Teaching GuideTerm
Higher Technical University College of Civil Engineering
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Grao en Tecnoloxía da Enxeñaría Civil
 Subjects
  Cálculo infinitesimal I
   Contents
Topic Sub-topic
I. THE REAL NUMBER. 1. Introduction. Necessary and sufficient condition.
2. Successive extensions of the concept of number.
3. Field structure. Ordered field.
4. Sequences in Q.
5. Properties of Q.
6. Extension of Q. The real numbers.
7. Properties of R.
8. Operations in R.
II. METRIC SPACES. 1. Distance.
2. Balls and neighborhoods
3. Notable points in a metric space.
4. Notable sets in a metric space.
5. Closed, open and compact sets.
6. The metric space (R,||).
III. NUMERICAL SEQUENCES. 1. Definition; limit concept; types of successions.
2. Properties of limits.
3. Monotone sequences.
4. Operations with limits. Types of indeterminacy.
5. Convergence criteria.
6. Infinites and infinitesimals.
7. Equivalent sequences.
8. Substitution by equivalent sequences.
9. Calculation of limits.
IV. FUNCTIONS IN R. A. GENERAL NOTIONS
1. Concept of function.
2. Operations with functions.
3. Types of functions.

B. FUNCTION LIMITS
1. Limit of a function.
2. One-side limits.
3. Extension of the concept of limit.
4. Sequential limit.
5. Properties of limits.
6. Operations with limits.
7. Infinites and infinitesimals.
8. Equivalent functions at a point.
9. Substitution by equivalent functions.

C. CONTINUITY OF FUNCTIONS
1. Continuous function.
2. One-sided continuity.
3. Discontinuities.
4. Operations with continuous functions.
5. Continuity of the elementary functions.
6. Composition of continuous functions.
7. Theorems of the continuous functions.
8. Uniform continuity.

D. DIFFERENTIABILITY OF FUNCTIONS
1. Derivability and differentiability.
2. The chain rule. Applications.
3. Derivative of the inverse function.
4. Mean value theorems: Rolle, Cauchy, Lagrange.
5. The derivative as a limit of derivatives.
6. L'Hôpital rules.
7. Successive derivatives.
8. Taylor and McLaurin polynomials.
9. Representation of curves.
V. CALCULUS OF PRIMITIVES. 1. Logarithms and hyperbolic functions.
2. Primitive of a function. immediate integrals.
3. Primitive calculation methods: semi-immediate; change of variables; parts; reduction formulas; rational; trigonometric; irrational.
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