# Calculus

## You are here

Credits
7.5
Types
Compulsory
Requirements
This subject has not requirements, but it has got previous capacities
Department
MAT
Introduction to the basic calculus tools for functions of one variable (continuity, derivation, integration and series) as well as some basic notions of calculus for functions of several variables. The presentation is used to introduce some basic numerical calculation tools.

## Teachers

### Person in charge

• Jordi Villanueva Castelltort ( )

### Others

• Jose Tomas Lazaro Ochoa ( )
• Rafael Ramirez Ros ( )

## Weekly hours

Theory
3
Problems
2
Laboratory
0
Guided learning
0
Autonomous learning
7.5

## Competences

### Technical Competences

#### Technical competencies

• CE1 - Skillfully use mathematical concepts and methods that underlie the problems of science and data engineering.

### Transversal Competences

#### Transversals

• CT5 - Solvent use of information resources. Manage the acquisition, structuring, analysis and visualization of data and information in the field of specialty and critically evaluate the results of such management.
• CT6 [Avaluable] - Autonomous Learning. Detect deficiencies in one's own knowledge and overcome them through critical reflection and the choice of the best action to extend this knowledge.

#### Basic

• CB1 - That students have demonstrated to possess and understand knowledge in an area of ??study that starts from the base of general secondary education, and is usually found at a level that, although supported by advanced textbooks, also includes some aspects that imply Knowledge from the vanguard of their field of study.

### Generic Technical Competences

#### Generic

• CG2 - Choose and apply the most appropriate methods and techniques to a problem defined by data that represents a challenge for its volume, speed, variety or heterogeneity, including computer, mathematical, statistical and signal processing methods.

## Objectives

1. Elementary functions, continuity, limit and associated concepts
Related competences: CT6, CB1,
2. The derivative and its use as a basic calculation tool.
Related competences: CE1, CT5, CT6, CG2, CB1,
3. Calculation of primitives and definite integrals.
Related competences: CE1, CT5, CT6, CG2, CB1,
4. Discussion of the convergence of improper integrals, sequences and series and calculation of their limit in simple cases where it is approachable.
Related competences: CE1, CT5, CT6, CG2, CB1,

## Contents

1. Functions
Rational and real numbers. Absolute value. Qualitative study of the most usual functions and their inverses. Limit and continuity. Theorem of Bolzano and theorem of the intermediate value.
2. Derivation
Derivative of a function. Direct applications of the derivative. Rolle's and mean value theorems. Rule of L'Hôpital. Taylor's formula and applications. Introduction to the functions of several variables. Numerical derivation. Numerical computation of zeros of functions.
3. Integration
Calculation of primitives. Definite integrals. Numerical integration. Improper integrals and their convergence criteria. Euler's Gamma function.
4. Sequences and series
Sequences and their limit. Numerical series and their convergence criteria. Power series. Taylor series.

## Activities

Activity Evaluation act

### Midterm exam

Objectives: 1 2
Week: 10 (Outside class hours)
Theory
2h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
5h

### Final exam

Objectives: 1 2 3 4
Week: 15 (Outside class hours)
Theory
2.5h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
10h

### Functions

Theory
9h
Problems
7h
Laboratory
0h
Guided learning
0h
Autonomous learning
22h

### Differentiability

Theory
12.5h
Problems
10h
Laboratory
0h
Guided learning
0h
Autonomous learning
31h

### Integrability

Theory
11h
Problems
8h
Laboratory
0h
Guided learning
0h
Autonomous learning
26.5h

### Sequences and series

Theory
8h
Problems
5h
Laboratory
0h
Guided learning
0h
Autonomous learning
18h

## Teaching methodology

Lectures introduce the concepts, algorithms and results needed to reach athe required level of understanding. These concepts are put into practice in the problem classes in which, due to its structure, it is easier to encourage the active participation of students. The practice note is aimed at encouraging the most creative and transversal aspects of the subject since it involves the completion of problems that involve the understanding of concepts and the use of tools that we could hardly fit into the regulated exhibition.

## Evaluation methodology

Final grade = max(0.1*NPract + 0.9*NExFinal, 0.1*NPract + 0.3*NExParcial + 0.6*NExFinal)
on
[NPract]: numerical methods exam
[NExParcial]: midterm exam

In case of reevaluation, the new grade will replace the previous.