Calculus II

Recommended Prerequisites

Mathematics A from Secondary School and Calculus I.

Teaching Methods

The lectures are presented using slides and graphic illustrations of some of the concepts presented. The theoretical presentation is illustrated with practical examples. The pratical classes run  accordingly to the lectures presentes before. In practical classes the students know, in advance, what exercises should they solve, so that they can put into practice the knowledge already presented in lectures. Thus, the classes work  as an active time of learning and exposure of concerns.

Learning Outcomes

Overall objectives

Understanding and solve problems related to economics and finance.

Specific objectives

Apply the techniques of integral calculus of real functions of real variables, series, differential equations and difference equations in exercises (that include economic and financial problems).

Generic competencies

The student should  be able to: analyze and  build a process of resolution of mathematical problems; interpret and solve real problems; communicate  the procedures used for solving the problems.

 

Specific competencies

The student should be able to: compute integrals and areas; analyse the nature of a serie and compute, if possible, its sum. Study series of functions and Taylor and MacLaurin series; solve differential equations of first order and real problems, involving differential equations, related to economic and financial issues; solve linear differential equations of order bigger than one, solve also linear difference equations.

Work Placement(s)

No

Syllabus

1. Integral (int.) calculus of real functions of real variable
1.1. Indefinite int.: definition and computation
1.2. Integrals: summations; definite int.: definition and properties; fundamental theorem of integral calculus; ilustrations; improper integrals

2. Series (S.)
2.1. Definitions
2.2. Geometric series
2.3. Series and improper integrals
2.4. Absolute convergence
2.5. Power series
2.6. Taylor/ MacLaurin series

3. First order differential equations (eq.)
3.1. Definition
3.2. Eq. of separated / separable variables
3.3. Some illustrations
3.4. Exact total differential eq.
3.5. Linear differential eq. of order 1
3.6. Change of variables: homogeneous eq. and Bernoulli Eq.

4.  Linear differential eq. and linear difference eq.
4.1 Linear differential eq.
4.2. Linear difference eq.
4.3. Homogeneous (homog.) linear differential eq.
4.4. Complex roots of the auxiliary eq.
4.5. Homog. linear difference eq.
4.6. Non-homog. eq. and non-homog.s difference eq.
4.7. Ilustrations

Head Lecturer(s)

Ana Margarida Machado Monteiro

Assessment Methods

Final Assessment
Exam: 100.0%

Continuous Assessment
Exercises report: 10.0%
Midterm exams: 20.0%
2 tests: 70.0%

Bibliography

BINMORE,Ken;DAVIES,Joan— Calculus:[concepts and methods].Cambridge:Cambridge University Press,2007.[BP 517 BIN]
BRAUN,Martin—Differential equations and their applications:an introduction to applied mathematics.4th ed.New York:Springer-Verlag,1993.[BP 517.9 BRA]
CHIANG,Alpha C.;WAINWRIGHT,Kevin—Fundamental methods of mathematical economics.4th ed. Boston:McGraw-Hill Book,2005.[BP 51-7 CHI]
LARSON,Ron;HOSTETLER,Robert P.;Edwards,Bruce H.—Cálculo.8ª ed.São Paulo:McGraw-Hill Interamericana do Brasil,2006.2 vol.[BP 517 LAR]
MURTEIRA,J.M.Ruas;SARAIVA,Paulo Manuel David Mota-Equações diferenciais ordinárias: introdução teórica exercícios e aplicações.Coimbra:Edições Almedina,2010.[BP 517 MUR]
PIRES, Cesaltina—Cálculo para economistas.Lisboa:Editora McGraw-Hill de Portugal,2001.[BP 51-7 PIR]
SARAIVA,Paulo Manuel David Mota-Cálculo II-apontamentos teóricos e folhas práticas.Coimbra : FEUC,2009 .[BP 517 SAR]