http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

Functions and Operators in MAPLE and in MATLAB

Matthias Kawski

Dept of Math & StatisticsArizona State UniversityTempe, AZ. U.S.A.

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

• Longstanding interest:How 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 reaction:“Not again”, “do they never learn/understand?”

…. but, after some reflection, ….

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

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)

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

The largest classes w/ engineers

Calculus I

VectCalc / Fourier PDELinear Algebra

Diff eqns

Calculus IIICalculus II

both

MATLAB

MAPLE

Some key concepts:

• Functions

• Derivatives and integrals (limits)

• Linearity

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

Functions

• Many different points of view

– abstractly: function = graph – covarying quantities

(relation satisfying vertical line test)– formulas y=f(x)– mappings – computable assignments– procedure, subroutine, method– …

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

Functions and their graphs

Adaptive grid – not at all transparent Making & arguing from table is so important!

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

Variables: CAS teaches precision

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

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

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

CAS: function composition

The most important operation is function composition.

– in CAS ?? – in MATLAB ??

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

CAS gem: Chain-rule w/o variables

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

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

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

MATLAB function example: Division

Division by matrix – if not invertible – yields, of course, least square solution! No warning message

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

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?

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

Traditional calculus is algebra, not analysisLimits 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!

Conclusion CAS matches traditional calculus – but it trivializes the exams. MATLAB is just a cute add-on – that is next-to-useless on the exams.

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Functions/operators in MAPLE MATLAB” Remarkable elta 03,Queenstown NZ

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 integrated

“integrability of MATLAB” just an INDICATORthat the courses match modern demands….!!!