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//>4882B1, 0130488224, Ewen, Dale, Gary, Joan S., Trefzger, James E., Technical Mathematics with Calculus, 2/E//> This book provides readers with necessary mathematics skills, including practical calculus. Mathematics provides the essential framework for and is the basic language of all the technologies. Mathematical, problemsolving, and critical thinking skills are crucial to understanding the changing face of technology. It presents the following major areas: fundamental concepts and measurement; fundamental algebraic concepts; exponential and logarithmic functions; righttriangle trigonometry; the trigonometric functions with formulas and identities; complex numbers; matrices; polynomial and rational functions; basic statistics; analytic geometry; differential and integral calculus with applications; partial derivatives and double integrals; series; and differential equations. An excellent learning aid and resource tool for engineers, especially computer software, hardware, and peripheral manufacturers. Its comprehensive appendices make this an excellent desktop reference.
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Preface
Technical Mathematics with Calculus provides the necessary comprehensive mathematics skills for students in an engineering technology program that requires a development of practical calculus.
The text presents the following major areas: fundamental concepts and measurement; fundamental algebraic concepts; exponential and logarithmic functions; righttriangle trigonometry, the trigonometric functions, and trigonometric formulas and identities; complex numbers; matrices; polynomial and rational functions; statistics for process control; analytic geometry; differential and integral calculus with applications; partial derivatives and double integrals; series; and differential equations.
KEY FEATURES
Illustration of Some Key Features
Examples: Since many students learn by example, a large number of detailed and wellillustrated examples are used throughout the text.
Exercises: To reinforce key concepts for students, we have provided a large variety of wellillustrated exercises.
Chapter End Matter: A chapter summary and a chapter review are provided at the end of each chapter to review concept understanding and to help students review for quizzes and examinations.
Calculator Story Boards: Calculator story boards, including screens, are used to show students the sequence of the stepbystep operations.
Illustrations and Boxes are abundantly and effectively used to highlight important principles.
TO THE FACULTY
The topics have been arranged with the assistance of faculty who teach in a variety of technical programs. However, we have also allowed for many other compatible arrangements. The topics are presented in an intuitive manner, with technical applications integrated throughout whenever possible. The large number of detailed examples and exercises is a feature that students and faculty alike find essential.
The text is written at a language level and a mathematics level that are cognizant of and beneficial to most students in technical programs. The students are assumed to have a mathematics background that includes one year of high school algebra or its equivalent and some geometry. The introductory chapters are written so that students who are deficient in some topics may also be successful. The material in this book should be completed in three or four semesters or equivalent and serves as a foundation for more advanced work in mathematics. This text is intended for use in Associate Degree programs as well as ABET (Accrediting Board for Engineering Technology) programs and BIT (Bachelor of Industrial Technology) programs.
Chapters 1 and 2 provide the basic skills that are needed early in almost any technical program. Chapters 3 through 8 complete the basic algebraic foundation, and Chapters 9 through 13 include the trigonometry necessary for the technologies. Chapters 14 through 17 include some advanced topics needed for some programs. Chapter 18 addresses the basics of statistics for process control. Chapter 19 (analytic geometry) completes a comprehensive mathematics background needed in many programs; some programs include this chapter at the end of the first year while other programs include this chapter at the beginning of the introductory calculus. Chapters 20 through 22 present intuitive discussions about the limit and develop basic techniques and applications of differentiation. Chapters 23 through 25 develop basic integration concepts, some appropriate applications, and more complicated methods of integration. Chapter 26 presents partial derivatives and double integrals. Chapters 27 through 29 provide an introduction to series and differential equations with technical applications.
We have included Appendix C on the basic graphing calculator and Appendix D on the advanced calculator so that faculty have the option of which, if any, graphing calculator is used in their course. Some graphing calculator uses are integrated into some of the examples in the text.
A companion Instructor's Manual with solutions for selected oddnumbered exercises, answers for evennumbered exercises, and sample chapter tests and answers is available.
TO THE STUDENT
Mathematics provides the essential framework for and is the basic language of all the technologies. With this basic understanding of mathematics, you will be able to quickly understand your chosen field of study and then be able to independently pursue your own lifelong education. Without this basic understanding, you will likely struggle and often feel frustrated not only in your mathematics and support sciences courses but also in your technical courses.
Technology and the world of work will continue to change rapidly. Your own working career will likely change several times during your working lifetime. Mathematical, problemsolving, and criticalthinking skills will be crucial as opportunities develop in your own career path in a rapidly changing world.
