Real Analysis I

Code:

M1017

Acronym:

M1017

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2017/2018 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=2061
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:F	79	Official Study Plan	1	-	9	84	243
			3
MI:EF	82	study plan from 2017/18	1	-	9	84	243

Teaching language

Suitable for English-speaking students

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 concepts and basic techniques of differential and integral calculus for real functions of a real variable;

- understand and be apply 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

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
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. Students should participate actively in classes.

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	80,00
Teste	20,00
Total:	100,00

Calculation formula of final grade

The assesment has the following components:

(a) 2 minitests during the semester with a total weight of 20%.

(b) Final exam with a weight of 80%. In the final exam students may also opt to answer groups of questions corresponding to each minitest with corresponding weight in the calculation of the final mark.

In order to obtain a final mark above 17, the student may be required to take a supplementary oral or written exam.

Special assessment (TE, DA, ...)

By oral and/or written exam. The marks of the minitests cannot be used for this purpose.

Classification improvement

By exam. The marks of the minitests cannot be used for this purpose.