1. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu Functions and Operators in MAPLE and in MATLAB Matthias Kawski Dept of Math & Statistics Arizona State University Tempe, AZ. U.S.A.

2. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu 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, ….

3. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu Institutional background Large state university w/ major research focus. 50 000 students Math: 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)

4. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu 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

5. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu 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 – …

6. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu Functions and their graphs Adaptive grid – not at all transparentMaking & arguing from table is so important!

7. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu Variables: CAS teaches precision

8. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu 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

9. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu CAS: function composition The most important operation is function composition. – in CAS ?? – in MATLAB ??

10. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu CAS gem: Chain-rule w/o variables

11. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu 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

12. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu MATLAB function example: Division Division by matrix – if not invertible – yields, of course, least square solution! No warning message

13. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu 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, or is it a meaningfully integrated professional tool?

14. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu Traditional calculus is algebra, not analysis Limits are fundamentally topological, yet in trad calculus exams they are all algebra. Differentiation is about approximability by linear objects, using limits. Derivations are purely algebraic operations: The traditional calc texts, and also the exams are about derivations! ConclusionCAS matches traditional calculus – but it trivializes the exams. MATLAB is just a cute add-on – that is next-to-useless on the exams.

15. Matthias Kawski “Functions/operators in MAPLE  MATLAB” Remarkable  elta 03,Queenstown NZ http://math.asu.edu/~kawski kawski@asu.edu 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 reform the calculus, diffeq, linalg, … curricula SO THAT MATLAB can easily be integrated “integrability of MATLAB” just an INDICATOR that the courses match modern demands….!!!