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Calculus

Code 12496
Year 1
Semester S1
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements There is no entry requirement.
Learning outcomes At the end of the course, students should be able to:
1) Compute limits and study the continuity of real functions of real variable.
2) Compute derivatives of real functions of real variable.
3) Obtain linear and differential approximations of real functions of real variable.
4) Use derivatives to the study a function, in the computation of limits and in mathematical models of Health and Life Sciences.
5) Primitivate and integrate real functions of real variable.
6) Apply integral calculus to the computation of areas and volumes and to study mathematical models of Health and Life Sciences.
7) Study the continuity of functions of several variables.
8) Determine partial and directional derivatives.
9) Determine linear approximations of functions of several variables.
10) Apply differential calculus to the study of mathematical models with functions of several variables .
11) Solve simple ordinary differential equations.
12) Understand and analyze mathematical models using ODEs.
Syllabus 1. Real functions of real variable - Definition and examples. Mathematical models; Composition of functions and Inverse of a function; Examples of functions; Limits and continuity.

2. Differential calculus in R - Definition; Derivative; Linear and differential approximations; Derivatives of higher order; Theorems of Rolle and Lagrange; Applications in mathematical models of Health and Life Sciences.

3. Integral calculus in R - Primitives; Riemann integral; The fundamental theorem of integral calculus; Change of coordinates; Applications in mathematical models of Health and Life Sciences.

4. Real functions of several variables - The space R^n; Functions of several variables; Limits and Continuity; Partial derivatives; Linear approximations; Mathematical models.

5) Introduction to ordinary differential equations - Definition and examples; Separable equations and first order linear equations; Mathematical models.
Main Bibliography - Stewart, J. (2006). Cálculo, Volumes I & II (5ª ed.). São Paulo: Pioneira Thomson Learning.
- Lima, E. L. (2004). Análise real (8ª ed). Rio de Janeiro: Instituto Nacional de Matemática Pura e Aplicada.
- Lima, E. L. (2007). Análise no espaço R^n. Rio de Janeiro: IMPA.
Teaching Methodologies and Assessment Criteria The classes will be theoretical-practical. The teacher presents the concepts and the results and illustrates the theory with examples and applications. The student is encouraged to participate in classes, interacting with the teacher and sometimes solving exercises and problems. Autonomous work, consisting mainly in solving the exercises, is encouraged. The assessment carried out over the teaching-learning period will consist of two written tests.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2016-06-09

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