Real Analysis I
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2019/2020 - 1S
Cycles of Study/Courses
Teaching language
Portuguese
Objectives
To develop the basic theory of differential and integral calculus for real functions of a real variable.
Learning outcomes and competences
On completion of the course, students should:
- know and be able to apply in problem solving the concepts and basic techniques of differential and integral calculus for real functions of a real variable;
- understand and be able to apply limiting processes in mathematical analysis;
- be able to distinguish between heuristic and rigorous arguments and, in simple situations, be able to construct rigorous arguments.
Working method
Presencial
Pre-requirements (prior knowledge) and co-requirements (common knowledge)
Knowledge corresponding to the Matemática A exam of Portuguese secondary education.
Program
1. The real numbers and the basic transcendent functions.
Supremum and infimum. Completeness of the real number system. Existence of the n-th root of a positive real number. Intervals and their characterization. Arc length in the unit circle. Trigonometric functions and their inverses. Exponential, logarithm and hyperbolic functions.
2. Limits and continuity
Sequences: convergence, basic properties and arithmetic of limits. The limit of a function of a real variable in an accumulation point of its domain. One-sided limits. Infinite improper limits. Characterization of limits using sequences according to Heine. Continuity of real functions of a real variable. Fundamental theorems on continuous functions: The Intermediate Value Theorem and Bolzano's Theorem, existence of extrema of a continuous function in a closed and bounded interval (Weierstrass).
3. Differential calculus
Differentiability and derivative of a function. Linear approximation. The derivative of sums and products of differentiable functions, the chain rule, the Inverse Function Theorem. Derivatives of exponential and trigonometric functions and their inverses. The Intermediate Value Theorem (Lagrange). Derivatives of higher order. Extrema and inflection points. Convexity. The n-th order Taylor polynomial. l'Hôpital's Rule. Newton's method. Antiderivatives. Uniqueness of the antiderivative, up to an arbitrary constant, of a function defined in an interval. Applications of the derivative (velocity, acceleration).
4. Integral calculus
The Riemann integral. Integrability of continuous functions. The Fundamental Theorem of the Calculus. Integration by parts and by substitution. Integration techniques. Improper integrals. Applications of the integral (area, volume, length, work). Introduction to separable differential equations.
5. Series
Infinite series: convergence and absolute convergence. Convergence criteria: comparison, quotient, root and integral. Alternating series. Uniform convergence of series of functions. Power series. Radius of convergence. Integration and differentiation of power series. Taylor's formula with integral remainder. Taylor series and analytic functions. Applications of power series (approximation, differential equations). The complex exponential function.
Mandatory literature
Lax, Peter D. and Terrell, Maria Shea; Calculus With Applications, Springer, 2014. ISBN: 978-1-4614-7946-8
Complementary Bibliography
Marsden Jerrold;
Calculus i. 2nd ed. ISBN: 0-387-90974-5
Marsden Jerrold;
Calculus ii. 2nd ed. ISBN: 0-387-90975-3
Stewart James 1941-;
Calculus. ISBN: 978-1-305-27237-8
Lima Elon Lages;
Analise real. vol. 1. 3ª ed. ISBN: 85-244-0048-X
Teaching methods and learning activities
Participation in classes and independent study.
keywords
Physical sciences > Mathematics > Mathematical analysis
Evaluation Type
Evaluation with final exam
Assessment Components
designation |
Weight (%) |
Exame |
100,00 |
Total: |
100,00 |
Amount of time allocated to each course unit
designation |
Time (hours) |
Estudo autónomo |
159,00 |
Frequência das aulas |
84,00 |
Total: |
243,00 |
Eligibility for exams
Attendance is not compulsory.
Calculation formula of final grade
Final exam: 100%
Special assessment (TE, DA, ...)
By oral and/or written exam.
Classification improvement
By exam.