Maple-esitys 24.10.2011, aiheitaSyys-lokakuu 2011
Maple15/esitysaiheita.mw
<Text-field style="Heading 1" layout="Heading 1">Newton (Student)</Text-field>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
<Text-field style="Heading 1" layout="Heading 1">Hints</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiP0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEuUGxvdHRpbmdHdWlkZUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0Y0Ri8=
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">K\303\244yr\303\244tutkimusta </Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">Esim. 1, Kofler s. 40-></Text-field>
<Text-field style="Heading 3" layout="Heading 3">Pohdiskelua
<Font bold="true" style="Text" size="14"> simplify</Font><Font encoding="UTF-8"> - yleissievent\303\244j\303\244, ensimm\303\244iseksi tarjoiltava.</Font></Text-field> expand - kertoo auki collect - kokoaa ? collect ja kokeile joitakin esimerkkej\303\244. Varsin hyv\303\244 komento monessa paikassa. factor - Jakaa tekij\303\266ihin.
convert - ... Monenmoisiin konversioihin sievennyksess\303\244 esim convert(lauseke,parfrac,muuttuja); Huomaa, ett\303\244 lauseke ei muutu, uusi tulos palautetaan. Jos halutaan p\303\244ivitt\303\244\303\244 lauseke, on komennetava lauseke:=convert(lauseke,parfrac,muuttuja); solve - ratkaisee yht\303\244\303\266n tai systeemin fsolve - ratkaisee numeerisesti(x^3+1)/(x^2-x+1);simplify(%);expand((x^2-4)*(x+1)*(x-2)*(x^2+x+1));factor(%);Yht\303\244l\303\266ist\303\244[SOL] s. 11-12[HAM] s. Luku 6 s. 133 - 142yhtalot:={2*x-5*y=12, 12*x+4*y=17}; # Kyseess\303\244 on joukkoratk:= solve(yhtalot, {x,y});Maple ei sijoita ratkaisuja muuttujille arvoiksi, vaan palauttaa sijoituss\303\244\303\244nn\303\266t. Ne voidaan antaa suoraan#subs-komennolle. Mieti seuraavan komentojonon logiikkaa!subs(ratk,x),subs(ratk,y);X:=subs(ratk,x); Y:=subs(ratk,y);Tarkistaminen k\303\244y vaivattomasti:subs(ratk,yhtalot);Voidaan my\303\266s k\303\244ytt\303\244\303\244 #lhs (left hand side) ja rhs (right hand side)-tapaa, mutta se on hiukan v\303\244hemm\303\244n elegantti ja my\303\266s altis Maplen harrastamalle j\303\244rjestyksen vaihtumiselle ratkaisujoukossa. Niinp\303\244 t\303\244t\303\244 tyyli\303\244 ei voida ajaa automaattisesti, vaan tuloksiin on puututtava interaktiivisesti, mik\303\244 voi olla kiusallista, jos on "vakavahenkisest\303\244" dokumentista kyse. Joka tapauksessa t\303\244m\303\244kin tyyli kannattaa omaksua.rhs(ratk[1]);lhs(ratk[1]);lhs(ratk[2])=rhs(ratk[2]);Ratkaisun sijoittaminen muuttujan arvoksiT\303\244ss\303\244 nyt on toisin sanoin selitetty samaa, pyyhi omasta dokustasi pois tai j\303\244rjest\303\244 paremmin.Tietysti n\303\244in:X:=rhs(ratk[1]);Y:=rhs(ratk[2]);Turvallisempi ja elegantimpi tapa (jota edell\303\244 mainostettiin): Suorita ratk-yht\303\244l\303\266iden ilmaisemat korvaamiset lausekkeessa (pelkk\303\244) x ja
suorita ratk-yht\303\244l\303\266iden ilmaisemat korvaamiset lausekkeessa (pelkk\303\244) y .Edellisess\303\244 y-yht\303\244l\303\266 j\303\244\303\244 vaille k\303\244ytt\303\266\303\244 (kun pelkk\303\244 x ei sis\303\244ll\303\244 y:t\303\244) ja j\303\244lkimm\303\244isella vastaavasta syyst\303\244 x-yht\303\244l\303\266. ratk-joukon j\303\244rjestys ei n\303\244yttele mit\303\244\303\244n osaa.X:=subs(ratk,x);Y:=subs(ratk,y);Tehokkain, mutta turvattomin tapa, joskus kyll\303\244kin tosi k\303\244tev\303\244: assign(ratk);assign(ratk);x;y;yhtalot;solve(yhtalot,{x,y}); # assign-komennon j\303\244lkeen x ja y eiv\303\244t ole vapaita muuttujia, josta syyst\303\244 solve menee virheeseen.assign(ratk); toimii ik\303\244\303\244nkuin ratk-yht\303\244l\303\266iss\303\244 yht\303\244l\303\266merkki (=) vaihdettaisiin sijoitusmerkkin (:=) Yht\303\244l\303\266iden toteutuminen saadaan hyvin k\303\244tev\303\244sti tarkistetuksi kirjoittamalla vain yhtalot; Sensijaan yht\303\244l\303\266iden ratkaisua ei voida toistaa, ennekuin muuttujat x ja y on vapautettu arvoistaan.x:='x':y:='y':solve(yhtalot,{x,y});
