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Mathematics 1 & 2 - A.Y.2017/2018

Università degli Studi Ca' Foscari of Venice,  A.Y.2017-2018, Department of Management, curriculum Business Administration and Management.

Textbook

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

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 - Notes for a crash-course in mathematics (this material is found on this web site).  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.
• 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.
• Basics of Financial Mathematics.
• Global maximum and minimum for two variables functions.
• Lagrangian multipliers (2017-2018 edition).
• The theorem of Rouché and Capelli. These notes are a supplement of the textbook.
• Linear dependence or independence.

Outline of the topics covered in class

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

• Lecture 1 - September 18, 2017: 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).
• Lecture 2 - September 19, 2017: still 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.
  • Quadratic functions (4.6).
  • Power functions (4.8).
  • Exponentials and logarithms (4.9, 4.10).
• Lecture 3 - September 20, 2017: still recalls of selected topics from chapters 1 to 5 of the textbook.  Introduction to limits.
  • Still exponentials and logarithms.
  • Properties of logarithms.
  • Piecewise defined functions. An economic example of such a function can be found in Pound Drop, on this web site.
  • 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. A summary of the introduction to limits, togheter with exercises, can be found in A brief introduction to limits.
• Lecture 4 - September 25, 2017: Limits.
  • Limits for one variable functions.
  • Continuity.
• Lecture 5 - September 26, 2017: limits.  Introduction to the concept of derivative.
  • Still algebra of limits: indeterminate forms.
  • Rules of limits.
  • The strength of infinities.
  • The slope of a curve: for this subject you can use the pdf A dash on derivatives, that you can found on  this web site.
• Lecture 6 - September 27, 2017: 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).
  • Composite functions (chain rule) (6.8).
• Lecture 7 - October 3, 2017: more on differentiation. One variable optimization. Mockups of first partial.
  • More on composite functions (chain rule) (6.8).
  • Compound or piecewise defined functions.
  • Kinks.
  • Increasing and decreasing functions (6.3).
  • Linear approximations (7.4).
  • Extreme points (8.1). A summary on one variable optimization can be found in "One variable optimization", a pdf you can find on this web site
  • Local extreme points (8.6).
  • Mockups of the first partial. All the proposed mockups, also for the next lectures, can be found on this web site in the page of A.Y. 2016/2017.
• Lecture 8 - October 4, 2017: more on differentiation. One variable optimization. Mockups of first partial.
  • Higher order derivatives (6.9).
  • Convex and concave functions (6.9).
  • Inflection points (8.7). 
  • Increasing and decreasing functions (6.3).
  • L'Hôpital's rule (7.12).
  • Higher order approximations (7.5).
  • Sample exercises.
  • Mockups of the first partial. All the proposed mockups can be found on this web site in the page of A.Y. 2016/2017.
• Lecture 9 - October 9, 2017: antiderivatives. Mockups of first partial. See, on this web site, the pdf A dash of integrals for a summary of integration theory.
  • Indefinite integrlas (9.1).
  • Some important integrals.
  • Some general rules.
  • Sample exercises.
  • Mockups of the first partial. All the proposed mockups can be found on this web site in the page of A.Y. 2016/2017.
• Lecture 10 - October 10, 2017: more on antiderivaties, definite integral.  Mockups of first partial.
  • Integration of composite functions.
  • Integration by parts (9.5).
  • Area and definite integrals (9.2).
  • A geometrical interpretation of definite integrals.
  • Sample exercises.
  • Mockups of the first partial. All the proposed mockups can be found on this web site in the page of A.Y. 2016/2017.
• Lecture 11 - October 11, 2017: conclusions on definite integrals, improper integrals. Mockups of first partial.
  • Properties of definite integrals (9.3).
  • More on areas and definite integrals.
  • Areas involving piecewise defined functions.
  • Improper integrals (9.7).
  • Infinite intervals of integration.
  • Integration of unbounded functions.
  • Sample exercises.
  • Mockups of the first partial. All the proposed mockups can be found on this web site in the page of A.Y. 2016/2017.
• Lecture 12 - October 16, 2017: basics of Financial Mathematics: use the pdf  Basics of Financial Mathematics, that you can find on this web site. Mockups of first partial.
  • Further formulas concerning the exponential and logarithmic functions with a general base.
  • Final observations about maxima and minima, in particular concerning the Extreme Value Theorem.
  • Areas between the graphs of two continuous functions.
