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# Mathematical Analysis I B

### Code

5000

Faculdade de Ciências e Tecnologia

### Department

Departamento de Matemática

7.0

### Teacher in charge

Bento José Carrilho Miguens Louro

6

78

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 theorem of calculus

### Prerequisites

Algebra
Simplifying
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:
exponential and logarithmic
Function composition and decomposition
Inverses
Graphs
points in the plane
graphs of the special functions above
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. 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.

### Bibliography

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.

### 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.

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