DIFFERENTIAL CALCULUS: EARLY TRASCENDENTALS

Our purpose in this edition of Differential Calculus Early Transcendental is to help the arriving students to the university to face up with exit the topics of calculus. To accomplish this goal, the precalculus material that appeared in the appendixes of the previous edition, was enlarged and presented in the three first chapters of this edition. The third chapter is entirely dedicated to the conics, which are developed thoroughly.
The three new chapters were written following the same scheme we use in our texts, balancing the theory and the practice. The theory is illustrated by many examples. A part of each section is dedicated to present, with all the details, solved problems. Most of the theorems are presented with his corresponding proofs. When the proof is complicated, this is presented as a solved problem.

CONTENT:

Chapter 1.PRELIMINARIES
PYTHAGORAS OF SAMOS
A short introduction to Logic and Set Theory
The Real Number System. Field Axioms
Radicals and Rational Exponents
Some Topics of Algebra
Polinomial Equations
Order Axioms for Real Numbers
Inequations
Absolute Value

Chapter 2.THE CARTESIAN PLANE AND THE LINE
REN DESCARTES
The Cartesian Plane
Graphs of Equations of two Variables
Symmetry and Translation Criterios
The line and the First Degree Equation


Chapter 3.THE CONIC
APOLONIO OF PERGA
Introduction
The Parabola
The Ellipse
La Hip rbola
The General Second Degree Equation
Rotation of Axes

Chapter 4.REAL FUNCTIONS
ARCHIMEDES
Real Functions and their Graph
Trigonometric Functions
New functions from old functions
Inverse Functions
Inverse Trigonometric Functions
Exponential Functions
Logarithmic Functions
Applications of the Exponential and Logarithmic Functions.
A Brief history of the Bernoulli family

Chapter 5.LIMITS AND CONTINUITY
Leonhard Euler
Limits, an Informal Introduction
Limits, a Rigorous Treatment
Trigonometric Limits
Continuity
Infinite Limits and Verticals Asymptotes
Limits at Infinite and Horizontal Asymptotes
The Number e as a limit
Oblique Asymptotes


Chapter 6.THE DERIVATIVE
ISAAC NEWTON
The Derivative
Basic Techniques of Differentiation
Derivatives of the Trigonometric Functions
Derivatives of the Exponential and Logarithmic
Functions
The Chain Rule

Chapter 7.MORE ON DERIVATIVES
GOTTFRIED WILHELD LEIBNIZ
Implicit Differentiation and the Derivative of an Inverse Function
Logarithmic Differentiation
Derivatives of Inverse Trigonometric Functions
Higher-order Derivatives, Velocity and Acceleration
Hyperbolic and Inverse Hyperbolic Functions
Rate of Change
Linear Approximations and Differentials
Nicholas Bourbaki: The mysterious story of a brilliant Mathematician that never existed

Chapter 8.APLICATIONS OF THE DERIVATIVE
GUILLAUME F. A. M. DE L HOSPITAL
Maximum and Minimum Values
The Mean Value Theorem
Monotony, Concavity and Derivative test for extrema
Indeterminate Forms. L H spital s Rule
Using Derivatives to Sketch the Graph of a Function
Optimization Problems
Newton-Raphson Method