  • The Fundamental Problem of Financial Mathematics.
  • Interest rate, Present Value, Future Value, accumulation factor, actualization factor.
  • Financial regimes and the compound interest.
  • Subdivision of the period in sub-periods and the effective rate of interest.
  • Continuous compounding.
  • Mockups of the first partial.
• Lecture 13 - October 17, 2017: basics of Financial Mathematics: use the pdf  Basics of Financial Mathematics, that you can find on this web site. Mockups of first partial.
  • Streams of Cash Flow.
  • Geometric Progressions.
  • Annuities. Ordinary annuities. Due annuities.
  • Mockups of the first partial.
• Lecture 14 - October 18, 2017: Mockups of first partial.
  • Mockups of the first partial.
• Lecture 15 - October 23, 2017: Mockups of first partial.
  • Mockups of the first partial.
• Lecture 16 - November 6, 2017: Introduction to the functions of two variables.
  • Natural domain of a two variables function (11.1).
  • Plotting the graph of a two variables function.
  • Partial derivatives (11.2) (skip the formal definition).
• Lecture 17 - November 7, 2017: local optimization in two variables.
  • More on partial derivatives.
  • More on geometric representation, graphs, level curves, geometric interpretation of the partial derivatives (11.3).
  • 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 9, 2017: global optimization in two variables. In addition to the textbook use the notes Global maximum and minimum for two variables functions.
  • Further observations on the theorem of Weierstrass.
  • 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). Exercises and examples.
• Lecture 19 - November 13, 2017: global optimization in two variables. In addition to the textbook use the notes Global maximum and minimum for two variables functions.
  • Examples and exercises involving global optimization in the case where the equation of the boundary can be used to reduce a two variables function to a single variable function.
  • Discussion on the first partial.
• Lecture 20 - November 14, 2017: constrained optimization in two variables. In addition to the textbook use the notes Lagrangian multipliers (2017-2018 edition).
  • Constraints.
  • The Lagrangian multiplier method (14.1).
  • Exercises and examples.
  • Discussion on the first partial.
• Lecture 21 - November 16, 2017: more on constrained optimization in two variables.
  • Bordered Hessian and local constrained maxima and minima.
  • The tangent plane to a surface.
  • Final exercises and examples on optimization problems.
• Lecture 22 - November 23, 2017: 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 27, 2017: 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  in general: minors, complementary minors, cofactors, expansion by cofactors (16.3, 16.4, 16.5,  partly).
  • Introduction to matrix inversion.
• Lecture 24 - November 28, 2017: matrix inversion; rank of a matrix.
  • The inverse of a matrix (16.6, partly).
  • The rank  of a matrix.
• Lecture 25 - November 30, 2017: 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. Use the notes: The theorem of Rouché and Capelli.
  • Solving linear systems with Cramer's rule.
  • Solving linear systems using the inverse matrix.
• Lecture 26 - December 4, 2017: solving general linear systems, parametric linear systems.
  • The solution of a linear system in the general case.
  • Parametric linear systems.
• Lecture 27 - December 5, 2017: more on parametric linear systems, linear dependence and independence.
  • More on the solution of a parametric linear system.
  • Introduction to linear dependence and independence of a set of vectors.
• Lecture 28 - December 7, 2017: linear dependence and independence.
  • Linear dependence and independence of a set of vectors (Use the notes Linear dependence or independence).
• Lecture 29 - December 11, 2017: tips for the second partial.
  • A brief summary on the theory of  two variables functions.
  • Mockups of the second partial.
• Lecture 30 - December 12, 2017: tips for the second partial.
  • A brief summary of Linear Algebra.
  •  Mockups of the second partial.

Homeworks.

• Exercises on linear dependence and independence and on parametric linear systems.
• A brief summary of two variables functions and linear algebra, with additional homeworks.

Old exams.

• All the exercises proposed in the first partial of november 4, 2017.
• All the exercises proposed in the second partial of december 21, 2017.
• First call A. First call B.

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.

See the page of A.Y. 2016/2017 for a collection of exercises, mockups and old exams.

Practice lessons.

You can find the exercises solved or proposed during practice lessons by prof. Somenzi Damiano Marino on his I.S.A. section of Unive Platform (http://static.unive.it/isa/index/docente/persona/10594999). You must use your username and password.

copyright 2017 et seq. luciano battaia

first published on 2017/09/18 - last updated on 2018/01/10