ACKNOWLEDGMENTS
We extend our sincere and special thanks to our reviewers: Joe Jordan, John Tyler Community College (VA); Maureen Kelly, North Essex Community College (MA); Carol A. McVey, FlorenceDarlington Technical College (SC); John D. Meese, DeVry Institute of Technology (OH); Kenneth G. Merkel, Ph.D., PE, University of NebraskaLincoln; Susan L. Miertschin, University of Houston; and Pat Velicky, FlorenceDarlington Technical College (SC). We would also like to express thanks to our Prentice Hall editor, Stephen Helba; to our Prentice Hall associate editor, Michelle Churma; to our production editor, Louise Sette; to Kirsten Kauffman (York Production Services); and to Joyce Ewen for her superb proofing assistance.
If anyone wishes to correspond with us regarding suggestions, criticisms, questions, or errors, please contact Dale Ewen directly at Parkland Community College, 2400 W. Bradley, Champaign, IL 61821, or through Prentice Hall.
Dale Ewen
Joan S. Gary
James E. Trefzger
Table of Contents
1. Fundamental Concepts.
The Real Number System. Zero and Order of Operations. Scientific Notation and Powers of 10. Measurement. Operations with Measurements. Algebraic Expressions. Exponents and Radicals. Multiplication of Algebraic Expressions. Division of Algebraic Expressions. Linear Equations. Formulas. Substitution of Data into Formulas. Applications Involving Linear Equations. Ratio and Proportion. Variation.
2. Review of Geometry.
Angles and Lines. Triangles. Quadrilaterals. Circles. Geometric Solids: Areas and Volumes.
3. RightTriangle Trigonometry.
The Trigonometric Ratios. Values of the Trigonometric Ratios. Solving Right Triangles. Applications of the Right Triangle.
4. Equations and Their Graphs.
Functions. Graphing Equations. The Straight Line. Parallel and Perpendicular Lines. The Distance and Midpoint Formulas.
5. Factoring and Algebraic Fractions.
Special Products. Factoring Algebraic Expressions. Other Forms of Factoring. Equivalent Fractions. Multiplication and Division of Algebraic Fractions. Addition and Subtraction of Algebraic Fractions. Complex Fractions. Equations with Fractions.
6. Systems of Linear Equations.
Solving a System of Two Linear Equations. Other Systems of Equations. Solving a System of Three Linear Equations. Determinants. Properties of Determinants. Solving a System of Linear Equations Using Determinants. Partial Fractions.
7. Quadratic Equations.
Solving Quadratic Equations by Factoring. Solving Quadratic Equations by Completing the Square. The Quadratic Formula. Applications.
8. Exponents and Radicals.
Integral Exponents. Fractional Exponents. Simplest Radical Form. Addition and Subtraction of Radicals. Multiplication and Division of Radicals. Equations with Radicals. Equations in Quadratic Form.
9. Exponentials and Logarithms.
The Exponential Function. The Logarithm. Properties of Logarithms. Common Logarithms. Natural Logarithms. Solving Exponential Equations. Solving Logarithmic Equations. Applications: Solving Exponential and Logarithmic Equations. Data Along a Straight Line.
10. Trigonometric Functions.
The Trigonometric Functions. Trigonometric Functions of Any Angle. Radian Measure. Use of Radian Measure.
11. Oblique Triangles and Vectors.
Law of Sines. The Ambiguous Case. Law of Cosines. Applications of Oblique Triangles. Addition of Vectors: Graphical Methods. Addition of Vectors: Trigonometric Methods. Vector Components. Vector Applications.
12. Graphing the Trigonometric Functions.
Graphing the Sine and Cosine Functions. Phase Shift. Graphing the Other Trigonometric Functions. Graphing Composite Curves. Simple Harmonic Motion.
13. Trigonometric Formulas and Identities.
Basic Trigonometric Identities. Formulas for the Sum and the Difference of Two Angles. Double and HalfAngle Formulas. Trigonometric Equations. Inverse Trigonometric Functions.
14. Complex Numbers.
Complex Numbers in Rectangular Form. Trigonometric and Exponential Forms of Complex Numbers. Multiplication and Division of Complex Numbers in Exponential and Trigonometric Forms. Powers and Roots.
15. Matrices.
Basic Operations. Multiplication of Matrices. Finding the Inverse of a Matrix. Solving a System of Equations by a Matrix Method.
16. Polynomials of Higher Degree.
Polynomial Functions. Real Solutions of Polynomial Equations. Complex Solutions of Polynomial Equations.
17. Inequalities and Absolute Value.
Inequalities. Equations and Inequalities Involving Absolute Value. Other Types of Inequalities. Inequalities in Two Variables.
