Prof. Morsella put some exercises with solutions on the topics he covered on his webpage.

1. The exam consists of a written test (with Moodle) and a compulsory oral test, both online.
2. To access the oral, a grade of at least 18/30 must be obtained on the written test. The exam is passed if the final mark is at least 18/30.
3. The written test and the oral test must take place in the same call.
4. Under penalty of exclusion, during written tests the use of electronic devices and applications except those required (MS Teams and possibly a web browser to access to the Moodle quiz) is not allowed. It is not possible to use books and notes.
5. It is necessary to have a webcam and a microphone to have test.

A simulation of the written test is available on the site.

• Winter session 1st Written test 24/1/20, 9:30, room C6, problems and solutions. 1st Oral assessment 29/1/20, 9:30, room C12.
• Winter session 2nd Written test 11/2/20, 9:30, room C6, problems and solutions. 2nd Oral assessment 18/2/20, 9:30, room C12.
• Summer session 1st Written test 30/06/20, 9:30 (online), problems and solutions. 1st Oral assessment 03/07/20, 9:30 (online).
• Summer session 2nd Written test 14/07/20, 9:30 (online), problems and solutions. 2nd Oral assessment 17/07/20, 9:30 (online).
• Autumn session 1st Written test 31/08/20, 9:30 (online), problems and solutions. 1st Oral assessment 31/08/20 (after written), 13:30 (online).
• Autumn session 2nd Written test 15/09/20, 9:30 (online), problems and solutions. 2nd Oral assessment 15/09/20 (after written), 13:30 (online).

• 9/23 Sequences of functions: pointwise and uniform convergence. Examples. Continuity of the limit and passage to the limit under the integral.
• 9/25 Series of functions: pointwise and uniform convergence. Continuity of the sum and term-by-term integration for uniformly convergent series. Weierstrass M-test. Power series. Examples. Existence of the radius of convergence. Differentiability of power series.
• 9/27 Exercises on sequences of functions and power series.
• 9/30 Taylor series. Unicity of the expansion in power series. Sufficient condition for the Taylor series expansion. Taylor series of sin x cosx, exp x. Solution of differential equations by power series. Binominal series.
• 10/2 Scalar and vector fields. Interior, exterior, boundary and accumulation points of sets in Rⁿ. Open and closed sets. Examples. Limit of a function of several variables. Operations with limits.
• 10/4 Exercises on Taylor series, solutions of differential equations by power series, open sets and continuous scalar fields in Rⁿ.
• 10/7 Directional and partial derivatives. Examples. Existence of partial derivatives does not imply continuity. Differentiability of scalar fields. Differentiability implies continuity and existence of directional derivatives. Sufficient condition for differentiability.
• 10/9 Curves and their parametrization, the tangent vector, Chain rule for scalar fields with examples. Level set of a function and examples.
• 10/11 Exercises on partial derivatives, gradients, chain rule and level sets.
• 10/14 Directional derivatives and differentiability of vector fields. Jacobian matrix. Chain rule. Examples and applications. Derivatives in polar coordinates. Derivatives of higher order. Commutativity of higher partial derivatives.
• 10/16 Solution of linear, first order partial differential equations with constant coefficients and of the one-dimensional wave equation.
• 10/18 Exercises on the chain rule and on partial differential equations.
• 10/21 Implicit functions and their partial derivatives.
• 10/23 Relative Minima, maxima and saddle points of scalar fields, Hessian matrix and a criterion for extrema.
• 10/25 Exercises implicit functions, stationary points and Hessian.