Overall objectives:

1 - To equip the students with some theoretical and practical knowledge in Mathematics which is necessary for general and academic purposes, thus giving continuity to the work developed at curricular unit General Mathematics I.
2 - To evaluate integrals of functions of one real variable using techniques such as immediate (or almost immediate) integration, integration by parts, by substitution and the simplification of rational functions. Apply the concept of integral in order to solve some practical problems (e.g. calculus of areas).
3 - To identify and solve different types of ordinary differential equations applying different methods (e.g. variable separation, variable change, variation of parameters). To know how to solve some systems of linear differential equations. To know some methods on qualitative analysis of solutions of differential equations.

Syllabus:

1 - Integral Calculus in R
1.1 - Immediate (or almost immediate) integration
1.2 - Integration by parts
1.3 - Integration by substitution
1.4 - Integration of rational functions
1.5 - Definite integral
1.6 - The Fundamental Theorem of Calculus
1.7 - Application of Integral Calculus on the calculation of areas
1.8 - Improper Integrals
1.9 - Integral calculus with the help of a computer
2 - Differential Equations
2.1 - Basic concepts on differential equations
2.2 - Ordinary differential equations and partial differential equations
2.3 - First-order differential equations
2.3.1 - Initial value problem
2.3.2 - Existence and uniqueness of solutions
2.3.3 - Integral curves
2.3.4 - Separable differential equation
2.3.5 - Linear differential equation
2.3.6 - Exact differential equation
2.3.7 - Homogeneous differential equation
2.4 - Qualitative analysis of solutions
2.4.1 - Equilibrium solutions
2.4.2 - Vector fields
2.4.3 - Integral curves
2.5 - Differential equations of order 2 and higher
2.6 - Systems of differential equations of 1st order
2.7 - Examples of problems (related to Biochemistry or other areas) involving differential equations
2.8 - Solving differential equations and analising their solutions with the help of a computer

Literature/Sources:

Apostol Tom M. , 1983 , Cálculo , Editora Reverté Ltda
Courant, R. John , 1989 , Introduction to Calculus and Analysis I , Springer
Ferreira, J. Campos , 2005 , Introdução à Análise Matemática , Fundação Calouste Gulbenkian
Piskounov, N. , 1990 , Cálculo Diferencial e Integral , Editora Lopes da Silva
McCann, Roger C. , 1982 , Introduction to ordinary differential equations , Harcourt Brace Jovanovich, New York

Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Use of the whiteboard for presentation, explanation of subject material and problem solving. Use of the computer and projector to help visualize and understand the concepts. Usage of adequate software (e.g. Geogebra) or graphing calculator to better visualize some concepts and / or to help in exploring problems with more complexity. Evaluation: two tests (one at the middle and one at the end of the semester), each one with a weight of 50% on the final grade. In the appeal season, students who have failed to pass the course during the normal season can choose between retaking one or the two tests, each corresponding to 50% of the final grade.