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

1. Definitions and Examples

2. Differentials of Higher Order

3. Taylor’s Formula

4. Expansions in Power Series

5. The Concept of Differential of Higher Order

6. Criterium of Extrema

7. Geometrical Interpretation of the Second Derivative

Bibliography

1. Thomas, G. et al (2005). Thomas’ Calculus:Early Transcendentals. 11th Edition: Pearson Education, Limited.

2. Stewart, J. Calculus Early Transcendentals. 8th edition. Belmont, CA: Brooks Cole.

3. Apostol, T. (1969). Calculus VOL. 1, 2d Ed. John Wiley and Sons..

4. Spivak, M. (2006). Calculus. 4th Ed, Cambridge University Press.

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