Mathematical Analysis I D
Code
10569
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Paula Alexandra da Costa Amaral, Paula Cristiana Costa Garcia Silva Patrício Rodrigues
Weekly hours
6
Total hours
84
Teaching language
Português
Objectives
The goals of the course include:
- a basic understanding of the special language, notation, and point of view of calculus
- the ability to solve basic computational problems involving derivatives and integrals
- a basic understanding of the fundamental theorems of calculus
Prerequisites
Algebra
Simplifying
exponents, radicals, logarithms
fractional expressions
Factoring polynomials
Solving equations
Solving inequalities
Functions
Domain and range
Evaluation
expressions such as f(x+h)
calculator use
Write one quantity as a function of another
Special functions:
linear, quadratic, polynomial
exponential and logarithmic
Function composition and decomposition
Inverses
Graphs
points in the plane
graphs of the special functions above
reading information
domain and range
increasing and decreasing behavior
maximum and minimum values
Special Topics
Translating verbal information into math symbols
Rates
Distance formula
Midpoint formula
Compound interest
Exponential growth and decay
Trigonometry
Radian and degree measurement of angles
The unit circle
Definitions of the six trig functions:
sine, cosine, tangent
cosecant, secant, cotangent
Graphs
Inverses
Basic identities
Pythagorean identities
Reciprocal identities
Subject matter
1. Topology, Mathematical Induction, Sequences: Basic topology of the real numbers. Order relation. Mathematical induction. Generalities about sequences. Convergence of a sequence and properties for calculus of limits. Subsequences. Bolzano-Weierstrass theorem.
2. Limits and Continuity: Convergence according to Cauchy and Heine. Calculus properties. Continuity of a function at a given point. Properties of continuous functions. Bolzano theorem. Weierstrass theorem. Continuity and reciprocal bijections.
3. Differentiability: Generalities. Fundamental theorems: Rolle, Lagrange and Cauchy. Calculus techniques for limits. Taylor formula and applications.
4. Indefinite Integration: Introduction. Indefinite integration by parts. Indefinite integration by substitution. Indefinite integration of rational functions.
5. Riemann Integration: Introduction. Fundamental theorems. Definite integration by parts and by substitution. Some applications. Improper integration.
Bibliography
Adopted Text
Ana Alves de Sá e Bento Louro, Análise Matemática I, FCT-UNL, 2011
Recommended Bibliography
1. Robert G. Bartle e Donald R. Sherbert, Introduction to Real Analysis, John Wiley & Sons Inc., 1999
2. Jaime Campos Ferreira, Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982
3. Rod Haggarty, Fundamentals of Mathematical Analysis, Prentice Hall, 1993
4. Carlos Sarrico, Análise Matemática, Leituras e Exercícios, Gradiva, 1997
Teaching method
Theoretical classes consists of an oral explanation which is illustrated by examples.
Practical classes consists on the resolution of exercises. Students have access to copies of the proposed exercises. Some of the exercises are solved in class, the remaining are left to the students as part of their learning process.
Evaluation method
Students must attend, at least, all problem-solving sessions, with the possible exception of three. The evaluation consists of three mid-term tests, or of a final exam. The three tests can replace the final exam, in case of approval. More detailed rules are available in the Portuguese version.