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: Derivatives 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. Limits and continuity. 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

The lectures will be given at the blackboard and structured in theoretical parts, examples and exercises.

Assessment methods

Final written exam of about two hours.
Mid term exam(s)/tests for 1st year students only (students from earlier years must take the full exams). The final grade will be based on an average of mid-term exams and final exams.

 

Teaching tools

Traditional blackboard lectures with occasional use of the projector for computer simulations and study of the graph of a function.

Office hours

See the website of Fabrizio Lillo

See the website of Andrea Santi