<Text-field style="Normal" layout="Normal"></Text-field>Olkoon 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 Selvit\303\244 m\303\244\303\244rittelyjoukko, \303\244\303\244riarvot, asymptootit, nollakohdat, k\303\244\303\244nnepisteet ym. Piirr\303\244.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y+Rj5GQEZDRj4=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 Nimitt\303\244j\303\244n nollakohdat: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 Huom! Jos haluat v\303\244ltt\303\244\303\244 denom-komennon, t\303\244ytyy nimitt\303\244j\303\244lauseke ensin kopioida ja sitten maalata ja hiiren oikea.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiP0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEkcXVvRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2V4ZWN1dGFibGVHRjRGLw== Vaakasuora asymptootti: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
<Text-field style="Heading 2" layout="Heading 2">Extreme value 1 p. 44-45</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRitGK0YrLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUZHNiVRInhGJ0ZKRk1GL0YvLUYsNi1RIj1GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGZW4tSSZtZnJhY0dGJDYoRlQtRiM2JS1JI21uR0YkNiRRIjhGJ0YvRkpGTS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZW8vJSliZXZlbGxlZEdGNC1GLDYtUSIrRidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjIyMjIyMjJlbUYnL0ZFRl5wLUZobjYoLUZdbzYkUSIyRidGLy1GIzYlRlRGSkZNRmBvRmNvRmZvRmhvLUYsNi1RIixGJ0YvRjIvRjZGTEY3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJ0YrRlQtRiw2LVEiPkYnRi9GMkY1RjdGOUY7Rj1GP0ZaRmZuLUZdbzYkUSMwLkYnRi9GK0YrRi8=M\303\244\303\244rit\303\244 3:n levyisen pystykaistaleen minipinta-ala.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 Ei tarvirta janojen piirtoa erikseen, selkeint\303\244 ja havainnollisinta samalla on t\303\244m\303\244: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
<Text-field style="Heading 2" layout="Heading 2">Israel Lab 10, Exa 1</Text-field>Tutki ja piirr\303\244: 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 Osoittajan nollakohdat:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkb3NvRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEmbnVtZXJGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJmRidGL0YyLUZTNiQtRiM2Iy1GLDYlUSJ4RidGL0YyRjlGOQ==JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1"></Text-field>JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">K\303\244yr\303\244 ja tangentti</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">Perustilanne</Text-field>restart:
with(plots): Selvit\303\244 itsellesi, mit\303\244 seuraavassa tehd\303\244\303\244n. Mieti tarkkaan, miksi jossain pit\303\244\303\244 antaa plot:lle argumentiksif(x) ja miksi taas jossain esim. tangentti. eik\303\244 tangentti(x). Voit sitten huvitella vaihtelemalla funktion m\303\244\303\244ritelm\303\244\303\244ja / tai pistett\303\244 x0. Kirjoitetaan pieni malliskripti, jonkalaisia voit tehd\303\244 moninaisissa yhteyksiss\303\244.x0:=0.5; f:=x->sqrt(x); # Vaihtuvat sy\303\266tteet.QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSJmR0YoNiNJInhHRigvRi07IiIhIiIiRjE= df:=diff(f(x),x); # Derivaattalauseke.
kk:=subs(x=0.5,df); # Tangentin kulmakerroin pisteess\303\244 x=1tangentti:=f(x0)+kk*(x-x0); Piirret\303\244\303\244n kuva niin, ett\303\244 talletetaan k\303\244yr\303\244n, tangentin ja sivuamispisteen kuvat omiin grafiikkamuuttujiinsa, jotka sitten n\303\244ytet\303\244\303\244n yhteisesti display-komennollafkuva:=plot(f(x),x=0..1,color=red): # Lopussa kaksoispiste, kun halutaan vain kuvan tallennus.
