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This book is intended to introduce calculus through examples to prepare the reader for the diverse problems he/she will have to solve using derivatives and integrals. The derivative is introduced right at the beginning of the book as the slope of a tangent. The reader is then exposed to a cross-section of problems that may be solved by knowing how to calculate slopes of tangents. College students have to master techniques of diffrentiation within a semester. The early introduction of derivatives gives them ample time to practice these techniques. The integral is introduced as a limit of sums with its applications to physical problems in mind. A number of problems that may be reduced to limits of sums are then addressed. Techniques of integration are also addressed as are methods of approximating definite integrals. Limits, which are the foundations of calculus are treated rigorously and most of the statements used in the course are proved. Finally, detailed solutions, (not just numerical answers), to about half the problems in the book are given. In particular, the solutions to all "Test - your - self" exercises are given. These are problems the student should attempt to do in the indicated time to test his/her mastery of some fundamental topics.
Es un libro único en su género, ya que a través del desarrollo de metodologías facilita el aprendizaje del cálculo integral, útil para alumnos que emprenden una carrera profesional. En su contenido contempla los temas: Diferenciales, la integral indefinida, técnicas de integración indefinida. la integral definida, integrales, impropias y aplicaciones de la integral.
Para el desarrollo del mismo se han instrumentado supuestos teóricos y técnicos del aprendizaje, tales como: Teoría tornado; teorías biogenéticas, teoría de los aprendizajes equiparables, teoría del bao cognitivo, técnica de los aprendizajes por justificandos y técnica de los aprendizajes por agrupamiento; todas ellas atendiendo las corrientes pedagógicas constructivistas.
También es un apoyo para el maestro ya que posee los lineamientos pedagógicos y anexa la instrumentación didáctica del curso, de tal forma que en todo momento sabe: que, como, cuando y con qué se imparte la clase, haciendo posible la programación del curso en el semestre y el control de avance programático del mismo. Además prevé los supuestos pedagógicos necesarios, así tenemos: Exposiciones globalizadas, crear confianza en los alumnos, educación en valores, tener conocimiento y control de los alumnos e identificación del grupo, inhibir la copia y recomendaciones a los alumnos.
The purpose of this book is to help students in calculus1 get a good practice for the midterms and final exams during their school year in calculus1. All the finals and midterms are real exams (with little changes) from several Universities around America ( USA, Canada, Puerto Rico, Mexico).
We believe to get a good grade in the midterms and final, the student should after reviewing his/her homework and notes pick some real midterms and final and do them.
We tried this idea with several students and it works very well.
We wish you all success in your studies.
Muslim Mathematical Society and Salah Abdel Hamid
The book begins with a discussion of finding the equation of a curve that passes through a fixed number of points. This technique is called curve fitting. This concept is then generalized to finding the equation of a curve that passes through an infinite number of points. This concept is in turn generalized to finding the equation of a curve that passes through a given curve over a specific interval. Such an equation and its curve that can be found to pass through the given curve over a particular interval is called a Fourier series. A Fourier series consists of an infinite number of sine and cosine terms added together in an infinite series whose coefficients must be determined. Such an infinite series of terms can be made to approximate the curve of an elementary function over a particular interval arbitrarily close. Also, since the terms of the Fourier series are Trigonometric terms the curve of their sum namely the Fourier series is periodic with a definite period. This brings us to the concepts of the periodic function. The curve of such a function keeps repeating itself over the same intervals. Numerous examples are provided for finding the Fourier series of various elementary functions over given intervals. The following concepts are also discussed-functions in general form, composite functions, and odd and even functions. The Fourier series of each of these types of functions is also found. Next. The equations for the Fourier series and its coefients are generalized to the complex number system. This allows is to derive the Fourier transform. Everything is logically derived with all of the steps included.
The book starts with the definition of the Laplace Transform and uses it to derive the Laplace Transforms of the elementary functions including; constant functions, polynomial functions, exponential functions, trigonometric functions, and hyperbolic functions. All steps in the derivations of the Laplace Transforms for these functions are included.
The concept of the Inverse Laplace Transform is then logically developed from the concept of the Laplace Transform. Numerous examples are provided for finding the Laplace Transforms of the various types of elementary functions and finding their corresponding Inverse Laplace Transforms.
The Product Rule is derived with all steps included. Also, the Laplace Transform of a derivative is derived with all steps included.
Finally, the following types of differential equations and their initial value problems are solved using both conventional methods and the Laplace Transform method:
•First-order homogenous linear differential equations with constant coefficients
•First-order non-homogenous linear differential equations with constant coefficients
•Second-order homogenous linear differential equations with constant coefficients
•Second-order non-homogenous linear differential equations with constant coefficients
This textbook is unique.
It provides all the required material for successful completion of AP Calculus AB even for a student working alone. Simple explanations of complex ideas are followed by a variety of worked examples. Each chapter contains many worksheets offering a very wide selection of problems; multiple-choice, calculator, non-calculator, practical, theoretical, graphical. Answers appear directly after each worksheet. There are far more problems than would normally be assigned and hence the textbook is quite enough on its own. Many of the problems are original accumulated over decades from actual classroom settings. The excellent exam results of our students, while using this book, are a testament to its value.
Roger Allen and Jack Koenka
Contact Jack Koenka at email@example.com or Roger Allen at firstname.lastname@example.org
Mathematical Methods for Partial Differential Equations is an introduction in the use of various mathematical methods needed for solving linear partial differential equations. The material is suitable for a two semester course in partial differential equations for mathematicians, engineers, physicists, chemistry and science majors and is suitable for upper level college undergraduates or beginning graduate students.
Chapter one reviews necessary background material from the subject area of ordinary differential equations and then develops solution techniques for some easy to solve partial differential equations. Chapter two introduces orthogonal functions and Sturm-Liouville systems. Chapter three utilizes orthogonal functions to develop Fourier series and Fourier integrals. The fourth, fifth and sixth chapters consider various applied engineering applications of partial differential equations. Selected applied topics are developed together with necessary solution methods associated with parabolic, hyperbolic and elliptic type partial differential equations. Chapter seven introduces transform methods for solving linear partial differential equations. Numerous examples associated with the Laplace, Fourier exponential, Fourier sine, Fourier cosine and selected finite Sturm-Liouville transforms are given. Chapter eight introduces Green's functions for ordinary differential equations and chapter nine finishes with applications of Green function techniques for solving linear partial differential equations.
There are four Appendices. The Appendix A contains units of measurements from the Système International d'Unitès along with some selected physical constants. The Appendix B contains solutions to selected exercises. The Appendix C lists mathematicians whose research has contributed to the area of partial differential equations. The Appendix D contains a short listing of integrals. The text has numerous illustrative worked examples and over 340 exercises.
The most difficult part of calculus for many students is the study of methods of integration and the solution of indefinite integrals. Differentiation of functions is, on the whole, not difficult. This book contains the worked solutions to 200 integrals and may be used as a supplement to any standard calculus text.
An index to the integrands is provided in chapter 3, referring the reader directly to worked solutions in chapter 4. Although it is impossible to cover all cases, the integrals have been selected to cover a range of problems likely to be found in junior college and senior high school examination papers. Definite integrals are not included since the proper solution to a definite integral depends on the limits of integration and the presence of discontinuities.
The integrands include algebraic, trigonometric, logarithmic and hyperbolic functions.
This book may be used by both teachers and students. Teachers will be able to select examination problems of varying degrees of difficulty. When used by a student, an honest attempt should be made to solve the problem before consulting the index and worked solution. It is hoped the student will develop responsible habits in this regard.