "" 0 "" {TEXT -1 0 "" }}{PARA 491 "" 0 "" {TEXT -1 110 "For \+ example, suppose I want to know the general form of the cubic equation through (x1,y1), (x2,y2), (x3,y3). " }}{PARA 494 "" 0 "" {TEXT -1 0 " " }}{PARA 493 "" 0 "" {TEXT -1 35 "First I make the Vandermonde matrix " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V:=matrix([[1,x1,x1^2], [1,x2,x2^2],[1,x3,x3^2]]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 290 31 "Then I define Y and solve AK=Y:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Y:=vector([y1,y2,y3]);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "K:=evalm(V^ (-1)&*Y)" }{TEXT -1 0 "" }}}{EXCHG {PARA 495 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 485 "" 0 "" {TEXT 289 10 "Proble m 5:" }{TEXT -1 68 " Finish this problem. That is, find p(x) so that p (xi)=yi, i=1,2,3. " }}{PARA 497 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 291 49 "Recall that if K=[c,b,a] then P(x)=a x^2+b x +c. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 496 "" 0 "" {TEXT -1 51 "Finish \+ by integrating your p(x) from x=x1 to x=x3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 489 "" 0 "" {TEXT -1 0 "" }}{PARA 488 "" 0 "" {TEXT -1 90 "This was a crucial step in deriving the formula for Simpson's r ule (the step we skipped). " }}{PARA 487 "" 0 "" {TEXT -1 0 "" }} {PARA 486 "" 0 "" {TEXT -1 53 "Can you do this without recourse to pap er and pencil?" }}}}}{MARK "0 10 0" 240 }{VIEWOPTS 1 1 0 1 1 1803 }