18. Progressions and the Binomial Theorem.
Arithmetic Progressions. Geometric Progressions. The Binomial Theorem.
19. Basic Statistics.
Graphic Presentation of Data. Measures of Central Tendency. Measures of Dispersion. The Normal Distribution. Fitting Curves to Data Sets. Statistical Process Control.
20. Analytic Geometry.
The Circle. The Parabola. The Ellipse. The Hyperbola. Translation of Axes. The General SecondDegree Equation. Systems of Quadratic Equations. Polar Coordinates. Graphs in Polar Coordinates.
21. The Derivative.
Motion. The Limit. The Slope of a Tangent Line to a Curve. The Derivative. Differentiation of Polynomials. Derivatives of Products and Quotients. The Derivative of a Power. Implicit Differentiation. Proofs of Derivative Formulas. Higher Derivatives.
22. Applications of the Derivative.
Curve Sketching. Using Derivatives in Curve Sketching. More on Curve Sketching. Newton's Method for Improving Estimated Solutions. Maximum and Minimum Problems. Related Rates. Differentials and Linear Approximations.
23. Derivatives of Transcendental Functions.
Derivatives of Sine and Cosine Functions. Derivatives of Other Trigonometric Functions. Derivatives of Inverse Trigonometric Functions. Derivatives of Logarithmic Functions. Derivatives of Exponential Functions. L'Hospital's Rule. Applications.
24. The Integral.
The Indefinite Integral. The Constant of Integration. Area Under a Curve. The Definite Integral.
25. Applications of Integrations.
Area Between Curves. Volumes of Revolution: Disk Method. Volumes of Revolution: Shell Method. Center of Mass of a System of Particles. Center of Mass of Continuous Mass Distributions. Moments of Inertia. Work, Fluid Pressure, and Average Value.
26. Methods of Integration.
The General Power Formula. Logarithmic and Exponential Forms. Basic Trigonometric Forms. Other Trigonometric Forms. Inverse Trigonometric Forms. Integration Using Partial Fractions. Integration by Parts. Integration Using Tables. Integration by Trigonometric Substitution. Integration Using Tables. Numerical Methods of Integration. Areas in Polar Coordinates. Improper Integrals.
27. ThreeSpace: Partial Derivatives and Double Integrals.
Functions in ThreeSpace. Partial Derivatives. Applications in Partial Derivatives. Double Integrals.
28. Series.
Series and Convergence. Ratio and Integral Tests. Alternating Series and Conditional Convergence. Power Series. Maclaurin Series. Operations with Series. Taylor Series. Computational Approximations. Fourier Series.
29. FirstOrder Differential Equations.
Solving Differential Equations. Separation of Variables. Use of Exact Differentials. Linear Equations of First Order. Applications of FirstOrder Differential Equations.
30. SecondOrder Differential Equations.
HigherOrder Homogenous Differential Equations. Repeated Roots and Complex Roots. Nonhomogenous Equations. Applications of SecondOrder Differential Equations. The Laplace Transform. Solutions by Method of Laplace Transforms.
Appendix A: Tables.
U.S. Weights and Measures. Conversion Tables. Physical Quantities and their Units.
Appendix B: The Metric System.
Introduction. Length. Mass. Volume and Area. Temperature, Time, Current, and Power. Other Conversions. Appendix B Review.
Appendix C: Using a Graphic Calculator.
Introduction to the Keyboard of the TI83 PLUS. Computational Examples. Graphing Features. Examples of Graphing. Trigonometric Functions and Polar Coordinates. Equation Solving and TABLE Features. The Numeric SOLVER. Matrix Features. LIST Features and Descriptive Statistics. The Line of Best Fit (Linear Regression). Calculus Features. Sequences and Series.
Appendix D: Using an Advanced Graphing Calculator.
Introduction to the TI89 Keyboard. Variables and Editing. The Home Screens Menus. The Keyboard Menus. Graphing Functions. Examples of Graphing. Trig Functions and Polar Coordinates. Numerical GRAPH and TABLE Features. Sequences and Series. The Numeric Solver. Matrix Features. The Data Editor and Descriptive Statistics. The Line of Best Fit (Linear Regression). Symbolic Algebra Features. Basic Calculus Features. Graphing in 3D. Advanced Calculus Features.
Appendix E: Table of Integrals.
Answers to OddNumbered Exercises and Chapter Reviews.
Index.
Preface
Preface
Technical Mathematics with Calculus provides the necessary comprehensive mathematics skills for students in an engineering technology program that requires a development of practical calculus.
