Mathematics I

Recommended Prerequisites

Mathematics A from Secondary Education.

Teaching Methods

The classes are essentially of expository nature and should include examples that lead the students to understanding and applying the material being taught. The classes should be focused on the teaching of the reasoning processes, so that the students learn how to manipulate the objects presented along the course and more easily find out by themselves how to reach other results by independent reading or problem solving. Some tutorial support will be available to help the students with the proposed tasks outside the classroom.

Learning Outcomes

Development of the skills on calculation of limits and derivatives of r.f.r.v.  Notions  of rate, related rates, differentials and derivatives in the solving of problems coming form applications. Calculation of integrals and its application in the calculation of areas, volumes, curve length and areas of surfaces of evolution. Identifying and solving differential equations of 1st order using graphical, numerical and analytical methods. Draw and identify curves in polar and parametric coordinates. Calculate curve length and areas of regions defined by curves in these coordinates. Posing and interpreting problems. Mathematical modelling in Chemistry.

 

Generic competencies:

Competencies in analysis and synthesis

Competencies in oral and written communication

Competencies in problems solving

Competencies in critical reasoning

Adaptability to new situations

Creativity

Competencies in applying theoretical knowledge in practical situations

Competencies in self-criticism and self-evaluation.

Work Placement(s)

No

Syllabus

1. Functions: Limits; continuity; derivatives (motivation, properties, implicit differentiation, rate of change, related rates, differentials, applications of the derivative, study of functions). What is mathematical modelling.

2. Integral calculus: antiderivatives; definition of definite integral and its properties; fundamental theorem of calculus; applications (areas, volumes, curve length and areas of surfaces of revolution); improper integrals; numerical integration.

3- Differential Equations: motivation; differential equations of first order; graphical method (direction field); numerical method (Euler method); analytical methods (differential equations of separable variables, linear and Bernoulli equations); logistic equation and Lotka-Volterra predator-prey model.

4. Parametric equations and polar coordinates: curve sketching; calculation of areas and curve length.

Head Lecturer(s)

Carlos Manuel Franco Leal

Assessment Methods

Assessment
Exam (100%) or Midterm (50%) + Test (40-50%) + Problem resolving report (0-10%).: 100.0%

Bibliography

SILVA, Jaime Carvalho e (1999). Princípios de análise Matemática Aplicada. McGraw-Hill.

SILVA, Jaime Carvalho e LEAL, Carlos (1996). Análise Matemática Aplicada. McGraw-Hill.

SILVA, Jaime Carvalho e, Apliquetas para Análise Matemática, http://www.mat.uc.pt/~jaimecs/am107/apliquetas.html

IME-USP-SP, e-cálculo, http://ecalculo.if.usp.br/.

Portal Educação, Videoaula | Cálculo Diferencial e Integral, http://www.youtube.com/watch?v=T239d170aNY

Jerison, David. 18.01SC Single Variable Calculus,Fall 2010. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 26 Jul, 2013). License: Creative Commons BY-NC-SA