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MAT 5930. Analysis for Teachers MAT 5930. Analysis for Teachers. Wm C Bauldry [email protected] Fall ’13 Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 1 / 147 Nordkapp, Norway. Today Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 2 / 147 Introduction and Calculus Considered 1 Course Introduction (Class page, Course Info, Syllabus) 2 Calculus Considered 1 A Standard Freshman Calculus Course B Topics List: MAT 1110; MAT 1120 B Refer to texts by Thomas (traditional), Stewart (very popular, but. ), and Hughes-Hallett, et al, or Ostebee & Zorn (“reform”). B Historical: L’Hopital’sˆ Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes, Cauchy’s Cours d’Analyse, and Granville, Smith, & Longley’s Elements of the Differential and Integral Calculus 2 An AP Calculus Course(AB,BC) 1 Functions, Graphs, and Limits Representations, one-to-one, onto, inverses, analysis of graphs, limits of functions, asymptotic behavior, continuity, uniform continuity 2 Derivatives Concept, definitions, interpretations, at a point, as a function, second derivative, applications, computation, numerical approx. 3 Integrals Concept, definitions, interpretations, properties, Fundamental Theorem, applications, techniques, applications, numerical approx. Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 3 / 147 Calculus Considered 2 (Calculus Considered) 3 Calculus Problems x1 Precalculus material: function, induction, summation, slope, trigonometry x2 Limits and Continuity: Squeeze Theorem, discontinuity, removable discontinuity, different interpretations of limit expressions x3 Derivatives: trigonometric derivatives, power rule, indirect methods, Newton’s method, Mean Value Theorem,“Racetrack Principle” x4 Integration: Fundamental Theorem, Riemann sums, parts, multiple integrals x5 Infinite Series: geometric, integrals and series, partial fractions, convergence tests (ratio, root, comparison, integral), Taylor & Maclaurin Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 4 / 147 A Practical Rationale: Growth in AP Calculus See: MAA-NCTM Joint Position Statement on Calculus of March, 2012. Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 5 / 147 AP Calculus 2010/11 Results North Carolina AP Score Distributions, 2010 – 2012 AB 2010 2011 2012 BC 2010 2011 2012 n 6;797 6;860 6;946 n 3;070 3;176 3;426 x¯ 2:63 2:63 2:78 x¯ 3:56 3:50 3:58 x ≥ 3 50:0% 50:3% 54:7% x ≥ 3 75:4% 74:3% 76:6% ASU: MAT 1110 credit (4 cr) MAT 1110 & 1120 (8) UNC-CH: MATH 231 (3 cr) MATH 231 & 232 (6) c 2013 The College Board. Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 6 / 147 A Little Background Our goal is A rigorous study of elementary calculus extending to a treatment of fundamental concepts of analysis involving functions of a real variable. (See “Course Objectives.”) What this course is not A “refresher course” in calculus. A “how to” course for teaching AP calculus. What this course does Consider calculus from an advanced view—that is, as real analysis. Relate analysis concepts to the calculus classroom for depth. Develop the profound understanding of fundamental mathematics . need[ed] to become accomplished mathematics teachers. — Liping Ma in Knowing and Teaching Elementary Mathematics. Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 7 / 147 Notes to Analysis Students: Learning Mathematics According to Paul Halmos, to learn mathematics, you have to know: The ‘What’: Know the definitions, statements of theorems, and formulas What is an open set? When does a function have to have a maximum? What is the limit formula for the derivative? The ‘How’: Apply techniques and use formulas to solve problems Find the minimum of the derivative on an interval. Determine where a function is continuous. Calculate the sum of the reciprocals of the squares. The ‘Why’: Understand why a technique works, why a theorem is true Why does a bounded infinite set have to have a limit point? Why does continuity imply integrability? Why can antiderivatives compute definite integrals? Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 8 / 147 Notes to Analysis Students: Doing Mathematics George Polya’s´ Four Steps in Problem Solving: Step 1: Understand the Problem (“see clearly what is required”) What is the unknown? What are the data? Step 2: Make a Plan (“the idea of the solution”) Do you know a related problem? How are the items (the data and the unknown) connected? Step 3: Carry Out the Plan Check each step. Can you see clearly and prove that the step is correct? Step 4: Look Back at the Completed Solution (“review and discuss it”) Reconsider, reexamine the result & the path that led to it. Can you check the result? The argument? For a complete version see “How to Solve It” Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 9 / 147 Notes to Analysis Students: Teaching Mathematics George Polya’s´ Ten Commandments for Teachers: 1 Be interested in your subject. 