The text presents the following major areas: fundamental concepts and measurement; fundamental algebraic concepts; exponential and logarithmic functions; righttriangle trigonometry, the trigonometric functions, and trigonometric formulas and identities; complex numbers; matrices; polynomial and rational functions; statistics for process control; analytic geometry; differential and integral calculus with applications; partial derivatives and double integrals; series; and differential equations.
KEY FEATURES
Illustration of Some Key Features
Examples: Since many students learn by example, a large number of detailed and wellillustrated examples are used throughout the text.
Exercises: To reinforce key concepts for students, we have provided a large variety of wellillustrated exercises.
Chapter End Matter: A chapter summary and a chapter review are provided at the end of each chapter to review concept understanding and to help students review for quizzes and examinations.
Calculator Story Boards: Calculator story boards, including screens, are used to show students the sequence of the stepbystep operations.
Illustrations and Boxes are abundantly and effectively used to highlight important principles.
TO THE FACULTY
The topics have been arranged with the assistance of faculty who teach in a variety of technical programs. However, we have also allowed for many other compatible arrangements. The topics are presented in an intuitive manner, with technical applications integrated throughout whenever possible. The large number of detailed examples and exercises is a feature that students and faculty alike find essential.
The text is written at a language level and a mathematics level that are cognizant of and beneficial to most students in technical programs. The students are assumed to have a mathematics background that includes one year of high school algebra or its equivalent and some geometry. The introductory chapters are written so that students who are deficient in some topics may also be successful. The material in this book should be completed in three or four semesters or equivalent and serves as a foundation for more advanced work in mathematics. This text is intended for use in Associate Degree programs as well as ABET (Accrediting Board for Engineering Technology) programs and BIT (Bachelor of Industrial Technology) programs.
Chapters 1 and 2 provide the basic skills that are needed early in almost any technical program. Chapters 3 through 8 complete the basic algebraic foundation, and Chapters 9 through 13 include the trigonometry necessary for the technologies. Chapters 14 through 17 include some advanced topics needed for some programs. Chapter 18 addresses the basics of statistics for process control. Chapter 19 (analytic geometry) completes a comprehensive mathematics background needed in many programs; some programs include this chapter at the end of the first year while other programs include this chapter at the beginning of the introductory calculus. Chapters 20 through 22 present intuitive discussions about the limit and develop basic techniques and applications of differentiation. Chapters 23 through 25 develop basic integration concepts, some appropriate applications, and more complicated methods of integration. Chapter 26 presents partial derivatives and double integrals. Chapters 27 through 29 provide an introduction to series and differential equations with technical applications.
We have included Appendix C on the basic graphing calculator and Appendix D on the advanced calculator so that faculty have the option of which, if any, graphing calculator is used in their course. Some graphing calculator uses are integrated into some of the examples in the text.
A companion Instructor's Manual with solutions for selected oddnumbered exercises, answers for evennumbered exercises, and sample chapter tests and answers is available.
TO THE STUDENT
Mathematics provides the essential framework for and is the basic language of all the technologies. With this basic understanding of mathematics, you will be able to quickly understand your chosen field of study and then be able to independently pursue your own lifelong education. Without this basic understanding, you will likely struggle and often feel frustrated not only in your mathematics and support sciences courses but also in your technical courses.
Technology and the world of work will continue to change rapidly. Your own working career will likely change several times during your working lifetime. Mathematical, problemsolving, and criticalthinking skills will be crucial as opportunities develop in your own career path in a rapidly changing world.
ACKNOWLEDGMENTS
We extend our sincere and special thanks to our reviewers: Joe Jordan, John Tyler Community College (VA); Maureen Kelly, North Essex Community College (MA); Carol A. McVey, FlorenceDarlington Technical College (SC); John D. Meese, DeVry Institute of Technology (OH); Kenneth G. Merkel, Ph.D., PE, University of NebraskaLincoln; Susan L. Miertschin, University of Houston; and Pat Velicky, FlorenceDarlington Technical College (SC). We would also like to express thanks to our Prentice Hall editor, Stephen Helba; to our Prentice Hall associate editor, Michelle Churma; to our production editor, Louise Sette; to Kirsten Kauffman (York Production Services); and to Joyce Ewen for her superb proofing assistance.
If anyone wishes to correspond with us regarding suggestions, criticisms, questions, or errors, please contact Dale Ewen directly at Parkland Community College, 2400 W. Bradley, Champaign, IL 61821, or through Prentice Hall.
Dale Ewen
Joan S. Gary
James E. Trefzger