Computer algebra-system-maple

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Computer Algebra System in EducationMaple

Azat Azhibekov

[email protected] of Computer Education and Instructional Technologies,Fatih

University,34500 Buyukcekmece,Istanbul,Turkey

May 23, 2015

FU BOTE

http://bote.fatih.edu.tr/?&language=EN

IntroductionGeneral Purpose CAS

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Symbolic Computation inMaple

Maple inEducation

Mathematics with Maple

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Computer Algebra System2 types

Computer Algebra System (CAS) is a type of softwarepackage that allows to manipulate mathematical objects.The primary goal of CAS is to manipulate automatetedious and sometimes difficult algebraic manipulationtasks.

Specialized onesFORM-for particle PhysicsFermat-for resultant computation and linear algebra withpolynomial entriesPARI/GP-for number theory

General Purpose onesMaple-used for teaching and scientific purposesMATLAB(matrix laboratory)-multi-paradigm numericalcomputing environment and fourth-generationprogramming language

Azat Azhibekov Computer Algebra System in Education May 23, 2015 2/31


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General Purpose CAS’s features

a user interfacea programming language and interpretera simplifier, which is a rewrite system for simplifyingmathematics formulasa memory manager,including a garbage collectoran arbitrary-precision arithmetic,needed by huge size ofintegers that may occura large library of mathematical algorithms



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Maple-a general purpose algebra system

Maple is one of the leading general-purpose commercialCAS and widely used in engineering,science,andmathematics. Maplesoft customers include:

FordBMWBoschBoeingNASACanadian Space AgencyCanonMotorolaMicrosoft ResearchBloombergDreamWorks



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Maple-a general purpose algebra system

Covering sectors:automotiveaerospaceelectronicsdefenceenergyfinancial servicesconsumer productsentertainmentEducation– Over 90% of advanced research institutionsand universities worldwide,includingMIT,Stanford,Oxford, the NASA Jet PropulsionLaboratory and the U.S. Department of energy,haveadopted Maplesoft solutions to enhance education andresearch activities.



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Maple

Maple is a very powerful interactive computer algebrasystem for doing maths and used for numerical andsymbolic computation

Definition (Symbolic Computation)

In mathematics and computer science,computeralgebra,also called symbolic computation or algebraiccomputation is a scientific area that refers to the study anddevelopment of algorithms and software for manipulatingmathematical expressions and other mathematical objects

Symbolic Computation in Maple(11)

∫ ∞−∞

e−x2dx =

√π



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Maple in Math Education

In every class there are places,where we can useTechnology to help advance understanding,and Maplemust be used to do that effectively.

Maple is a tool for doing mathematicsMaple allows you do math in your own wayMaple makes you love itself as well as mathematicsResearchers recommend that teachers use CASfeatures to focus on concepts,personalize curricular to fitstudent needs,and emphasize meaningful math tasks.



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Commands useful in Calculus I

>with(Student): A Maple package is called bywith(packagename):>diff(f ,x$n): Finds the nth derivative of f with respect to x(f @g)(x): f (g(x)) (composition function)

>f := x → 2x3 + 5:>g := x → x2

>diff(f (x), x); 6x2

>diff(f (x), x$3); 12>diff(((x2 + x7)/x5), x); −3+2x5

x4

>(f @g)(x); 2x6 + 5>(g@f )(x); (2x3 + 5)2

NOTE: All results in this presentation were computed in Maple11 Classic WorksheetAzat Azhibekov Computer Algebra System in Education May 23, 2015 8/31


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>f := x → F : arrow notation to define f as a function ofx>evalf(a): evaluates the expression a using floating pointarithmetic>Digits:=n: sets the number of digits used for floatingpoint numbers to n (the default is 10)

>f := x → 3x + 5:>f (2); 11>f (5x); 15x + 15>Digits:=6:>evalf(Pi); 3.14159>evalf(exp(1)); 2.71828



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Some Symbols in Maple

Like any good computing system,Maple has a certaincollection of of well-used mathematical constants andfunctions available.