tangkuva:=plot(tangentti,x=0..1,color=blue):
p0kuva:=plot([[x0,f(x0)]],style=point,symbol=circle,symbolsize=15,color=black):display([fkuva,tangkuva,p0kuva]); # Puolipiste, kun kuva halutaan n\303\244hd\303\244.Kokeile (mutta vain kerran el\303\244m\303\244ss\303\244si, silloinkin syv\303\244sti katuen, ett\303\244 olit yllytyshullu), mit\303\244 tapahtuu, kun vaihdat (:) -> (;) vaikkapa fkuva:=; ... yll\303\244. PLOT-tietorakenne paljastuu.- No, mene edit-valikkoon ja -> remove output, kyll\303\244 se siit\303\244.JSFH
<Text-field style="Heading 1" layout="Heading 1">Linear approx, Newton's method (Israel s. 52 ->)</Text-field>
<Text-field style="Heading 2" layout="Heading 2">Esim: Tutkitaan funktion <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2"></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2"></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field bookmark="Piirtoa" style="Heading 1" layout="Heading 1">10. Piirtoa</Text-field>[SOL] ss. 10 - 11[HAM] Luku 3 ss. 89 - 100
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Tavallinen k\303\244yrien piirto</Font></Text-field>K\303\244yr\303\244parven voi sulkea joko joukkosulkuihin { } tai listasulkuihin [ ] .
J\303\244lkimm\303\244inen lienee suositeltavaa, koska j\303\244rjestys on silloin k\303\244ytt\303\244j\303\244n hallinnassa (esim jos halutaan v\303\244rit hallitusti, tms).plot([sin(x),cos(x)],x=0..4*Pi);Ota hiiren vasemmalla kiinni kuvasta, saat kehyksen ja uudet valikot. Valitse sama mittakaava akseleilla (1:1)Jos grafiikka alkaa hidastaa ty\303\266arkin selausta, kannattaa valita EDIT-valikon alimmasta,
Remove output. Sit\303\244 kannattaa k\303\244ytt\303\244\303\244 muutenkin silloin t\303\244ll\303\266in. Ty\303\266arkki tiivistyy (ja nopeutuu) kummasti. Outputit saa takaisin saman EDIT-valikon toiseksi alimmaisesta execute worksheetHuomaa: Pi on Iso P, pieni i . (pi kirjoittuu oikean n\303\244k\303\266isesti, mutta Maple ei tunnista sit\303\244
muuksi kuin kreikkalaiseksi kirjaimeksi. ) [No johan tuosta huomauteltiin.]
<Text-field style="Heading 2" layout="Heading 2">Parametrimuotoinen piirtomatrixplot(logtaulu);
</Text-field>Syntaksi poikkeaa aika v\303\244h\303\244n, t\303\244m\303\244 t\303\244ytyy vain oppia (tai katsoa aina uudelleen helpist\303\244).Aina ei ole hauskaa joutua valitsemaan hirell\303\244, samaskaalaiisuus (ja monia muita) voidaan antaa plot-komennon tarkentimena.plot([cos(t),sin(t),t=0..2*Pi],scaling=constrained);Pisteet voidaan yht\303\244 hyvin ajatella kompleksitason pistein\303\244. complexplot osaa k\303\244sitell\303\244 suoraan reaalimuuttujan kompleksiarvoista funktiota. T\303\244ss\303\244 on sekin mukava piirre, ett\303\244 syntaksi on luonnollisempi ja siten helpompi muistaa kuin tuossa parametrimuotoisessa R^2-piirrossa.with(plots):complexplot(cos(t)+I*sin(t),t=0..2*Pi,scaling=constrained,color=blue);Edellinen voidaan ilmaista lyhyemmin eksponenttifunktion (Eulerin kaavan) avulla. Piirret\303\244\303\244n vaihteeksi pelk\303\244t pisteet.complexplot(exp(I*t),t=0..2*Pi,scaling=constrained,color=cyan,style=point);
<Text-field style="Heading 2" layout="Heading 2">Taulukoidun datan piirto ("listaplotti")</Text-field>Sometimes we will be interested in functions defined in terms of a discrete
table of values rather than a formula. For example, consider the following table of
temperatures recorded at various times on a spring day in Raleigh.
Time: | 6:00 a.m. | 10:00 a.m. | 12:00 p.m. | 4:00 p.m. | 5:00 p.m.
----------------------------------------------
Temp: | 45 deg. | 57 deg. | 65 deg. | 67 deg. | 66 deg.
