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# Mathematics 1 & 2 - A.Y.2016/2017

Università degli Studi Ca' Foscari of Venice,  A.Y.2016-2017, Department of Economics, curriculum Economics, Markets and Finance.

***** Ca' Foscari - Harvard Summer School *****

On Monday November 28 prof. Paolo Pellizzari, during our class, introduced the Ca' Foscari Harvard Summer School, in particular the Course Math 1b, of which you can download the Syllabus. Other info on the web page www.unive.it/pag/10033/. All parts of the prerequisites have been studied in our course, except for trigonometry, so no problem if you want to take this great and important opportunity. If any student wants to enroll in this course (as I hope!!) and has the need to review trigonometry, please tell me: I can help and, among other things, I can write a brief summary of the needed concepts of trigonometry.

Textbook

Knuth Sydsæter, Peter Hammond, Arne Strøm & Andrès Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2016 (V edition).

Office hours

Mathematics 1: Tuesday and Wednesday 3.15pm to 3.45pm and 5.15pm to 5.45pm, room D0004, complex D, only during the period of the lectures.

Mathematics 2: Tuesday and Thursday 3.15pm to 3.45pm and 5.15pm to 5.45pm, room D0004, complex D, only during the period of the lectures.

Teaching material

• Approaching the course.
• Saying maths: hints for reading numbers and mathematical formulas in english and american english (this material is found on this web site).
• Precalculus, by Luciano Battaia, Giacomo Bormetti and Giulia Livieri: A prelude to calculus with exercises (this material is found on this web site). Notes for a crash-course in mathematics. This booklet can help in revising basic maths concepts, and contains mainly the same topics as in the first five chapters of the textbook.
• A brief introduction to limits. These notes supplement the concept of limit, treated in paragraphs 6.5 and 7.9 of the textbook.
• Limits and continuity: just the gist: a concise introduction  to limits and continuity, that can be used to deepen the same concepts treated in "A brief introduction to limits"  (this material is found on this web site).
• A dash on derivatives. These notes are to be considered as a summary of some concepts included in chapters 6 and 7 of the textbook.
• One variable optimization. These notes are to be considered as a summary of chapter 8 of the textbook.
• A dash of integrals. These notes are to be considered as a summary of chapter 9 of the textbook.
• Global maximum and minimum for two variables functions. These notes are to be considered as a supplement of paragraph 13.5 of the textbook.
• Lagrangian multipliers. These notes are to be considered as a supplement of paragraph 14.1 of the textbook.
• Constrained Optimization in two variables. Notes, in italian, used for another course at Ca' Foscari: you can find here additional info and solved exercises.
• The theorem of Rouché and Capelli. These notes are a supplement of the textbook.
• Linear dependence or independence. These notes are a supplement of the textbook.

Outline of the topics covered in class

References to the paragraphs of the textbook are given in parentheses, when possible.