I;Pi;exp(1);infinity;alpha;theta;lambda;gamma;omega;Omega;

sqrt(-1) (imaginary number)π (pi;π,but it’s not defined thereforeMaple doesn’t know its value)e (Euler number)∞α

θ

λ

γ (Euler’s constant)ω

Ω



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>expand(f ): expands the expression f using the laws ofalgebra and trigonometry>factor(f ): factors the given expression>fsolve(f = a, x): solves the equation f = a for x . Theanswer is given in decimal form>rationalize(a): rationalizes denominator

>expand((a + b)2); a2 + 2ab + b2

>factor(x2 − 2x − 63); (x − 9)(x + 7)>fsolve(r3 + 4 = 45, r ); 3.448217240>rationalize(2/(2−

√2)); 2 +

√2



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Some important commands

?name: help descriptions of syntax, data types andfunctions#comment : All characters that follow a poundcharacter on a line are considered to be part of acomment.%: gives the previously computed result,Mapleremembers previous three (%,%%,%%%):= assignment operator;(semi-colon): (each instruction to Maple mustend with colon or semi-colon) output is printed:(colon): output is not printedunassign(’a’): unassigns names



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>Slope(p1,p2): computes the slope of the linethrough the points p1 and p2>limit(f , x = a): finds the limit of f as x approaches a>solve(f = a, x): solves f (x) for x>root(x ,n): nth root of x

>limit(sin(x)/x ,x = 0); 1>limit(exp(b),infinity); ∞>limit(−1/x , x = 0,right); −∞>solve(sin(x) + y = 2, x); −arcsin(y − 2)>solve(x2 − 9 = 0, x); ±3>root(32,5); 2>Slope([0,0], [1,2]); 2



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>int(a, x): computes the indefinite integral ofexpression a with respect to x>int(a, x = b..c): computes the definite integral of awith respect to x>maximize(a,opt1,opt2..optn): computes globalmaximum value of a>minimize(a,opt1,opt2..optn): computes globalminimum value of a

>int(2 + x , x); 2x + x2

2

>int(4x2 − 2/x5 + 7, x); 4x3

3 + 12x4 + 7x

>int(exp(x)/2, x); 12ex

>maximize(x3 − 3x2 − 9x + 5, x = 0..4,location);5, [x = 0,5]>minimize(x2 + 1, x = −1..2); 1Azat Azhibekov Computer Algebra System in Education May 23, 2015 14/31


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Some important commands II

restart: clears all previously assigned variables,makesMaple act like as if just started

: executes the current expression

: executes the worksheet[list ] (ordered): z := [c,a,b]: >z [1]; cset (unordered,no duplicates): >a,b,a, c; a,b,c>=: ≥ (greater than or equal)<>: 6= (not equal)<=: ≤ (less than or equal)@: composition operator(composition function)"text": assigns nothing but only text



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>subs(x = a, f ): substitutes a for x in the expression f>convert(expr,form): converts expression to anotherform>collect(expr,xn): collects coefficients of like powers ofx orxn

>coeff(p,xn): extracts a coefficient of a polynomial in x

>subs(x = y3, x2 + 9x); y6 + 9y3

>subs(x = 0, y = −1, z = Pi , x + y + cos(z)); −1 + cos(π)>convert(Pi,degrees); 180

>convert(9,binary); 1001>collect(a3x − x + a3 + a, x); (a3 − 1)x + a3 + a>coeff(yx3 + x2y − x2y2 − xy − 2x2y2x − y − y2 − y3x +y3 + y4, x2); −y2 + y



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Basic Functions

+: addition(plus)−: subtraction(minus)∗: multiplication(times)/: division(fraction)^: exponentiation(power)abs(x): absolute valuesqrtx : square rootn!: factorialsin(x): sinecos(x): cosine

tan(x): tangentsec(x): secantcsc(x): cosecantcot(x): cotangentlog(x): general logarithmln(x): natural logarithmexp(x): exponentialfunctionsinh(x): hyperbolic sinecosh(x): hyperbolic cosinetanh(x): hyperbolic tangent



