Mathematical Analysis I
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2008/2009 - 1S
Cycles of Study/Courses
Teaching language
Portuguese
Objectives
1- BACKGROUND
Almost every aspect of professional work in the world involves mathematics. A solid knowledge of mathematical analysis is required for any engineering degree namely mechanics and industrial management.
2- SPECIFIC AIMS
Development of the reasoning capacity of the students and understanding essential mathematical concepts. Students should get solid theoretical and practical formation on the main concepts and results of sequences, differential and integral calculus real functions of one real variable.
3- PREVIOUS KNOWLEDGE
High school math. Functions and graphs. The limit concept, the concept of continuity at a point and the derivative of a function. Differentiation rules.
4- PERCENT DISTRIBUTION
Scientific component:75%
Technological component:25%
5- LEARNING OUTCOMES
Knowledge and Understanding- Theoretical concepts and practical formation on differential and integral calculus real functions of one real variable. Polynomial approximation for real-valued functions using Taylor's polynomials and the error concept. Calculation of areas in cartesian and polar coordinates.Volume calculations.
Engineering analysis-Application of differentiation and integral calculus of one real variable function.
Engineering design- Engineering problems of one real variable.
Investigations- Practical formation on the main concepts and results of differential and integral calculus in R.
Engineering practice- Applications in Physics and Mechanics.
Transferable skills- Knowledge of differential and integral calculus of one real variable.
Program
A. Differential Calculus in R:
Review of fundamentals of differentiation.
Increments, differentials and linear approximations. The mean-value theorem for derivatives.
Polynomial approximations to functions: Taylor's polynomials generated by a function.
Taylor’s formula with remainder. Estimates for the error in Taylor’s formula.
The Taylor series as a limit of Taylor polynomials.
Numerical series: properties, convergence criteria, alternating series.
Reference of functional series. Concept of convergence interval.
B. Integral Calculus in R:
Riemann sums and the integral. Integrability of bounded monotonic functions. The integrability theorem for continues functions. Properties of the integral. Mean-value theorem for integrals.
The derivative of an indefinite integral. The first fundamental theorem of calculus.
Primitive functions and the second fundamental theorem of calculus.
Integration by substitution. Integration by parts.
Areas of plane regions. Polar coordinates. Area calculation in polar coordinates. Volume calculations by the method of cross sections.
Integration by rational partial fractions. Rational trigonometric integrals. Integrals containing quadratic polynomials.
C. Additional topics:
Hyperbolic functions.
Improper integral.
First order differential equations.
Mandatory literature
Carlos A. Conceição António; Análise Matemática 1, Texto de apoio, AEFEUP, 2007/2008
Carlos C. António, Catarina F. Castro, Luísa C. Sousa, M. Graça Pinto; Exercícios propostos para as aulas práticas de AM I, AEFEUP, 2007/2008
Larson, Hostetler & Edwards; Cálculo, McGraw-Hill Interamericana, 2006. ISBN: 85-86804-56-8
Larson, Hostetler & Edwards; Cálculo, McGraw-Hill Interamericana,, 2006. ISBN: 85-86804-82-7
Demidovitch, B. 340;
Problemas e exercícios de análise matemática. ISBN: 972-9241-53-8
Complementary Bibliography
Spivak, Michael;
Calculus, N. ISBN: 0-914098-89-6
Apostol, Tom M.;
Cálculo, N. ISBN: 84-291-5015-3 (vol.1)
Edwards, C. Henry;
Calculus. ISBN: 0-13-736331-1
J. Campos Ferreira; Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 8ª edição
Teaching methods and learning activities
Theoretical-practical classes are based on written and oral expositions. Application examples are frequently presented, especially at the end of each unit. Students are encouraged to solve specific exercises during class.
Evaluation Type
Distributed evaluation without final exam
Assessment Components
Description |
Type |
Time (hours) |
Weight (%) |
End date |
Subject Classes |
Participação presencial |
65,00 |
|
|
Examinations |
Exame |
6,00 |
|
2009-02-16 |
|
Total: |
- |
0,00 |
|
Amount of time allocated to each course unit
Description |
Type |
Time (hours) |
End date |
Study time for examinations |
Estudo autónomo |
18 |
2009-02-16 |
Study time following classes |
Estudo autónomo |
73 |
2009-02-16 |
|
Total: |
91,00 |
|
Eligibility for exams
To attain frequency, the student must:
1) Not exceed the absence limit allowed in Article 4 of the General Evaluation Rules of FEUP; and
2) To attend at least one of the two tests with minimal classification of 6 out of 20.
Calculation formula of final grade
The grade of the final classification is obtained taking into consideration the grade of the distributed evaluation.
Examinations or Special Assignments
N/A
Special assessment (TE, DA, ...)
This evaluation will be exclusively the result of the written exam according to General Evaluation Rules of FEUP.
Classification improvement
Classification improvement will be exclusively the result of the written exam according to General Evaluation Rules of FEUP.
Observations
Language of instruction: Portuguese