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ProblemSolvingwithMaple

Ahandbookforcalculusstudents

CarlEberhart,carl@ms.uky.edu

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December12,2003

Contents

1Raisond'Maple4

2AnintroductiontotheMaplelanguage9

3SettingUpandSolvingProblems28

4MoreworkedProblems41

5Dierentiationanditsuses.49

1

6MoreMax-minProblems59

7EarlyIntegration.67

7.5Modelingthe

8MomentsandCenterofMass77

9DenitionsandTheoremsofCalculusI80

10InverseFunctions83

11IntegrationTechniquesandApplications94

12Taylor'sTheorem104

2

13SequencesandSeries109

13.3.1TheSnow

14Dierentialequations121

3

1Raisond'Maple

1.1FourPropertiesofMaple

Forexample,thecommand>

displayed). 4 language,andfosterourexperimentalurges.

3*(4+5*3)*(7+6;Syntaxerror,`;`unexpected

occursandmakethecorrection s:=3^23+4^12; ifactor(s);s:=94159956043 (727)(129518509)5 theinputcell,andreexecuteit. yourworksheetiftheneedarises.> plot(x^3-x+4,x=-2..2);±20246810 ±2

±112x

plot(x^3-x+4,x=-2..2);

±20246810

±2

±112x

TextorCommentCells:

6 document.

1.3Gettoknowthelanguage

,etc.YoucanlearnaboutthemwithonlineHelp.

Forexample,tondoutaboutfactorjusttype>

examplesoftheusageoftheword,inmanycases.

1.3.1Problems:

7

1.4Experiment!

1.4.1Problems

example(list); intheexamplesheetforsetandlist. 8

2AnintroductiontotheMaplelanguage

2.1Arithmetic

2-3+4/5*6^7;1119739

5 isthesameasentering> (2-3)+(4/5)*(6^7);1119739 5

1/3+1/2;5

returnadecimalanswer.> entering> ofacircleofradius3,youwouldenter>

Pi*3^2;99

whenris3,weenter> oftheradiusr.

2.3Functions

numbers,butthenotionoffunctionismuchmore exiblethanthat.

Asanexpression:Theassignment>

plot(area,r=0..4);10

01020304050

1 234r
r=0tor=4. arrowoperator.Forexample,> statement> plot(area,0..4);

01020304050

1 234

Notethatthevariablerisomittedhere.

assignment> pol:=x^2+4*x-1;pol:=x2+4x111 thentheassignment> theprocedure,withtheMaplewordERROR. area:=proc(r)

Pi*r^2fiend;area:=proc(r)

endproc> area(3);9>

Notetheif..then..controlstatementhere.

plot3d(V,0..4,0..4,axes=boxed);12 0 1 2 3 40
1 2 3 40

50100150200

moreandmoreastimegoesby.

Piecewisedenedfunctions:

Hereisanexampletoshowusage.>

cos(x),3*x);f(x):=8 :x 3+8x1

7+2xx2

11cos(x)x4

3xotherwise>

f(2);f(2) g(2);11 plot(g,-3..6,style=point);

±15±10±5051015

±2246

13 infunctions,type> thex-axis.> f:=x->10*x^5-30*x+10;f:=x!10x530x+10> plot(f,-3..3);±2000±1000010002000 ±3

±2±1123

14 plot(f,-1.5..1.5);±20±10010203040

±1.5±1±0.50.511.5

asolution.> whattype(1/2);fraction> whattype(a+b);+ whattype(x^2+x=2*x-1);= example,> ;x+(sin@cos@(x!x2+3))z 15 viola[6];x+(sin@cos@(x!x2+3))z List. ;x+(sin@cos@(x!x2+3))z nops(explist);7 list.Thisdeviceisusedagainandagain.> nops([3,4,a]);3 multipliedbyanumber.> p:=[1,2];q:=[-3,1];p:=[1;2] q:=[3;1]> plot(ab);16

11.522.53

11.522.53

restrictingthexandycoordinates.> plot([1+4*cos(t),5+4*sin(t),t=0..Pi], scaling=constrained); 6789

±2024

thesequenceoflists> cos(t);1+12 sin(t);t=0::2];[3+12 cos(t);1+12 sin(t);t=0::2]; [3+12 cos(t);3+12 sin(t);t=0::2];[1+12 cos(t);3+12 sin(t);t=0::2]17 0.5

11.522.533.5

tosplitofromabtheoddandeventerms{> wecangetittodowhatwewant.> plot(newab,color=black);18

1.522.53

±10

±8±6±4±20246810

Sets listit.> plot(fx^2-2,2*x+5g,x=-5..5);

±505101520

±4

±224x

youhavenamed. 19 pl1:=plot(fx^2-2,2*x+5g,x=-5..5): pl2:=plot([[2,1],[3,20],[0,0],[2,1]]): plots[display]([pl1,pl2]);±505101520 ±4

±224x

TablesandArrays

exible.Thepackagesofspecial exceptforarrays. multiplication&*.> sincos#> ang:=evalf(Pi/180*31);ang:=0:5410520681> [0:3421292258;1:372205376]]20 plot(f[[0,0]],ab,rotabg);01234

±0.50.511.522.53

2.7Maplecontrolstatements

for..from..by..to..while..do..od; aniterativealgorithm.

Example:Adduptherst100numbers.>

s:=0:forifrom1to100dos:=s+iod: s;5050 doitagain,storingtheminanarray.

Solutionwithlists:>

locube:=NULL:#startwiththeemptyexprseq forifrom1to5do locube:=locube,i^3od:

Solutionwitharrays:>

aocube:=array(1..5):#initializethearray. aocube2:=8 aocube3:=2721 aocube4:=64 aocube5:=125> aocube[3]:=0;aocube3:=0> print(aocube);[1;8;0;64;125]> if..then..elif..else..;

Asolution:>

myabs(-23);23 toseewhatitlookslike.22 my absolute value

00.511.52

±2

±112

2.8ABriefVocabularyofMapleWords

browser.>

D(cos);#thedifferentialoperatorsin>

y:='y';#makesyavariableagain.y:=y> expand((x+b)^7);#expandstheproductx

Int(x*exp(x),x=0..1);#Apassiveintegral.Z

1

0xexdx>

±3±2±1123x>

f(3);#thenreturns9.9 24
2+2> iquo(23,2);#dividestheintegers11 irem(23,2);#givestheintegerremainder1 4ya i:='i';#makesiavariableagaini:=i>

2.9TroubleShootingNotes

workinglikeitshould. plotx^2;Error,missingoperatoror`;` 25
intoaninputcell,ifyouareexpectingoutput. x-3. thisistokeepaninputcellofvariablesused. inputcells. referencethelastcomputation. end;Thisoftenworks. yourworksheet. 26
outforyourinspection.quotesdbs_dbs7.pdfusesText_13