2 Know your subject. 3 Know about the ways of learning: The best way to learn anything is to discover it by yourself. 4 Try to read the faces of your students, try to see their expectations and difficulties, put yourself in their place. 5 Give them not only information, but “know-how,” attitudes of mind, the habit of methodical work. 6 Let them learn guessing. 7 Let them learn proving. 8 Look out for such features of the problem at hand as may be useful in solving the problems to come—try to disclose the general pattern that lies behind the present concrete situation. 9 Do not give away your whole secret at once—let the students guess before you tell it—let them find out by themselves as much as is feasible. 10 Suggest it, do not force it down their throats. Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 10 / 147 Fundamental Concept: Function Who First used the term function? – Leibniz1 (1646 –1716) Introduced f(x) notation and originally thought a function must be a single algebraic expression? – Euler (1707–1783) Shocked the math world by presenting a bizarre function? – Weierstrass (1815 –1897) Introduced the formal definition of function in terms of sets of ordered pairs? – Homework! When did the function concept become central in school math? – 1920’s See: “The Function Concept in School Mathematics,” H Hamley, Math Gazette, 18(229), 169-179 Homework: Read “Subject-Matter Knowledge and Pedagogical Content Knowledge: Prospective Secondary Teachers and the Function Concept,” R Even, JRME 24, No. 2. (1993), pp. 94-116. 1See: Jeff Miller’s website Earliest Known Uses of Some of the Words of Mathematics Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 11 / 147 The Definition of Function Definition (Function) When two variables are so related that the value of the first variable is determined when the value of the second variable is given, then the first variable is said to be a function of the second. From Elements of the Differential and Integral Calculus by Granville, Smith, & Longley (1934). Definition (Function) Let X and Y be nonempty sets. Let f be a collection of ordered pairs (x; y) with x 2 X and y 2 Y: Then f is a function from X to Y if to every x 2 X there is assigned a unique y 2 Y: From Calculus and Analytic Geometry by Thomas (1968). Definition (Function) A function is a rule for assigning to each member of one set, called the domain, one (and only one) member of another set, called the range. From Calculus from Graphical, Numeric, and Symbolic Points of View by Ostebee & Zorn (2002). Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 12 / 147 Elementary Properties of Functions Definition Let f :X ! Y be a function. Well-defined: If x1 = x2; then f(x1) = f(x2): One-to-one: If f(x1) = f(x2); then x1 = x2; or if x1 6= x2; then f(x1) 6= f(x2): Onto: For each y; there is an x such that f(x) = y: Inverse image: If B ⊆ Y; then f −1(B) = fx 2 X j f(x) 2 Bg: Graph: The set of pairs f(x; f(x)) j x 2 Xg ⊂ X × Y is the graph of f: Even: If f(−x) = +f(x) for each fx; −xg ⊆ X: Odd: If f(−x) = −f(x) for each fx; −xg ⊆ X: Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 13 / 147 Interlude: Domains Matter Example Is the statement +1 = p+1 p+1 = p+1 +1 = ( 1) ( 1) = p 1 p 1 = 1 × × − × − − × − − correct? p If yes, then +1 = 1 − If no, where’s the flaw? Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 14 / 147 The Principle of Mathematical Induction Theorem (Mathematical Induction) Let P (n) be a statement about the natural number n: If P (1) is true, and whenever P (k) is true, then P (k + 1) must also be true, then P (n) is true for all n N: 2 Theorem (Strong Induction) Let P (n) be a statement about the natural number n: If P (1) is true, and whenever P (k);P (k 1);P (k 2); ..., P (2); and P (1) are true, then P (k + 1) must also be− true, − then P (n) is true for all n N: 2 Wm C Bauldry ([email protected]) MAT 5930. Analysis for Teachers. Fall ’13 15 / 147 Summations and Induction, I n n(n+1) Example ( k=1 k = 2 by Picture) P Wm C Bauldry ([email protected]) MAT 5930.
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