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). Thomas' Calculus:Early Transcendentals. 11th Edition: Pearson Education, Limited.
Stewart, J. Calculus Early Transcendentals. 8th edition. Belmont, CA: Brooks Cole.
Apostol, T. (1969). Calculus VOL. 1, 2d Ed. John Wiley and Sons..
Spivak, M. (2006). Calculus. 4th Ed, Cambridge University Press.
