**What are you going to learn?**

Evaluate the limit of a function, including one-sided and two-sided, using numerical and algebraic techniques and the properties of limits. Determine whether a function is continuous or discontinuous at a point. Calculate the derivative of an algebraic function using the formal definition of the derivative. Find derivatives of functions using the constant, power, sum, difference, product, quotient, and chain rules, and using implicit differentiation. Determine maxima and minima in optimization problems using the derivative. Sketch the graph of functions using the first and second derivatives. Apply the concept of derivative to solve applied problems.

**Content**

**Chapter 1. ****Basic Concepts of Sets Theory**

Introduction

Notation

Subsets

Operations

Neghborhood Apaces and Fuction Spaces

Introduction

Integers and Rational Numbers

The Field Axioms

The Order Axioms

**Chapter 2. ****The Real Numbers System**

**Chapter 3. Functions**

Representation of Functions

Mathematical Models

Transformation of Functions

Exponential Functions

Inverse Functions and Logarithms

**Chapter 4. ****Limits**

Tangent and Velocity Problems

The Limit of a Function

Calculation of Limits

The Precise Definition of a Limit

Limits at Infinity

**Chapter 5. ****Continuity**

Definition

Cauchy Characterzation of Continuity

Continuity of Elementary Functions

General Properties of Continuous Functions

Continuity of Inverse Functions

**Chapter 6. ****Derivatives and the Differentiation Rules**

Derivatives of First Order. Definition

Differentiation of Elementary Functions

Differentiation of Inverse Functions

Differentiation of Composite Functions

**Chapter 7. ****Applications of Differentiation**

Extrema of Functions. THe Rolle Theorem

Lagrange and Cauchy Theorems

Geometrical Interpretation of the Derivative Sign

Indeterminate Expressions

Asymptotes

The Concept of Differential

**Chapter 8. ****Derivatives of Higher Order**

Definitions and Examples

Differentials of Higher Order

Taylor's Formula

Expansions in Power Series

The Concept of Differential of Higher Order

Criterium of Extrema

Geometrical Interpretation of the Second Derivative

### Bibliography

Thomas, G. et al (2005).

*T**homas' Calculus:Early Transcendentals**.*11th Edition: Pearson Education, Limited.Stewart, J.

. 8th edition. Belmont, CA: Brooks Cole.*Calculus Early Transcendentals*Apostol, T. (1969).

*Calculus VOL.**1*, 2d Ed. John Wiley and Sons..Spivak, M. (2006).

. 4th Ed, Cambridge University Press.*Calculus*