# APPLIED MATHEMATICS

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Versione italiana
2018/2019
Teacher
LORENZO PARESCHI
Credits
8
Didactic period
Primo Semestre
SSD
MAT/08

#### Training objectives

The goal of the course is twofold. On one side the course will give to the students a basic preparation in elementary mathematics, on the other side the course will introduce new mathematical tools based on differential calculus and linear algebra.

Knowledge
Fundamentals of differential calculus , the notion of limit , concept of derivative . Integral as an area. Matrices and linear systems , linear transformations . Simple differential equations and series expansions .

Skill
Ability to qualitatively analyze the behavior of a function and to perform the calculation of some integrals. Solving linear systems .
In addition , students must be able to construct and analyze simple mathematical models ( also connected to differential equations ) with which to formulate problems of Structural Design, Technical Physics and Economy applied to the city and the territory.

#### Prerequisites

The course requires no prerequisites. In order to standardize the basic preparation of students it is organized an elementary math course called MINIMAT for students who have more gaps in their mathematical education. The verification of this knowledge is carried out through a written test at the beginning of the course.

#### Course programme

1. Getting Started
Sets. Numerical sets. Logic and mathematical method. Operations and functions between sets. Real numbers passing through the rational numbers. Absolute value. Intervals. Inequalities. Polynomials and roots.

2. Functions
Generality. The square root function and sign function. composite functions. Even and odd functions. Inverse functions. monotone functions. elementary transcendental functions. trigonometric functions. Exponential and logartitmiche functions.

3. Limits and Continuity
Definition and examples. main theorems. Extensions, limits from the right and left. limits to infinity. infinite limits. continuous functions. Property. Theorems on continuous functions.

4. Tangent lines and derivation
tangent line. Newton quotient. normal lines. Definition of derivative. right and left derivative. Operations with derivatives. Derivative of a product. The concept of induction. Derivative of the inverse function. Derivative of a quotient. Derivative of a composite function. Higher order derivatives. Antiderivative and indefinite integral. Derivatives of elementary functions.

5. Graphs of functions and approximations
Mean value theorem and its consequences. Critical points and extreme values. Tests of the first derivative. Concavity and points of inflection. Testing of the second derivative. Graphic Design. Asymptotes. Examples. of de L'Hôpital rules.

6. Calculation of integrals
Area of ​​a trapezoid. lower and upper Riemann sums. The Riemann integral. Properties of Riemann integral. The fundamental theorem of calculus. The definite integral. Mean value theorem for integrals. Integration by substitution and by parts. Integration of rational functions.

7. Improper integrals and series
Improper integrals, convergence criteria, Numerical series, Convergence criteria, Convergence and absolute convergence, Power series, Taylor series.

8. Differential Models
Introduction to differential equations, introductory examples, Existence and uniqueness, linear equations of the first order, homogeneous equations with constant coefficients.

9. Matrices and linear systems
Definition of matrix and vector operations between matrices, inner product, linear systems, Gaussian elimination.

10. Eigenvalues ​​and eigenvectors
Eigenvalues ​​and determinants, Complex numbers, definitions, calculation of eigenvalues ​​and eigenvectors, Diagonalization and applications, symmetric matrices, least squares, Norms of vectors and matrices.

11. Geometric vectors
Vectors in geometry, free vectors and applied vectors, circular and spherical coordinates, scalar and vector product.

#### Didactic methods

The course is based on a theorical and a practical part. The practical part through exercises will guide the students in solving problems which are at the basis of the written exam.

Teaching is divided into :
- Basic theoretical knowledge ;
- Practical exercises in the classroom ;
- Verification of learning during the course through individual exercises evaluated by the teacher .

#### Learning assessment procedures

During the course two written tests or one final written examination. The written test is aimed at verifying the student's ability to solve simple practical problems similar to those seen during exercises in the classroom . The grade of the written test must be integrated with that of the oral exam. The oral examination is based on the verification of theoretical concepts seen during the course and has less impact on the final grade than the written test . There are two types of oral test .
-Type A oral exam (grade larger or equal to 17.5). Two written answers out of six questions proposed. If positive it will confirm the grade obtained in the written part with a maximum increment of 10%. If negative the exam is not sufficient and also the written part must be repeated.

-Type B oral exam (grade larger or equal to 14.5). A single oral exam on the blackboard with two or three questions. If positive it will confirm the grade obtained in the written part with a maximum increment of 30%. If negative the exam is not sufficient and also the written part must be repeated.

#### Reference texts

1. G.Aletti, G.Naldi, L.Pareschi, Calcolo differenziale e Algebra lineare, McGraw-Hill, 2005 (main reference text)
2. R. A. Adams, Calcolo differenziale Vol.1, Seconda Edizione, Casa Editrice Ambrosiana, Milano 1999
3. S. Salsa, Squellati, Esercizi di matematica Vol. 1, Calcolo infinitesimale e Algebra lineare,Ed. Zanichelli, 2001 (collection of exercises)
4. Teacher's handouts