• Lecture 1 - September 19, 2016: recalls of selected topics from chapters 1 to 5 of the textbook. It is important to remember that all the topics contained in these chapters are considered known.
• Approaching the course.
• Informations about the software packages that can help during the course, in particular Wolfram Alpha and Geogebra. As far as Geogebra is concerned the software can be used directly on line in a browser's window or downloaded and installed to your personal computer. There is also a version suitable for use on portable media without leaving setting files on the host computer: see the web site geogebra.org.
• Basic definitions about functions (4.2, 4.3).
• Linear functions (4.4).
• Power functions (4.8).
• Lecture 2 - September 20, 2016: still recalls of selected topics from chapters 1 to 5 of the textbook.
• Exponentials and logarithms (4.9, 4.10).
• Some very simple exponential and logarithmic inequalities.
• Piecewise defined functions (or compound functions) (5.4).
• New functions from old: sums, products and quotients (5.2, partly)
• Graphs of equations, in particular circles: differences compared to the graphs of functions (5.4).
• Lecture 3 - September 21, 2016: conclusion of recalls of selected topics from chapters 1 to 5 of the textbook. Introduction to limits.
• Injective functions.
• Surjective functions.
• Bijective or one-to-one functions; properties of graphs of such functions.
• Composite functions (5.2, partly).
• A graphical approach to limits (6.5, partly). The limits will be treated in a somewhat different way from the textbook. Geogebra has been used to illustrate in a simple way what a limit is. You can download the files used: FirstExampleOnLimits, SecondExampleOnLimits.
• Lecture 4 - September 26, 2016: limits and continuity. Introduction to derivatives.
• Definition of limit, continuity, horizontal and vertical asymptotes, always using a graphical approach, as discussed in "A brief introduction to limits".
• First exercises on limits and continuity.
• A graphical approach to the concept of derivative: tangents to curves. For this topic you can use the geogebra file SecantTangent. Directions for using this software: using the mouse move the point P on the given curve, the point Q moves accordingly as the difference between x(P) and x(Q) is given by the number a. Changing the value of a on the slider (click on the slider and then use the left and right arrows on the keyboard), Q moves accordingly. When a is different from 0, you can see the secant line (in black) through P and Q. When a reaches 0, in what case Q is over P, the secant disappears and you can see a new line in red: this line is exactly the tangent to the curve. By tiping a new function in the input line you can experiment with different curves.  Pay attention while typing a new function: the abscissa of the point P must be in domain of the new function, otherwise the point P itself (and consequently Q) becomes undefined. Functions to test (type exactly the written formulas in the input line, or copy and paste them):
• f(x)=ln(x)
• f(x)=exp(x)
• f(x)=sqrt(x)
• f(x)=1/x
• f(x)=x^3-2x^2+1/2
• Lecture 5 - September 27, 2016: differentiation. A concise summary on differentiation can be found in "A dash on derivatives".
• Slopes of curves (6.1).
• Tangents and derivatives (6.2).
• Simple rules for differentiation (6.6).
• Sums, products and quotients (6.7).
• Exponential functions (6.10, partly).
• Logarithmic functions (6.11, partly).
• Lecture 6 - September 28, 2016: more on differentiation.
• Composite functions (chain rule) (6.8).
• Compound or piecewise defined functions.
• Kinks: to understand the graphical meaning of "kink" you can use the geogebra file Kink.
• Increasing and decreasing functions (6.3).
• Sample exercises.
• Lecture 7 - October 3, 2016: limits in practice.
• Rules for limits (6.5).
• Indefinite forms and simple resolution techniques.
• Sample exercises.
• Lecture 8 - October 4, 2016: more on derivatives.
• Linear approximations (7.4)
• Higher order derivatives (6.9).
• Convex and concave functions (6.9).
• Inflection points (8.7).
• L'Hôpital's rule (7.12).
• Sample exercises.
• Lecture 9 - October 5, 2016: one-variable optimization.
• Extreme points (8.1).
• Tests for extreme points (8.2).
• The extreme value theorem (8.4).
• Local extreme points (8.6).
• Sample exercises.
• Lecture 10 - October 10, 2016: antiderivatives.
• Indefinite integrlas (9.1).
• Some important integrals.
• Some general rules.
• Integration of composite functions.
• Sample exercises.
• Lecture 11 - October 11, 2016: more on antiderivaties, definite integral.
• Integration by parts (9.5).
• Area and definite integrals (9.2).
• A geometrical interpretation of definite integrals.
• Area of the region between two curves.
• Sample exercises.
• Lecture 12 - October 12, 2016: conclusions on definite integrals, improper integrals.
• Properties of definite integrals (9.3).
• Improper integrals (9.7).
• Infinite intervals of integration.
• Integration of unbounded functions.
• Sample exercises.
• Lecture 13 - October 17, 2016: polynomial approximations, review exercises.
• Polynomial approximations (7.5).
• Exam type exercises.
• Lecture 14 - October 18 2016: review exercises.
• Exam type exercises
• Lecture 15 - October 19, 2016: review exercises.
• Exam type exercises.
• Lecture 16 - November 7, 2016: functions of two variables.
• Functions of two variables (11.1).
• Partial derivatives with two variables ( 11.2) (skip the formal definition).
• Geometric representation, graphs, level curves, geometric interpretation of the partial derivatives (11.3).
• Lecture 17 - November 8, 2016: local optimization in two variables.
• The Young's theorem (11.6, partly).
• Two variables optimization: necessary conditions (13.1).
• Two variables optimization: sufficient conditions (13.2).
• Local extreme points (13.3).
• Lecture 18 - November 10, 2016: global optimization in two variables. In addition to the textbook use the notes Global maximum and minimum for two variables functions.
• The equation of the tangent plane to a surface (12.8, partly).
• Bounded, open and closed subsets of the plane and the extrem value theorem (13.5).
• Finding global maximum and minimum using the equations of the boundary (13.5 partly).
• Lecture 19 - November 14, 2016: constrained optimization in two variables. In addition to the textbook use the notes Lagrangian multipliers.
• Global maximum and minimum for a two variables function in unbounded sets in simple cases.
• The Lagrangian multiplier method (14.1).
• Exercises.
• Lecture 20 - November 15, 2016: constrained optimization in two variables.
• Bordered Hessian and second order sufficient conditions for local constrained maxima and minima (14.5 partly and 16.2 partly).
• Exercises.
• Lecture 21 - November 17, 2016: multivariable optimization.
• Use of level curves for problems of two variables optimization, both constrained and not.
• Exercises.
• Lecture 22 - November 22, 2016: matrices and operations between matrices.
• Systems of linear equations (15.1).
• Matrices and matrix operations (15.2).
• Matrix multiplication (15.3).
• Rules for matrix multiplication (15.4).
• Lecture 23 - November 24, 2016: matrices and determinants.
• The transpose of a matrix (15.5).
• Symmetric matrices (15.5).
• Vectors and operations on vectors (15.7).
• The inner product (15.7).
• Determinants of order two (16.1) (execpt geometric interpretation).
• Determinants of order three (16.2) (except geometric interpretation).
• Determinants  in general: minors, complementary minors, cofactors, expansion by cofactors (16.3, 16.4, 16.5,  partly).
• Introduction to matrix inversion.
• Lecture 24 - November 28, 2016: matrix inversion; rank of a matrix.
• The inverse of a matrix (16.6, partly).
• The rank  of a matrix.
• Lecture 25 - November 29, 2016: linear systems and matrix form.
• The matrix form of a linear system (15.3, partly).
• Coefficients matrix and augmented matrix.
• The Rouché-Capelli's theorem.
• Solving linear systems with Cramer's rule.
• Solving linear systems using the inverse matrix.
• Lecture 26 - December 1, 2016: solving general linear systems.
• Lecture 27 - December 5, 2016: parametric linear systems.
• The solution of a parametric linear system.
• Lecture 28 - December 6, 2016: linear dependence and independence.
• Lecture 29 - December 12, 2016: mockups of second partial.
• Solutions of sample exercises for the second partial: real functions of two variables.
• Lecture 30 - December 13, 2016: mockups of second partial.
• Solutions of sample exercises for the second partial: linear algebra.

