37292 - Mathematics

Learning outcomes

At the end of the course the student will be capable of using the techniques of Linear Algebra; furthermore he will have acquired a working knowledge of First Year Calculus, together with the related applications in Finance and Economics.

Course contents

This course offers a comprehensive overview of key concepts of Mathematics. At the end of the course, students will be capable to using the main techniques of Calculus, Multivariate Calculus and Linear Algebra with the related applications in Finance and Economics. The first part of the course is related to Calculus, the second part to Linear Algebra, while the third part is dedicated to Multivariate Calculus.

Structure of the course:

1 Introduction to Calculus. Sets and numbers.

2 Functions of one variable.

3 Graphs of elementary functions and properties.

4 Limits of functions of one variable.

5 Limits: computation of elementary forms.

6 Indeterminate forms and notable limits.

7 Continuous functions: definitions, properties and theorems.

8 Differentiation. Definitions and rules of calculus of derivatives.

9 Higher-order derivatives. Taylor’s formula.

10 Local and global extremes. Study of the graph of a function.

11 Integration. Indefinite integrals. Integrals of elementary functions.

12 Definite integral. Definition and properties.

13 Method of integration. Integration by parts and by substitution.

14 Generalized integrals. Definition, properties and applications.

15 Introduction to linear algebra. Vectors. Operations among vectors.

16 Matrices. Definitions and operations among matrices.

17 Product of matrices. Determinants: definitions and properties.

18 Calculus of determinants. Inverse of a square matrix.

19 Rank of a matrix. Definition and properties. Determination of the rank of a matrix.

20 Linear systems of equations. Definitions and theorems.

21 Solution of linear systems of equations.

22 Multivariate calculus: real-valued functions of many variables.

23 Limits of functions of two variables: restrictions and existence.

24 Differentiation of functions of many variables: partial derivatives.

25 Composite functions and rules of partial differentiation.

26 Unconstrained optimization.

27 Constrained optimization: Lagrange’s multiplier method.

28 Constrained optimization: necessary and sufficient conditions.

29 Differential equations: separable differential equations.

30 Differential equations: first-order linear differential equations.

Readings/Bibliography

Required Readings:

- R.A. ADAMS, C. ESSEX. Calculus, a complete course, 9th Edition, 2018, Pearson.

- K. SYDSÆTER, P. HAMMOND, A. STRØM, A. CARVAJAL, Essential Mathematics for Economic Analysis, 5th Edition, 2016, Pearson.

Teaching methods

The main instructional approaches used in the course are:

- Lectures

Assessment methods

Though not compulsory, class attendance and participation is strongly encouraged.

Course evaluation will be as detailed in the following:

Component Weight (%)

Midterm Exam 100

Or:

Component Weight (%)

Midterm Exam 1 33.3

Midterm Exam 2 33.3

Midterm Exam 3 33.3

- Exam structure:

The final exam is a written exam where students have to solve different exercises on the course topics. To each exercise a given maximum number of points is associated, and to get it the student has to solve correctly the exercise and all the steps must be justified. The theoretical maximum number of points achievable in case of a perfect exam is 32.

The evaluation depends on the correctness, completeness and rigor of the answers.

- Exam policy:

Students can attend 3 midterm exams (not compulsory) during the break at the end of teach sub-cycle. The grades of the three midterm exams will be averaged to compute the “final written exam” grade.

Those who positively each midterm exam (i.e. grade greater or equal to 18), have the chance to take the next midterm exam. Therefore, if a student does not one of midterm exams s/he can attend only total exams from that moment on.

Registration on Almaesami

It is possible to reject the grade of the 3 midterms and take the full exam. It is possible to reject the final exam grade 2 times.

Grading scale:

< 18: failed

18-23: sufficient

24-27: good

28-30: very good

30 e lode: outstanding

Students with disability or specific learning disabilities (DSA) are required to make their condition known to find the best possibile accommodation to their needs.

Teaching tools

Tools, platforms, or resources used during the course:

- Learning platform: Virtuale (virtuale.unibo.it) contains the slides of the lessons

- Communication tools: Email; Teams; Forum on Virtuale

Office hours

See the website of Fabio Gobbi