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Also write the names of any other students \+ you are working with. Don't forget to number your problems and to typ e restart at the beginning of each problem." }}{PARA 0 "" 0 "" {TEXT -1 59 "If you need more space to work than you are given, you can " } {TEXT 259 16 "get a new prompt" }{TEXT -1 100 " by selecting the promp t button right below the word \"spreadsheet\". It has the symbol \"[> \" on it. " }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 504 "Suppose we want to solve f(x)=0. We already know tha t we can use solve or fsolve. However, we do not really unerstand how Maple finds these solutions. When we type solve, Maple uses formulas like the quadratic formula and algebraic techniques just like you hav e all learned in algebra. When we type fsolve Maple uses numerical te chniques similar to the ones we will learn today to find the decimal e xpansion of a solution. These techniques will also give you an introd uction to computer programming." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 260 "" 0 "" {TEXT -1 48 "Problem I: The Method of Tabulation an d Loops: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 265 "Suppose we want to \+ find the real roots of a continuous function. We know that if a conti nuous function is positive at one point and negative at another point \+ then it must be zero somewhere in between the two points. This is ca lled the intermediate value theorem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "Let f(x)= x^3-x-1; We want to find a \+ real solutions of f(x) = 0 using the intermediate value theorem. Hit \+ enter on the following lines:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " f := x -> x^3 - x - 1; " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "plot(f(x), x=-10 .. 10);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "The plot shows there is only one real root, in (-2, 3). It is not clear where the graph is positive or negative be tween -2 and 3, so we can just check:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f(-2), f(-1),f(0), f(1), f(2), f(3);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "And see that f(1)<0 and f(2)>0 so there i s a solution between 1 and 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 238 "If we check every number 1.0, 1.1, 1.2,. . 1.9, 2.0, then we can find out what the solution is with an accuracy of one decimal place. This would take a while to work out by hand, b ut a computer can do it quickly. In fact we can write a \"" }{TEXT 264 14 "Loop\" Program " }{TEXT -1 26 "and tell the computer that" } {TEXT 260 6 " for x" }{TEXT -1 20 " = numbers starting " }{TEXT 267 7 "from 1 " }{TEXT -1 1 " " }{TEXT 261 3 "by " }{TEXT -1 21 "taking step s of size " }{TEXT 262 4 "0.1 " }{TEXT -1 13 "until you get" }{TEXT 266 8 " to 2 do" }{TEXT -1 13 " the command " }{TEXT 263 7 "x, f(x)" } {TEXT -1 88 ". Recall that the words for, from, by, to, do and od ar e all commands that you can use" }}{PARA 0 "" 0 "" {TEXT -1 88 "to wri te a loop program. do and od are placed on either side of the list of commands. " }{TEXT 277 1 " " }{TEXT 278 94 "Be sure to save your fil e before clicking enter on this loop in case it crashes the computer! \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "for x from 1 by 0.1 to 2 do x, f(x) od;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "The output shows that there is a solution of f(x) = 0 in (1.3, 1.4). To pinpoint the solution, test f(x) in (1.3, 1.4) , taking step = 0.01:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for x fro m 1.3 by 0.01 to 1.4 do x, f(x) od;" }}}{PARA 0 "" 0 "" {TEXT -1 78 "Since 1.32 < c < 1.33, the solution c = 1.3 is accurate to 1 \+ decimal place. " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "Now you repeat this process until you get a solution which is accurate to 3 decimal \+ places. Write comments explaining what you are doing." