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

Foundations: real numbers; sequences and limits; functions and continuity

Differentiation: derivative and its geometrical interpretation. Left and right derivative. Continuity of differentiable functions. Differentiation rules and theorems (Rolle, Lagrange, Cauchy). Local and global extrema of a function. Stationary points and extrema. Higher order derivatives. Second order derivative and convexity. Taylor's expansion. Higher order derivatives criterion for maxima and minima. Indeterminate forms and De l'Hôpital's theorems.The study of the graph of a function. Asymptotes.

Integrals: Partitions. The Riemann integral and its geometrical interpretation as area. Primitives and indefinite integrals. First and second fundamental theorems of calculus. Integration by parts. Integration by substitution. Improper integrals.

Linear algebra: Vector spaces. Matrices and their properties. Rank and determinant. Systems of equations. Existence of solutions. Case of one solution and of infinitely many solutions. Triangular systems. Linear maps, associated matrices, matrix of basis change. Invertible linear maps, image, kernel, Dimension Theorem. Eigenvectors, eigenvalues, characteristic polynomial, diagonalizability. Symmetric matrices, Spectral Theorem (over the real numbers).

Series: Sequences and series. Telescopic series, geometric series, harmonic series. Convergence criteria: comparison test, root test, ratio test, absolute convergence.

Multi-variable calculus: Limits and continuity for scalar functions of more than one variable. Directional derivatives. Partial derivatives. Gradient. Derivative of composite functions. Total derivative. Differentiability. Continuity of differentiable functions. Extremum points: maxima, minima and saddle points. Second derivatives test. Lagrange's multipliers for constrained extrema.

Readings/Bibliography

Tom M. Apostol, Calculus (vol. 1 and 2) Wiley

Robert A. Adams, Christopher Essex , Calculus, a complete course. Pearson.

M. Spivak, Calculus, 4th edition, Publish or Perish

Teaching methods

Lectures on the blackboard and structured in theoretical parts, examples and exsercises.

Assessment methods

Final written exam. First year and former students can take intermediate exams and the final grade will be the average of the exams taken.

Below the rules for mid-term exams.

The written exam will take 2 hours and will consists of two parts.

• In the first part (30 minutes long) the student is asked to answer 15 questions (5 statements, each of which consisting of 3 true/false items).

  One point is awarded for every correct answer, one point is taken off for each wrong answer, zero points for non-given answers.

  The total amount of points, that can be collected in this first part, ranges between -15 and +15.

  The usage of books, notes, tablets, computers, mobiles, and so on, is not allowed.

• In the second part (90 minutes long) the student is asked to solve some exercises, each associated with a certain score.

   The total score in this second part ranges between 0 and 20.

   The usage of books and notes is allowed.

The total score ranges, therefore, between -15 and 35.

The second part will be corrected only if the student obtains at least 5 points in the first part.

Students can access to the next mid-term exam only if their total score is at least 15 (5 in the first part and 10 in the second part).

Teaching tools

Occasional use of the projector to illustrate the study of functions and computer simulations.

Office hours

See the website of Barbara Di Fabio