Functions and Operators in MAPLE and in MATLAB

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Functions and Operators in MAPLE and in MATLAB

• Matthias Kawski
• Dept of Math Statistics
• Arizona State University
• Tempe, AZ. U.S.A.

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• Longstanding interestHow do different new
  technologies interfere with the development of
  key concept images.
• Immediate trigger (for this article)Our
  engineers again asking to replace MAPLE by
  MATLAB in our introductory calculus
• Immediate reactionNot again, do they never
  learn/understand?
• . but, after some
  reflection, .

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Institutional background

• Large state university w/ major research
  focus.50 000 studentsMath 13 000 /
  semester.All in small classes!!!
• Calculus and above - main client engineering
  students
• Engineering Aggressive reforms of instructional
  practices,largely driven by demanding
  accreditation reviews (ABET).Some key points
  Teamwork, problem solving, PROFESSIONAL tools
  (technology)

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The largest classes w/ engineers
Calculus I
Calculus III
Calculus II
Diff eqns
VectCalc / Fourier PDE
Linear Algebra
MAPLE
both

• Some key concepts
• Functions
• Derivatives and integrals (limits)
• Linearity

MATLAB
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Functions

• Many different points of view
• abstractly function graph
• covarying quantities (relation satisfying
  vertical line test)
• formulas yf(x)
• mappings
• computable assignments
• procedure, subroutine, method

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Functions and their graphs
Adaptive grid not at all transparent
Making arguing from table is so important!
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Variables CAS teaches precision
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CAS (pure) functions
Math no need for an x in defn of function,
plot, composition , derivative The sciences are
different e.g. always differentiate W.R.T some
QUANTITY
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CAS function composition
The most important operation is function
composition. in CAS ?? in MATLAB ??
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CAS gem Chain-rule w/o variables
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MATLAB functions

• Everything is a matrix, tables of numeric
  values
• Predefined functions usually heavily
  overloaded(i.e. domains are unions, MATLAB
  guesses what the user must have intended)
• Functions as subroutines or external m-files,no
  arrow analogue or function w/o variable names
• Composition OK for procedures, not for tables
• Inverse only numerically as procedures

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MATLAB function example Division
Division by matrix if not invertible yields,
of course, least square solution! No warning
message
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Summarizing observations

• CAS and MATLAB support very different aspects of
  the function concept.
• Integration versus add-on The students
  under-stand what matters is what is on the final
  exam.Allowing MATLAB / CAS on all exams ?is
  the tool useless, does it trivialize the exam,
  oris it a meaningfully integrated professional
  tool?

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Traditional calculus is algebra, not analysis
Limits are fundamentally topological, yet in
trad calculus exams they are all
algebra. Differentiation is about
approximability bylinear objects, using
limits.Derivations are purely algebraic
operations The traditional calc texts,
and also the exams are about derivations!
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Final thought

• Maybe, maybe,when our engineers ask that we use
  MATLAB in calculus, and that we INTEGRATE it
  into the curriculum,maybe they really mean
  that we should reformthe calculus, diffeq,
  linalg, curriculaSO THAT MATLAB can easily
  be integratedintegrability of MATLAB just an
  INDICATORthat the courses match modern
  demands.!!!