Table of Temperatures on a Spring Day in Degrees Fahrenheit
To plot this data you must first define the points as a list. Since the time is recorded in a cyclic fashion, we will plot the times in the so-called "military style", i.e. 6:00 a.m. is 0600 and 6:00 p.m. is 1800.
ListOfPoints := [[600,45], [1000,57], [1200,65], [1600,67], [1700,66]];The plot command allows you to plot a list of points and to connect them with straight lines (unless you choose another option). This is illustrated in the next command. Notice also the option which produces a title on the plot.
plot(ListOfPoints, title="Temperatures at Various Times");
While the data can be plotted in this way, because the data is given for a discrete set of points you may prefer the style=POINT option as illustrated in the next command. Note also the effect of the symbol=CIRCLE option.
plot(ListOfPoints, style=POINT, symbol=CIRCLE,symbolsize=15, title=`Temperatures at Various Times`);
<Text-field style="Heading 3" layout="Heading 3">Exercise</Text-field>1. Plot the graph of 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 over the interval [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUYsNiVRJSZwaTtGJy8lJ2l0YWxpY0dGOkY1RjUtRjI2LVEiLEYnRjVGOC9GPFEldHJ1ZUYnRj1GP0ZBRkNGRS9GSFEmMC4wZW1GJy9GS1EsMC4zMzMzMzMzZW1GJ0ZMRjU=] .2. Make a plot of the points in the following table:
X: | -1 | 0 | 1 | 2 | 3
-------------------------------------------
Y: | 2.72 | 1 | 0.368 | 0.135 | 0.0498
Make two plots, one with the points connected and the other with only the data points.
Miten saadaan k\303\244tev\303\244sti koordinaattiparien lista [[x1,y1],[x2,t2],...] ?Olkoon annettu numeeriset listat (Matlabissa sanoisimme vektorit) X ja Y. Matlabin plot toimii tyyliin plot(x,y);Maplessa t\303\244ytyy muodostaa parien lista vaikkapa t\303\244h\303\244n tapaan:a:=-Pi:b:=Pi:N:=20:h:=(b-a)/N:x:=seq(evalf(a+i*h),i=0..N);Olkoot y-arvot vaikkapa sin-funktion arvoja x-pisteiss\303\244. Parijono saataisiin nyt n\303\244in:xy:=[seq([x[i],sin(x[i])],i=1..N+1)];plot(xy);plot(xy,style=point,symbol=circle);Huom: plots-pakkauksessa on listplot ja pointplot. Maplelle luonteenomaista on, ett\303\244 siin\303\244 on joukko redundantteja funktioita. Mielest\303\244ni on parempi oppia k\303\244ytt\303\244m\303\244\303\244n harvempaa ydinfunktiojoukkoa kuin omaksua monenlaisia synonyymej\303\244. Niiss\303\244 voi toki olla joitakin uusia mahdollisuuksia, mutta esim. n\303\244iss\303\244 ei v\303\244ltt\303\244m\303\244tt\303\244 ole (kuka noita kaikkia tuhansia ehtii penkoa).TaulukointiT\303\244ss\303\244 teimme itse asiassa taulukon, joka on havainnollisempi usein matriisina.txy:matrix(xy):linalg[transpose](xy):Tilan s\303\244\303\244st\303\244miseksi otetaan v\303\244hemm\303\244n dataa esimerkkiimme. J\303\244tet\303\244\303\244np\303\244 my\303\266s evalf pois.a:=-Pi:b:=Pi:N:=8:h:=(b-a)/N:x:=seq(a+i*h,i=0..N);
xy:=[seq([x[i],sin(x[i])],i=1..N+1)];
matrix(xy);linalg[transpose](xy);array(xy); # Taas synonyymi, mutta tiettyj\303\244 erojakin yleisyydessa / matriisilaskukyvyiss\303\244.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Grafiikoiden yhdist\303\244minen, plots[display] ja textplot</Font></Text-field>T\303\244ss\303\244 tarvitaan lis\303\244grafiikkapakkaus plotsrestart:with(plots): Selvit\303\244 itsellesi, mit\303\244 seuraavassa tehd\303\244\303\244n. Mieti tarkkaan, miksi jossain pit\303\244\303\244 antaa plot:lle argumentiksif(x) ja miksi taas jossain esim. tang. Voit sitten huvitella vaihtelemalla funktion m\303\244\303\244ritelm\303\244\303\244ja / tai pistett\303\244 x0. Kirjoitetaan pieni malliskripti, jonkalaisia voit tehd\303\244 moninaisissa yhteyksiss\303\244.x0:=1: f:=x->x^3: # Vaihtuva sy\303\266te df:=diff(f(x),x): kk:=subs(x=1,df); x0:=1: tang:=f(x0)+kk*(x-x0);fkuva:=plot(f(x),x=0..2,color=red): tangkuva:=plot(tang,x=0..2,color=blue): p0kuva:=plot([[x0,f(x0)]],style=point,symbol=circle,symbolsize=15,color=black):display([fkuva,tangkuva,p0kuva]);Kokeile (mutta vain kerran el\303\244m\303\244ss\303\244si, silloinkin syv\303\244sti katuen, ett\303\244 olit yllytyshullu), mit\303\244 tapahtuu, kun vaihdat (:) -> (;) vaikkapa fkuva:=; ... yll\303\244. PLOT-tietorakenne paljastuu.- No, mene edit-valikkoon ja -> remove output, kyll\303\244 se siit\303\244.Sometimes it will be desirable to print text on a plot. The plots procedure textplot will allow this. The following commands will serve as examples of the textplot and display routines. Make sure you notice where colons and semicolons are used in these commands.
