MATHEMATICAL ANALYSIS I

General Module

Course Sheet

Course code:

TO001A

Academic Year of enrolment:

2023

Year of supply:

2023/2024

Type of Course:

Basic

Degree:

Course of 1st Cycle Degree in INGEGNERIA BIOMEDICA

Disciplinary Sector:

MATHEMATICAL ANALYSIS

Is part of:

CALCULUS 1 AND PHYSICS 1

Location:

PESCARA

Credits:

Programme Year:

Professor and Collaborators:

DORIA Serena

Cycle:

First semester

Hours of classroom activity:

Individual study hours:

Prerequisites:

First degree equations; integer and fractional first level inequalities; second degree equations; integer and fractional second level inequalities; equation systems; systems of inequalities.

Objectives

The student who successfully completes this course will have the following skills:

-Knowledge and ability to understand:

to use the rules of logic to study the way in which mathematical arguments are constructed
to use the structure of topological space
to analyze the structure of mathematical proofs and illustrate proof techniques by means of examples
to define the limit of a function and continuity of a function
to prove and to work with theorems relating to continuous functions beyond those found in elementary calculus
to define the derivative of a function andto establish properties of differentiable functions
to define the Riemann integral and to establish properties of integrable functions

-autonomy of judgment
to analyse new problems and to solve them

-communication skills

to use the mathematical language to describe and to solve problems

Contents

Functions, limits, continuity, differentiability, development by Taylor polynomial Riemann integrability, differential equations with separable variables, linear differential equations, second order differential equations with constant coefficients.

Extended Syllabus

• Elementary functions - Invertible functions. Exponential function and logarithm. Sine and cosine functions and their inverse. Tangent and cotangent functions and their inverse.
• Limits and functions in a topological space. Topological spaces. Examples of topological spaces. Notable metric and topological spaces. Function limits. Theorems of uniqueness of the limit, of the compound function and of the restriction. Special cases of expanded R and R.
• Continuous functions - Definitions and first examples. Calculation of limits for substitution. Theorems on continuous functions.
• Derivation - Incremental ratio and derivative . Examples. Derivable functions. Left derivative, right derivative. Theorem of the left and right derivative. Tangent to a curve.
• Examples and applications in geology - Sedimentation frequency.
• Rules of derivation - Linearity theorem. Derivative of the product and the quotient. Derivative of the compound function. Derivative of the inverse function. Derivatives of elementary functions. Derivatives of order superior to the first.
• Applications of differential calculus - Increment, decrescence, maxima and minima. Increasing, decreasing, strictly increasing and strictly decreasing function. Relative and absolute maxima and minima.Transmissions on the derivative in the points of crescenza and decrescence, maxima and minima.
• Notable theorems on the derivative - Rolle's theorem. Lagrange's theorem and its consequences. Cauchy's theorem. Theorems of the Hospital. Taylor's formula with the rest of Peano. Taylor's formula with the rest of Lagrange. Use of the Taylor formula for the theorems related to growth, decrescence, maxima and minima and for the approximate calculation of the values assumed by a function.
• Complements and exercises on derivation - Asymptotes. Convex and concave functions. Flexed. Theorems for the search for points of concavity and convexity and inflections. Study of the graph of a function. Calculation of limits with the help of Hospital theorems and Taylor's formula.
• Integrals of functions with a variable. Definition and geometric interpretation. Some properties of integrals. Comparison and average theorem. Applications of the integral. The fundamental theorem of integral calculus. Theorems on primitives.
• Examples and applications in geology - Sedimentation accumulation

Recommended Bibliography

S.Doria, V. Piattelli
Matematica - Elementi di Teoria e guida alla risoluzione degli esercizi, Aracne Editrice

Teaching Methods

Lectures and exercises given by the teacher in the classroom using the blackboard.

Evaluation methods

Verification of learning:

fractional evaluations in periodic evaluations during the course.
There will be two written evaluations and one oral evaluation.
The final grade will be the arithmetic mean of the three evaluations.

Università degli Studi
"G. d'Annunzio"

Chieti - Pescara

Aree

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