Homeworks

Selected exercises from the texbook.

• Chapter 6: 6.6.3; 6.7.3, 6.7.4; 6.8.3; 6.10.1, 6.10.4; 6.11.3,6.11.6,6.11.7;  Review exercises n. 3, 5, 14, 15.
• Chapter 7: 7.12.1, 7.12.2, 7.12.3, 7.12.4, 7.12.5.
• Chapter 8: 8.2.1, 8.2.5, 8.2.6, 8.2.7; 8.4.2; 8.6.2, 8.6.3; 8.7.2, 8.7.3; Review exercises n. 1, 6, 7.
• Chapter 9: 9.1.1, 9.1.2, 9.1.4, 9.1.5, 9.1.6, 9.19, 9.1.10,9.1.11, 9.1.12; 9.2.3, 9.2.4, 9.2.5; 9.3.1, 9.3.2, 9.3.3; 9.5.1, 9.5.2; 9.7.1, 9.7.9; Review exercises n. 1, 2, 3, 4.
• Chapter 11: 11.1.6, 11.1.7; 11.2.4, 11.2.5; Review exercises n. 6, 7, 9, 10, 12.
• Chapter 13: 13.3.1, 13.3.2, 13.3.3; 13.5.1, 13.5.2; Review exercises: 4.
• Chapter 14: 14.1.1, 14.1.3.c, 14.1.4.

Exercises sheets used during practice sessions.

Mockups of partial examinations.

Old Exams.

Tips.

Notebooks and pdfs used during the lectures.

The following teaching materials have been used during the lectures. They contain no explanation, so they can't be used without the support of the classroom lesson. There are two kinds of materials: the Notebooks are dinamical pages that must be downloaded and then opened with the program Mathematica of Wolfram (commercial and copyrighted software!) or with CDF Player, a viewer that can be downloaded (about 700 Mb) from https://www.wolfram.com/cdf-player/ and installed on your computer; the pdfs are normal pdf files that reproduce the same content of the notebooks, but without interactivity. Many files have explanations in italian, as they have been used during other courses at Ca' Foscari University.

first published on 2016/09/18 - last updated on 2017/02/06