# MATHEMATICS

**Academic Year 2024/2025**- Teacher:

**DANIELA FERRARELLO**

## Expected Learning Outcomes

The course has a twofold target: on the one hand, it aims to provide basic calculus tools, useful for other courses, on the other hand it aims to train the skills of reasoning and problem solving, typical skills of a mathematical education.

We will always start with a practical problem and then we provide the mathematical knowledge useful in solving the posed problem.

The exercises will be aimed, in a special way, at developing a problem-solving aptitude.

EXPECTED RESULTS (ER):

- Knowledge and understanding: knowledge on goniometry and trigonometry, functions, differential calculus, integral calculus.
- Applying knowledge and understanding: knowing how to manipulate goniometric basis, knowing how to study functions, knowing how to interpret graphs of functions.
- Making judgements: knowing how to give mathematical interpretations of real problems, knowing how to deduce information on real problems starting from mathematical data, knowing how to make judgments on real facts starting from mathematical considerations.
- Communication skills: knowing how to communicate in a rigorous way mathematical topics, knowing how to communicate effectively mathematical meanings.
- Learning skills: ability to study and understand both in groups and independently, being able to connect the studied topics among them, ability to grasp connections between mathematical topics and other disciplines (lateral transfer), ability to understand also more complex mathematical topics (vertical transfer).

## Course Structure

The class hours (63) are divided into lectures (21 hours) and exercises (42 hours).

In the case of merged mode (distance learning – face to face learning), face to face lectures will be used only for practical exercises (ER b.)

Topics will be mediated by a visual and practical approach, also with the aid of software with a great didactic impact (ER a. and b.), to reach later on formal definitions (ER d.), through participated lessons (ER d.).

Examples of real applications will be provided. (ER c. and e.).

In case of mixed or remotely mode, necessary changes will be provided.

**Information for students with disabilities and/or Learning Disorders.**

As a guarantee of equal opportunities and in compliance with current laws, interested students can ask for a personal interview in order to plan any compensatory and/or dispensatory measures, based on their specific needs and on teaching objectives of the discipline.

It is also possible to ask the departmental contacts of CInAP (Center for Active and Participatory Inclusion - Services for Disabilities and/or Learning Disorders), in the persons of professor Anna De Angelis.

## Required Prerequisites

Basic math cultural prerequisites:

- Arithmetic (numbers, operations, percentages, approximations);
- Geometry (polygons, Pythagorean theorem);
- Algebra (polynomials, first degree equations and inequalities, second degree equations and inequalities).

## Attendance of Lessons

Attendance at the course is strongly recommended, especially for exercises, which will actively involve students and promote their learning.

Attendance will be recorded, only for statistical and course evaluation purposes.

## Detailed Course Content

- Basic goniometry and trigonometry.
- Functions: monotone functions, linear and polynomial functions, exponential and logarithmic functions, limits.
- Differential calculus: derivative of a function and theorems;
- Basic statistics.

## Textbook Information

Dario Benedetto, Mirko Degli Esposti, Carlotta Maffei. Matematica per scienze della vita. Terza edizione

Casa Editrice Ambrosiana. Distribuzione esclusiva Zanichelli.

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Functions | From [1]: Cap. 4.1; Cap. 4.2. |

2 | Monotone functions | From [1]: Cap. 4.3. |

3 | Linear functions | From [1]: Cap. 5.1 |

4 | Polynomial functions | From [1]: Cap. 5.2; Cap. 5.3. |

5 | Exponential functions | From [1]: Cap. 6.1. |

6 | Logarithmic functions | From [1]: Cap. 6.2. |

7 | Goniometric functions | From [1]: Cap. 6.3 |

8 | Limits | From [1]: Cap. 7.1; Cap. 7.2; Cap 7.3. |

9 | Derivatives | From [1]: Cap. 8.1; Cap. 8.2; Cap. 8.3. |

10 | Statistics | From[1]: Cap. 12.1; Cap. 12.2; Cap. 12.3. Cap. 12.5 |

## Learning Assessment

### Learning Assessment Procedures

**In itinere** (multiple-choice) tests will be administered during the course.

**Modalities for those who use the in itinere tests: **

In itinere tests are worth 20 points out of 30 and the final written (with exercises) is worth 10 points out of 30.

Those who score at least 18 with itinere tests and the written, may decide to confirm their written grade.

**Modalities for those who do not make use of the in itinere tests:**

The final test consists of a written test (with exercises) and an oral test.

Final exams may also be carried out on-line, if conditions require it.

### Examples of frequently asked questions and / or exercises

EXAMPLES OF EXERCISES FOR THE WRITTEN TEST

Find the domain of the following function: f(x)= log(1-x2)

Calculate the limit for x tending to infinity of the function f(x)= (x2-1)/(x2)

Study the behaviour and produce the graph of the function f(x) = x * e^x

A population of bacteria follows the trend of the logistic function p(t)= 1000/(1 +e^(-2t) ) , with t>=0, where the variable t expresses the time in days and p(t) the numerosity of the bacterial colony. Calculate the limit lim_(t→+∞) p(t) . What do we infer from the calculated limit?

Describe the behavior of the function p(t) = 1 / [1+ and -2t] that expresses the trend of a population

In the first three math partial tests, Liliana's grades were 24, 18, 27. The grade on the fourth and final test did not change the (arithmetic) mean . What is the grade of the fourth test?

What can be inferred from a data set if the arithmetic mean is less than the median?

Write a data set whose arithmetic mean is greater than the median.

Determine mean and median of the following data set: 1, 2, 3, 6, 7, 8, 9.

EXAMPLES OF ORAL QUESTIONS

Define the goniometric functions sine, cosine, tangent.

Enunciate the sine theorem.

How do you solve a right triangle?

How can a phenomenon with a linear behavior be represented on the Cartesian plane? What kind of equation will be associated with the representation on the plane?

How is a function defined? In what cases is it appropriate to use a function to model a real phenomenon? Provide examples

What phenomenon might an exponential function describe? What about a logarithmic function?

Give the definition of a finite limit of a real variable function

What can be the use of calculating an infinite limit for a real variable function?

Give the definition of derivative of a real function of real variable and explain its geometric meaning

Why is useful to understand the trend of the derivative of a function? Bring examples

Say what the mean and median of a set of data are. When is it best to use one or the other?

What is the standard deviation?