Università degli Studi di Urbino Carlo Bo / Portale Web di Ateneo


Analisi Matematica

A.Y. Credits
2014/2015 12
Lecturer Email Office hours for students
Renzo Lupini Mon-Tue from 10:30 am 11:00 am and from 01:00 pm to 01:30 pm.

Assigned to the Degree Course

Date Time Classroom / Location

Learning Objectives

The aim of the course is to give the basis of calculus, starting from the numeric sets to infinite sequences and series, functions of one variable and of several variables, theory of differentiation and integration, ordinary differential equations.


01. Generalities on sets, relations and structure.
01.01 The algebra of sets.
01.02 Relations.
01.03 Functions.
01.04 Binary operations.
01.05 Natural numbers.

02 The rational line and plane.
02.01 The algebraic system of natural, integer and rational numbers.
02.02 Points, translations and vectors in the cartesian plane.
02.03 Vector algebra in the plane.
02.04 Cartesian representation of lines, half-lines and segments.

03. Rational functions in one variable
03.01 Graph of a function.
03.02 Analytic properties of a function and properties of the graph.
03.03 Restrictions of functions.
03.04 Algebra of functions and composition of functions.
03.05 The graph of linear functions.
03.06 The graph of quadratic functions.

04. Real numbers:
04.01 The y = x^2 equation and the problem of the inverse function.
04.02 The fixed point equation and the Newton algorithm for appoximating 2^1/2.
04.03 Monotone bounded sequences of rational and irrational numbers.
04.04 The algebraic system of real numbers.
04.05 Real line and real plane.
04.06 Convergence of sequences of real numbers. Completeness.
04.07 Series of real numbers: sums and convergence criteria.

05. Real functions in one variable:
05.01 Power and exponential functions.
05.02 Piecewise defined functions.

06. Limits and continuity:
06.01 Limit in a point.
06.02 Infinitesimal functions.
06.03 Continuity.
06.04 Weierstrass Theorem.

07. Differentiability:
07.01 Taylor's formula at first order. Tangent line to the graph.
07.02 Derivative functions.
07.03 Singular points. Discontinuity and angular point of a graph.

08. Theorems of Differential Calculus:
08.01 Classification of regular stationary points.
08.02 Rolle's and Lagrange's theorems on finite increases.

09. Higher order derivatives:
09.01 Higher derivatives.
09.02 Infinitely differentiable functions.

10. Definite and indefinite integration:
10.01 The problem of area and the problem of inversion of derivation.
10.02 Riemann sums and definite integral.
10.03 Properties of definite and indefinite integration.
10.04 Primitives and Leibniz-Newton's theorem. Lagrange's theorem on mean value.

11. Circular functions:
11.01 The unit circle.
11.02 Sine and cosine and tangent functions.
11.03 Plane geometry.

12. Taylor's formula and local analysis:
12.01 Taylor's formula at first order.
12.02 Second derivative and Taylor formula at second order.
12.03 Classification of regular stationary points.

13. Regular curves in the plane:
13.01 Kinematics and regular curves in the plane.
13.02 Ellipses and hyperbolas.
13.03 Closed and open simple curves.

14. Regions of the plane:
14.01 Closed and open simple curves and regions of the plane.
14.02 Boundedness and connectedness.

15. Plane scalar fields and functions in two variables:
15.01 Scalar fields and plane vector fields. Algebra of plane fields.
15.02 Contour lines and topography of a plane scalar field.

16. Limits, continuity and differentiability of scalar fields:
16.01 Infinitesimal fields: continuity and differentiability.
16.02 Taylor's formula at first order and gradient.
16.03 Differential Calculus in two variables.

17. Plane vectorial fields:
17.01 Plane vector fields.
17.02 Algebra of plane fields.
17.03 Infinitesimal fields and local approximation.
17.04 Differentiability of vector fields and second differentiability.

18. Local analysis at second order:
18.01 Hessian matrix and Taylor's formula at second order.
18.02 Local analysis in regular stationary points.

19. Optimization:
19.01 Optimization with constrain on a curve.
19.02 Optimization on a region.
19.03 Optimization of quadratic functions on the plane.
19.04 Least squares line for a finite system of points.

20. Three-dimensional space:
20.01 Algebra and geometry of points and vectors in space.
20.02 Equations of lines and planes in space.
20.03 Graph of a function in two variables. Coordinate and contour lines.
20.04 Closed and open simple surfaces and regions in space.
20.05 Projectable regions: curvilinear cylinders and parallelepipeds.

21. Double integrals:
21.01 Approximation of volumes of projectable three-dimensional regions.
21.02 Riemann sums for functions in two variables.
21.03 Properties of double integrals.
21.04 Volumes and double integrals.
21.05 Simple iterated integrals.
21.06 Integration in polar coordinates.

22. Complex Variables:
22.01 Complex algebra and the complex plane. Fundamental theorem of Algebra.
22.02 Euler's formula and complex exponential.

23. Improper integrals:
23.01 Improper integrals in one variable and areas of unbounded plane regions.
23.02 Improper double integrals and volumes of unbounded regions of space.

Bridging Courses

Although there are no mandatory prerequisites for this exam, students are strongly recommended to take it after Discrete Structures and Linear Algebra.

It is also worth noticing that the topics covered by this course will be used  in Algorithms and Data Structures, Digital Signal and Image Processing, Physics I and Probability and Statistics.

Teaching, Attendance, Course Books and Assessment


Theory lectures and exercises, both face-to face and on-line.


Although recommended, course attendance is not mandatory.

Course books

Lupini, "Matematica" parte 1 e parte 2, Quattroventi, 2005.

Bramanti, Pagani, Salsa, "Matematica", Zanichelli, 2000.

Thomas, Finney, "Calculus and Analytic Geometry", Addison Wesley Publishing Company, 1998.


Written and oral exam.

The written exam is passed if the mark (which is valid for the exam calls of the same academic year) is at least 18/30. The oral exam, which must be taken if the written exam is passed, determines a spread between -12/30 and 12/30 of the previous mark, thus yielding the final mark. There are no limitations to the number of trials per session per year.


The course is offered both face-to-face and on-line within the Laurea Degree Program in Applied Computer Science.

For additional lecture notes and information see http://www.sti.uniurb.it/lupini/

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