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>iscont(f , x = a..b,′ openor closed ′): tests continuity off on an interval a..b>gcd(a,b): finds greatest common divisor of a andb(polynomials)>lcm(a,b): finds lowest common multiple(polynomials)>discont(f , x): finds the discontinuities of f over thereals

>iscont((x + 2)/(x − 2), x = 1..2,’closed’); false>iscont((x + 2)/(x − 2), x = 1..2,’open’); true>discont((x2 + 1)3/(1− x2), x); 1,−1>gcd((X 2 − y2)/(x3 − y3)); −y + x>lcm(x2 + 2x + 1, x + 1); x + 1



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>piecewise(cond1, f1, cond2, f2..condn, fn):piecewise-continuous functions>simplify(a): simplifies expression a

>piecewise(x2 > 4and x < 8, f (x));f (x), 4 < x2andx < 80, otherwise

>simplify(%); f (x), x < −20, x ≤ 2f (x), x < 80, 8 ≤ x



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Plotting 2D Graphs

>plot(f , x): creates two-dimensional plot>plot(f , x = a..b, y = c..d ,opt1,opt2..optn): the plotcommand has many options>plot(f1, f2..fn, x = a..b,options):

>plot(x2, x = −1..2, y = −1..2, title =

”Plot example(Graphof x2)”, thickness = 2, color =

blue, style = point);>plot(piecewise(x ≤ 1, x3 − 3, x > 1,2x + 4),x = −11..11, y =

−5..30,discont = true, color = green, title =

”Graphof piecewise function);



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2D Graph



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Some plot options

You can change the view of graphs by applying plot options

axes="boxed,frame,none or normal"

color="Orange,Silver,Gold,Green,Coral,BlueViolet,Lime,Yellow,DeepSkyBlue,etc."

discont=true,false

filled=true,false

labels=[x,y]

labeldirections=[horizontal,vertical]

linestyle=solid,dot,dash,dashdot,longdash,spacedash,spacedot

numpoints=n (default is 50 points)

style=line,point,patchnogrid,patch

thickness=nAzat Azhibekov Computer Algebra System in Education May 23, 2015 22/31


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2D Graph

>plot([sqrt(x),3log(x)], x = 0..400,numpoints =

1000, thickness = 3);



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Plotting 3D Graphs

For some special plots you need special commands thatare in plots package

>plot3d(f , x = a..b, y = c..d): createsthree-dimensional plot>plot3d([f ,g,h], s = a..b, t = c..d):>plot3d([f ,g,h],a..b, c..b):>plot3d([exprf ,exprg,exprh], s = a..b, t = c..d):

>plot3d(x2, x = −2..2, y = 1..5,axes = boxed , scaling =constrained , color = ”SkyBlue”, style = patch, title =”3Dgraphof x2);



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3D Graph



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3D Graph

>plot3d(xexp(−x2 − y2), x = −2..2, y = −2..2, color = x);>plot3d((1.3)xsin(y), x = −1..2Pi , y = 0..Pi , coords =spherical , style = patch);



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Some 3D Plot Options

axes=boxedcaption="c"coords=polar,spherical,cylindrical,conical,bisphericalfont=[family , style, size]

family=TIMES,HELVETICA,COURIER,SYMBOLTIMES→ style=ROMAN,BOLD,ITALIC,BOLDITALICHELVETICA and COURIER→style=BOLD,OBLIQUE,BOLDOBLIQUESYMBOL→ style=no style

lightmodel=none,light1,light2,light3,light4scaling=constrained,unconstrainedstyle=surface,patch,contour,patchcontour,line,pointsymbol=asterisk,box,circle,diagonalcross,diamondpoint,solidsphere,spheresymbolsize=n(default=10)



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Potting points

>plot([[x1, y2], [x2, y2], [x3, y3]..., [xn, yn]]): Plots points

>L:=[[0,0], [1,1], [2,3], [3,2], [4,−2]]:

>plot(L);



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Ford Motor Company uses Maple

PROBLEMFord Motor Company wrestled with -incessant noise andvibration in chain noise. Ford detected a severe1800− 1900Hz chain noise,and sound pressure levelswere 10− 15db over nominal values and the cause was

unknown.



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