Mathematical Analysis I B
Faculdade de Ciências e Tecnologia
Departamento de Matemática
Teacher in charge
Bento José Carrilho Miguens Louro
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 theorem of calculus
exponents, radicals, logarithms
Domain and range
expressions such as f(x+h)
Write one quantity as a function of another
linear, quadratic, polynomial
exponential and logarithmic
Function composition and decomposition
points in the plane
graphs of the special functions above
domain and range
increasing and decreasing behavior
maximum and minimum values
Translating verbal information into math symbols
Exponential growth and decay
Radian and degree measurement of angles
The unit circle
Definitions of the six trig functions:
sine, cosine, tangent
cosecant, secant, cotangent
1. Topological in R. Mathematical indution. Sequences of real numbers.
2. Single real variable functions: limits and continuity. Properties of continuous functions; Bolzano’s theorem.
3.Differential calculus. Fundamental theorems: Rolle, Darboux, Lagrange and Cauchy. Indeterminate forms. Cauchy and L’Hospital rules. Taylor’s formula. MacLaurin’s formula. Extrema, concavity and inflection points.
4. Primitives. Primitivation by parts and by change of variables. Computation of primitives of rational, irrational and transcendent functions.
5. Integral calculus. Riemann integral. Fundamental theorem of calculus; Mean Value Theorem; Barrow’s formula. Computation of areas of plane figures.
6. Improper integrals.
The book used in classes is:
Howard Anton,Irl Bivens, Stephen Davis- Cálculo, 8ª edição, Artmed/Bookman (edição Brasileira),vol 1.
It can be used, as a complement, any other calculus book.
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.