Computer algebra-system-maple

www.usm.my
Computer Algebra System in Education
Maple
Azat Azhibekov
azatazhibekov@gmail.com
Department of Computer Education and Instructional Technologies,Fatih
University,34500 Buyukcekmece,Istanbul,Turkey
May 23, 2015
FU BOTE

Introduction
General Purpose CAS
Maple in Use
Symbolic Computation in
Maple
Maple in
Education
Mathematics with Maple
Basic Function
Visualisation
Plots
Extra(Research)
Reference
Computer Algebra System
2 types
Computer Algebra System (CAS) is a type of software
package that allows to manipulate mathematical objects.
The primary goal of CAS is to manipulate automate
tedious and sometimes difﬁcult algebraic manipulation
tasks.
Specialized ones
FORM-for particle Physics
Fermat-for resultant computation and linear algebra with
polynomial entries
PARI/GP-for number theory
General Purpose ones
Maple-used for teaching and scientiﬁc purposes
MATLAB(matrix laboratory)-multi-paradigm numerical
computing environment and fourth-generation
programming language
Azat Azhibekov Computer Algebra System in Education May 23, 2015 2/31

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General Purpose CAS’s features
a user interface
a programming language and interpreter
a simpliﬁer, which is a rewrite system for simplifying
mathematics formulas
a memory manager,including a garbage collector
an arbitrary-precision arithmetic,needed by huge size of
integers that may occur
a large library of mathematical algorithms

Introduction
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Maple-a general purpose algebra system
Maple is one of the leading general-purpose commercial
CAS and widely used in engineering,science,and
mathematics. Maplesoft customers include:
Ford
BMW
Bosch
Boeing
NASA
Canadian Space Agency
Canon
Motorola
Microsoft Research
Bloomberg
DreamWorks

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Maple-a general purpose algebra system
Covering sectors:
automotive
aerospace
electronics
defence
energy
ﬁnancial services
consumer products
entertainment
Education– Over 90% of advanced research institutions
and universities worldwide,including
MIT,Stanford,Oxford, the NASA Jet Propulsion
Laboratory and the U.S. Department of energy,have
adopted Maplesoft solutions to enhance education and
research activities.

Introduction
General Purpose CAS
Maple in Use
Maple
Maple in
Education
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Maple
Maple is a very powerful interactive computer algebra
system for doing maths and used for numerical and
symbolic computation
Deﬁnition (Symbolic Computation)
In mathematics and computer science,computer
algebra,also called symbolic computation or algebraic
computation is a scientiﬁc area that refers to the study and
development of algorithms and software for manipulating
mathematical expressions and other mathematical objects
Symbolic Computation in Maple(11)
∞
−∞
e−x2
dx =
√
π

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Maple in Math Education
In every class there are places,where we can use
Technology to help advance understanding,and Maple
must be used to do that effectively.
Maple is a tool for doing mathematics
Maple allows you do math in your own way
Maple makes you love itself as well as mathematics
Researchers recommend that teachers use CAS
features to focus on concepts,personalize curricular to ﬁt
student needs,and emphasize meaningful math tasks.

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Commands useful in Calculus I
>with(Student): A Maple package is called by
with(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 Maple
11 Classic Worksheet

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>f := x → F: arrow notation to define f as a function of
x
>evalf(a): evaluates the expression a using floating point
arithmetic
>Digits:=n: sets the number of digits used for floating
point 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 certain
collection of of well-used mathematical constants and
functions available.
I;
Pi;
exp(1);
inﬁnity;
alpha;
theta;
lambda;
gamma;
omega;
Omega;
sqrt(-1) (imaginary number)
π (pi;π,but it’s not deﬁned therefore
Maple 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 of
algebra and trigonometry
>factor(f): factors the given expression
>fsolve(f = a, x): solves the equation f = a for x. The
answer 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 and
functions
#comment: All characters that follow a pound
character on a line are considered to be part of a
comment.
%: gives the previously computed result,Maple
remembers previous three (%,%%,%%%)
:= assignment operator
;(semi-colon): (each instruction to Maple must
end with colon or semi-colon) output is printed
:(colon): output is not printed
unassign(’a’): unassigns names

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>Slope(p1,p2): computes the slope of the line
through 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 of
expression a with respect to x
>int(a, x = b..c): computes the deﬁnite integral of a
with respect to x
>maximize(a, opt1, opt2..optn): computes global
maximum value of a
>minimize(a, opt1, opt2..optn): computes global
minimum value of a
>int(2 + x, x); 2x + x2
2
>int(4x2 − 2/x5 + 7, x); 4x3
3 + 1
2x4 + 7x
>int(exp(x)/2, x); 1
2 ex
>maximize(x3 − 3x2 − 9x + 5, x = 0..4,location);
5, {[{x = 0}, 5]}
>minimize(x2 + 1, x = −1..2); 1

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Some important commands II
restart: clears all previously assigned variables,makes
Maple act like as if just started
: executes the current expression
: executes the worksheet
[list] (ordered): z := [c, a, b]: >z [1]; c
{set} (unordered,no duplicates): >{a, b, a, c}; {a,b,c}
>=: ≥ (greater than or equal)
<>: = (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 another
form
>collect(expr,xn): collects coefﬁcients of like powers of
x 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 value
sqrtx: square root
n!: factorial
sin(x): sine
cos(x): cosine
tan(x): tangent
sec(x): secant
csc(x): cosecant
cot(x): cotangent
log(x): general logarithm
ln(x): natural logarithm
exp(x): exponential
function
sinh(x): hyperbolic sine
cosh(x): hyperbolic cosine
tanh(x): hyperbolic tangent

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>iscont(f, x = a..b, openor closed ): tests continuity of
f on an interval a..b
>gcd(a, b): finds greatest common divisor of a and
b(polynomials)
>lcm(a, b): finds lowest common multiple(polynomials)
>discont(f, x): finds the discontinuities of f over the
reals
>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((X2 − 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 < x2
andx < 8
0, otherwise
>simplify(%); 


f(x), x < −2
0, x ≤ 2
f(x), x < 8
0, 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 plot
command 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 piecewisefunction);

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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=n

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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 that
are in plots package
>plot3d(f, x = a..b, y = c..d): creates
three-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)x sin(y), x = −1..2Pi, y = 0..Pi, coords =
spherical, style = patch);

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Some 3D Plot Options
axes=boxed
caption="c"
coords=polar,spherical,cylindrical,conical,bispherical
font=[family, style, size]
family=TIMES,HELVETICA,COURIER,SYMBOL
TIMES→ style=ROMAN,BOLD,ITALIC,BOLDITALIC
HELVETICA and COURIER→
style=BOLD,OBLIQUE,BOLDOBLIQUE
SYMBOL→ style=no style
lightmodel=none,light1,light2,light3,light4
scaling=constrained,unconstrained
style=surface,patch,contour,patchcontour,line,point
symbol=asterisk,box,circle,diagonalcross,diamondpoint,
solidsphere,sphere
symbolsize=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
PROBLEM
Ford Motor Company wrestled with -incessant noise and
vibration in chain noise. Ford detected a severe
1800 − 1900Hz chain noise,and sound pressure levels
were 10 − 15db over nominal values and the cause was
unknown.

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