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 27 "Problem 2: The while loop:" }}{PARA 0 "" 0 "" {TEXT -1 297 "N ow suppose you have a slow computer and you don't want it to bother do ing all the computations. After all, you know that for the f in probl em 1, f(1)<0 and f(2) >0. You need only check f(1), f(1.1) and so on \+ until you get a negative answer. So you use a \"while loop\" and tell the computer that" }{TEXT 268 6 " for x" }{TEXT -1 20 " = numbers sta rting " }{TEXT 273 7 "from 1 " }{TEXT -1 1 " " }{TEXT 269 3 "by " } {TEXT -1 21 "taking steps of size " }{TEXT 270 4 "0.1 " }{TEXT -1 13 " until you get" }{TEXT 272 21 " to 2 while f(x)<0 do" }{TEXT -1 13 " th e command " }{TEXT 271 11 "x, f(x). " }{TEXT -1 117 " This means tha t the computer should only keep working while f(x)<0 and stop when f(x ) is bigger than or equal to 0. " }{TEXT 283 1 " " }{TEXT 276 99 "Be s ure to save your file before clicking enter on the while loop in case \+ it crashes the computer! " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for x \+ from 1 by 0.1 to 2 while (f(x) < 0) do x, f(x) od;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 264 "This output tells us that the computer g ot f(1.3)<0 but f(1.4)>0. It didn't bother to do the command 1.4, f( 1.4) because the answer f(1.4) wasn't negative anymore. Now you can c ontinue with another while loop to find the decimal expansion to two d ecimal places." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 27 "Problem 3: N ewton's Method:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 546 "Using the Met hod of Tabulation described above you have to run the for or while loo ps over and over, one time for each extra digit of accuracy. This is \+ not very efficient because you are just dividing the intervals up into ten pieces every time and doing lots of computations. If the functi on you are working with is a differentiable function then you can use \+ another technique called Newton's Method. It won't always work, but i t often works faster than the method of tabulation. The idea is to us e tangent lines to point towards solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "We are going to use the same f unction again so we won't type restart, but we do need to clear the va lue for x. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:='x';" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Recall that for a function f, the \+ slope of the tangent line at a point " }{TEXT 284 1 "a" }{TEXT -1 4 " \+ is " }{TEXT 285 11 "m:= D(f)(a)" }{TEXT -1 17 " and the line is " }} {PARA 0 "" 0 "" {TEXT 286 26 "y= (D(f)(a)) (x-a) +f(a). " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 123 "So for example, if we take the fun ction f from problem 1, we can find the tangent line at 1 and graph it with the function:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(\{ f(x) , D(f)(1)*( x-1 ) + f(1) \}, x=-2..3);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Recall from last week that the tangent line at " }{TEXT 287 1 "a" }{TEXT -1 38 " is the linear a pproximation for f at " }{TEXT 288 1 "a" }{TEXT -1 24 ". This means t hat near " }{TEXT 289 1 "a" }{TEXT -1 22 " the tangent line and " } {TEXT 290 2 "f " }{TEXT -1 198 "are rather close to each other. In pa rticular, in the graph you've just plotted, you can see that the tange nt line crosses the x axis near where the function crosses the x axis. So we can find out" }{MPLTEXT 1 0 1 " " }{TEXT -1 45 "where it cross es doing some simple algebra: " }}{PARA 0 "" 0 "" {TEXT 291 72 "(D(f) (1))*(x-1)+f(1)=0 (D(f)(1))(x-1) = - f(1) x-1= -f(1)/D((f)(1)) " } }{PARA 0 "" 0 "" {TEXT -1 4 "so " }{TEXT 292 21 "x = 1- f(1)/(D(f)(1) )" }{TEXT -1 67 " . We can call this x where the line crosses the x a xis, \"crossa\" " }{TEXT 293 91 "(In computers it is common to use an \+ entire word for a variable instead of just one letter)" }{TEXT -1 1 ": " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "cros sa:= 1-f(1)/(D(f)(1));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "and find the tangent line to f at a point a = crossa:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a:= eva lf(crossa); plot(\{f(x), (D(f)(a))*(x-a)+f(a)\}, x= -2..3);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "Now t his tangent line crosses very close to the same place f does. It is e asy to repeat the process and choose a new crossa where this tangent l ine crosses: We will call it \"crossa\" again. The old crossa is jus t called \"a\" now." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "crossa:=a-f( a)/(D(f)(a)); evalf(crossa);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "N ow lets find the tangent line to f at a= this new crossa." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a:= evalf(crossa ); plot(\{f(x), (D(f)(a))*(x-a)+f(a)\}, x= -2..3);" }}{PARA 13 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Now we check wh ere it crosses, and can use the same formula as before to get a third \+ crossa:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "crossa:=a-f(a)/( D(f)(a)); evalf(crossa);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Lets repeat this again using the same commands:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a:= evalf(crossa); plot(\{f(x), (D(f)(a)) *(x-a)+f(a)\}, x= -2..3);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "crossa:=a-f(a)/(D(f) (a)); evalf(crossa);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 214 "Note that we've already found an answer which is accurate up t o 2 decimal places, 1.32. To compute the error it is standard to comp are the old crossa (which is now called a) with the new cross a. We n eed to find " }}{PARA 0 "" 0 "" {TEXT -1 85 "|crossa -a | and since th e command in Maple for the absolute value is abs we write: " } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "err:= a bs(crossa-a);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 214 " Now suppose we want an error which is less than .0000001. We need to \+ keep repeating these commands. One way to do this would be to use a f or loop and a counter j which counts the number of repetitions (up to \+ 10):" }}{PARA 261 "" 0 "" {TEXT -1 133 "for j from 1 by 1 to 10 do plo t (tan line at a), crossa:=a-f(a)/(D(f)(a)); evalf(crossa); error:=abs (crossa-a); a:=evalf(crossa); od;" }}{PARA 0 "" 0 "" {TEXT -1 406 "Not e that we must start by giving the first a and error values to start w ith. Then the computer will graph the tan line find the cross compute \+ the error and set the new a to be the crossa before repeating the proc ess. However there is no reason to make the computer do j steps if t he error is alread small enough. There is also no reason to keep plot ting the graph. So we use the following while loop. " }{TEXT 275 100 " Be sure to save your file before clicking enter on the while loop in case it crashes the computer! " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a:=1; err:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 " for j from 1 by 1 to 10 while (err>0.0000001) do j; crossa:=a-f(a)/(D(f)(a)); err:=evalf(abs(crossa-a)); a:=evalf(crossa); od;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Notice that it only takes 6 steps! Now what would happen if we started with a different a:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a:=0; err:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 126 " for j from 1 by 1 to 30 while (err>0.0000001) do j;crossa:=a-f(a)/(D(f)(a)); err:= evalf(abs(crossa-a)); a:=evalf(crossa); od;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 348 "This time 10 steps hav e gone by and the error is still large. So increase the number of ste ps. If you increase it to 20 steps the error is small but not small e nough yet. If you increase it to 30 steps it is done in 22 steps and \+ it says that the error is 0. The error is probably not zero but some \+ number which has 0's for the first ten places." }}{PARA 0 "" 0 "" {TEXT 279 13 "Exploration: " }{TEXT -1 79 " Cut and paste the Newton' s Method while loop so you can use it for your work:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 10 "Problem A:" }{TEXT -1 109 " Try f(x)=sin(x) a nd starting with a=2 use Newton's method to find the value of Pi up to 8 decimal places. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 12 "Problem B: " }{TEXT -1 571 "If you would like to design some while an for loops of your own, do them in another file and be c areful to save regularly. Loops can ask too much of a computer and ca use it to crash if they are accidently written incorrectly. If you wa nt to try this at home, remember that if your computer freezes you can type Ctrl Alt Delete and you will be asked if you would like to End a Task. Choose Maple and the computer will just crash Maple and not al l the other programs. Everything you did after your last save will be lost. You can open Maple again and continue working." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 344 "If you are intere sted in a career designing programs that perform calculations you shou ld major in both Mathematics and Computer Science. It is not importan t to complete two majors but to complete all the requirements for each major. Then plan to get a masters in Numerical Methods (this can be \+ done either in a math dept or a comp sci dept)." }}}}{MARK "40" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }