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More About This Textbook
Overview
Taking a fresh approach while retaining classic presentation, the Tan Calculus series utilizes a clear, concise writing style, and uses relevant, real world examples to introduce abstract mathematical concepts with an intuitive approach. In keeping with this emphasis on conceptual understanding, each exercise set in the three semester Calculus text begins with concept questions and each endofchapter review section includes fillintheblank questions which are useful for mastering the definitions and theorems in each chapter. Additionally, many questions asking for the interpretation of graphical, numerical, and algebraic results are included among both the examples and the exercise sets. The Tan Calculus three semester text encourages a real world, application based, intuitive understanding of Calculus without comprising the mathematical rigor that is necessary in a Calculus text.
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Meet the Author
Soo T. Tan has published numerous papers in Optimal Control Theory and Numerical Analysis. He received his S.B. degree from Massachusetts Institute of Technology, his M.S. degree from the University of WisconsinMadison, and his Ph.D. from the University of California at Los Angeles. "One of the most important lessons I learned from my early experience teaching these courses is that many of the students come into these courses with some degree of apprehension. This awareness led to the intuitive approach I have adopted in all of my texts."
Table of Contents
0. PRELIMINARIES. Lines. Functions and Their Graphs. The Trigonometric Functions. Combining Functions. Graphing Calculators and Computers. Mathematical Models. Chapter Review. 1. LIMITS. An Intuitive Introduction to Limits. Techniques for Finding Limits. A Precise Definition of a Limit. Continuous Functions. Tangent Lines and Rates of Change. Chapter Review. ProblemSolving Techniques. Challenge Problems. 2. THE DERIVATIVE. The Derivative. Basic Rules of Differentiation. The Product and Quotient Rules. The Role of the Derivative in the Real World. Derivatives of Trigonometric Functions. The Chain Rule. Implicit Differentiation. Related Rates. Differentials and Linear Approximations. Chapter Review. ProblemSolving Techniques. Challenge Problems. 3. APPLICATIONS OF THE DERIVATIVE. Extrema of Functions. The Mean Value Theorem. Increasing and Decreasing Functions and the First. Derivative Test. Concavity and Inflection Points. Limits Involving Infinity; Asymptotes. Curve Sketching. Optimization Problems. Newton's Method. Chapter Review. ProblemSolving Techniques. Challenge Problems. 4. INTEGRATION. Indefinite Integrals. Integration by Substitution. Area. The Definite Integral. The Fundamental Theorem of Calculus. Numerical Integration. Chapter Review. ProblemSolving Techniques. Challenge Problems. 5. APPLICATIONS OF THE DEFINITE INTEGRAL. Areas Between Curves. Volumes: Disks, Washers, and Cross Sections. Volumes Using Cylindrical Shells. Arc Length and Areas of Surfaces of Revolution. Work. Fluid Pressure and Force. Moments and Centers of Mass. Chapter Review. ProblemSolving Techniques. Challenge Problems. 6. THE TRANSCENDENTAL FUNCTIONS. The Natural Logarithmic Function. Inverse Functions. Exponential Functions. General Exponential and Logarithmic Functions. Inverse Trigonometric Functions. Hyperbolic Functions. Indeterminate Forms and L'HