Course: 2020/2021

Calculus I

(15526)

Competences and skills that will be acquired and learning results. Further information on this link

a. To understand the concept of real number and its implications, mainly the concept of limit.
b. To understand and manipulate series of real numbers.
c. To identify functions, their dependence on variables and their basic properties (monotony, parity, continuity, differentiability).
d. To master the basic operations of Calculus: limits, derivatives, integrals and Taylor expansions.
e. To interpret the derivative as rate of variation of a function, and the integral as an area.
f. To understand the Taylor polynomial as the best polynomial local approximation for a sufficiently smooth function, and to apply that approximation to simple cases.
g. To be able to graph simple functions.
h. To be able to solve simple optimization problems.

Description of contents: programme

Part I: Real Numbers and Functions
Chapter 1: The Real Line
1.1 Ordered Fields
1.2 Number Systems
1.3 Absolute value, bounds, and intervals
Chapter 4: Real Functions
2.1 Definition and basic concepts
2.2 Elementary functions
2.3 Operations with functions
Part II: Sequences and Series
Chapter 3: Sequences
3.1 Sequences of real numbers
3.2 Limit of a sequence
3.3 Number e
3.4 Indeterminacies
3.5 Asymptotic comparison of sequences
Chapter 4: Series
4.1 Series of real numbers
4.2 Series of nonnegative terms
4.3 Alternating series
4.4 Telescopic series
Part III: Differential Calculus
Chapter 5: Limit of a Function
5.1 Concept and definition
5.2 Algebraic properties
5.3 Asymptotic comparison of functions
Chapter 6: Continuity
6.1 Definition, properties, and continuity of elementary functions
6.2 Discontinuities
6.3 Continuous functions in closed intervals
Chapter 7: Derivatives
7.1 Concept and definition
7.2 Algebraic properties
7.3 Derivatives and local behaviour
Chapter 8: Taylor expansions
8.1 Asymptotic comparison of functions
8.2 Taylor¿s polynomial
8.3 Calculating limits
8.4 Remainder and Taylor¿s theorem
8.5 Taylor series
8.6 Numerical approximations
8.7 Local behaviour of functions
8.8 Function graphing
Part IV: Integral Calculus
Chapter 9: Primitives
9.1 Integration by parts
9.2 Primitives of rational functions
9.3 Change of variable
Chapter 10: Fundamental Theorem of Calculus
10.1 Riemann¿s integral
10.2 Properties of the integral
10.3 Riemann¿s sums
10.4 Fundamental theorem of calculus
Chapter 11: Geometric Applications of Integrals
11.1 Area of flat figures
11.2 Area of flat figures in polar coordinates
11.3 Volumes
11.4 Length of curves

Learning activities and methodology

The methodology will be the usual one for classes in the classroom, writing on the blackboard, with the occasional help of some resources on-line to illustrate some graphic or computational aspects of the course. Also, the classroom notes will be uploaded in Aula Global at the end of each chapter, along with the problem sheets that will be solved and discussed in the small groups.

Assessment System

- % end-of-term-examination 60
- % of continuous assessment (assigments, laboratory, practicals...) 40

Basic Bibliography

- J. Stewart. Single variable calculus: early transcendentals. Brooks-Cole . 1999
- R. Larson, R.P. Hostetler & B.H. Edwards. Calculus. Brooks-Cole. 2005
- S.L. Salas, G.J. Etgen & E. Hille. Calculus: One and Several Variables. Wiley. 2006

Additional Bibliography

- H. Anton, I.R.L. Bivens and S. Davis. Calculus: Early Transcendentals . Wiley. 2012
- T.M. Apostol. Calculus vol. 1. Wiley. 1991