65851 - Mathematics and Statistics

Learning outcomes

The student acquires a good knowledge of mathematical and technical tools they use, and is able to set and solve problems and to assimilate new concepts from the experience and knowledge. Moreover the student is able process statistical data.

Course contents

A) Prerequisites
1. Polynomial expressions
2. Linear equations in one variable
3. Factorization of polynomial expressions.
4. Simplification of rational expressions
5. Square Roots

B) Teaching units

Preliminaries
Theoretical contents (4 hours)
Equations of lines and circles.
Linear systems of equations in two variables
Inequalities of the second degree.

Tutorials relating to theoretical content (3 hours)

Knowledge acquired in the teaching unit 1

The student knows the equations of lines and circles in the plane.
Interpret graphically and solve a linear system of two equations in 2 unknowns.
Solve second degree inequalities..

2.Percentages

Theoretical contents (2 hours)
How to calculate percentage, what is it and how to use it. Methods to calculate the percentage with several examples


Exercises related to the theoretical content (2 hours)

Knowledge acquired in the course unit 2

•      Knowing how to calculate percentages
• Ablility in recognizing  formulas to solve problems on percentages.
  

Tutorials relating to theoretical content (2 hours)

Knowledge acquired in the course unit 2

The student is able to work using simple operations with sets.
The student uses the logical connectives and quantifiers and is able to distinguish between hypothesis and theis.






3. Real numbers and real functions of a real variable.

Theoretical contents (4 hours)

The sets of natural numbers, integers, rational numbers.
The set R of real numbers: algebraic structure and ordered structure;
existence of irrational numbers (eg square root of 2).
Intervals and rays in R.
Functions between sets: domain, codomain, image, injective, surjective and bijective.
The real functions of a real variable and their graphs;
Composition of functions.
Invertible functions and the inverse function.
Maximum, minimum, for a real function, the points of maximum and absolute minimum.
Monotone functions.
Even and odd functions.

Tutorials relating to theoretical content (2 hours)

Knowledge acquired in the teaching unit 3

The student knows the elementary language of set theory
The student knows the properties of operations on real numbers in relation to the properties of order.
The student knows recognizes the graph of a function domain and image.
The student knows recognizes the graphs of even and odd functions .
The student knows how to compose decomposing given functions.


4. Elementary functions

Theoretical contents (6 hours)

The absolute value function;
The power functions with exponent integer, rational, real.
The exponential function.
The logarithm function, properties of logarithms, logarithmic and exponential inequalities.
The trigonometric functions

Tutorials relating to theoretical content (4 hours)

Knowledge acquired in the teaching unit 4:
The student knows and recognizies graphs of elementary functions.
The student operates on graphs of elementary functions with translations, symmetries, compositions and absolute value.



5. Derivatives and their applications

Theoretical contents (3 hours)

Notion of derivative at a point for real functions and equation of the tangent line.
Derivative of the sum, difference, product and quotient.
Local maxima and minima; stationary points.
Maximum and absolute minimum of differentiable functions on a closed interval limited and Fermat's Theorem.

Tutorials relating to theoretical content (2 hours)

Knowledge acquired in the teaching unit 5:

The student knows the geometrical meaning of the derivative at a point.
The student lculates the tangent line to the graph of a function at a point.
The student lculates the maximum and minimum for a differentiable function defined on a closed and bounded interval.

Readings/Bibliography

Edizioni Zanichelli

Teaching methods

Lessons and exercises in the classroom.

Assessment methods

The teaching of Mathematics is part of the Integrated Course : Elements Of Mathematical Statistics  with the following other teaching: Statistics.
Therefore, the evaluation of the course takes into account jointly the level of knowledge and skills acquired by the student in relation to the contents of all of the above teachings. The knowledge and skills imparted by the teaching of mathematics are assessed through the following ways:

   a final exam, written with a part of multiple-choice questions and two years on the main parts of the program: that is, graphs of elementary functions, calculus of maxima and minima and a question related to the theory of activity (1 hour and 20 minutes).

For the test statistic:

Written test organized into two parts:

- A test of 10-15 multiple choice questions

- The performance of 3-4 problems


The final evaluation will be given by the average of the results obtained in tests of mathematics and statistics.

Teaching tools

The teaching material can be found in the website of the teacher.

Office hours

See the website of Francesca Cagliari

See the website of Stefania Curti