In the next commands it is important to use colons to punctuate the first two statements. Otherwise Maple will ouput an entire page of text describing the plot structure rather than the graph. The textplot in the second command causes the text x=0.6356,y=0.5000 to be printed at the point (.6356,.5000). The statement align=RIGHT aligns this text to the right of the point. f := x -> exp(-x)*sin(3*x);
plot1 := plot(f, 0..3):# tai plot1:=plot(f(x),x=0..3):
plot2 := plots[textplot]([0.6356, 0.5000, "x=.6356, y=.5000"], align=RIGHT):
plots[display]({plot1, plot2});Huom:plots-pakkauksen (kuten muidenkin pakkausten) funktioita voidaan k\303\244ytt\303\244\303\244 my\303\266s lataamattakoko pakkausta, t\303\244ll\303\266in pakkauksen nime\303\244 ik\303\244\303\244nkuin indeksoidaan ao. funktion nimell\303\244tyyliin plots[display]
<Text-field style="Heading 2" layout="Heading 2">3d-grafiikkaa, animaatiot</Text-field>Otetaan l\303\244mm\303\266njohtumisesimerkki, jossa valaistaan3d-pinnan piirtoa Animaatiota3d-kuvan projektiok\303\244yr\303\244parven piirtoaTarkastellaan sivuiltaan l\303\244mp\303\266eristetty\303\244 sauvaa, jonka p\303\244\303\244t upotetaan hetkell\303\244 t=0 j\303\244\303\244vesis\303\244ili\303\266ihin (0 astetta) ja jonka
alkul\303\244mp\303\266tilajakauma on
f:=x->100*sin(Pi*x/80);Huomaa funktiom\303\244\303\244ritys.Olkoon sauvan pituus L=80.
T\303\244ss\303\244 tapauksessa l\303\244mp\303\266yht\303\244l\303\266n ratkaisuna olevasta Fourier-sarjasta tulee vain yksi termi. Ratkaisu on (sopivalla l\303\244mm\303\266njohtumiskertoimella)
u:=(x,t)->100*sin((Pi*x)/80)*exp(-0.001785*t);
<Text-field style="Heading 3" layout="Heading 3">Pintapiirros</Text-field>with(plots):plot3d(u(x,t),x=0..80,t=0..400);Klikkaa hiirell\303\244 kuvaan (ja tarvittaessa valitse "boxed") ty\303\266kalunauhasta.Kierr\303\244 kuvaa hiirell\303\244.Kokeile STYLE-valikosta PATCH ja PATCH with contour valintoja.
<Text-field style="Heading 3" layout="Heading 3">Animaatio</Text-field>animate(u(x,t),x=0..80,t=0..300,frames=30,color=blue);
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">L\303\244mp\303\266tilaprofiilit ja korkeusk\303\244yr\303\244t</Font></Text-field>plot({seq(u(x,t),t=[0,10,50,100,200,300,400])},x=0..80);contourplot(u(x,t),x=0..80,t=0..400);
<Text-field style="Heading 3" layout="Heading 3">Implicitplot</Text-field>T\303\244m\303\244 on periaatteessa sama kuin yhden korkeusk\303\244yr\303\244n piirto.implicitplot([x^2+y^2=1,x^3+y^3=1,x^10+y^10=1],x=-2..2,y=-2..2,color=[blue,red,green]);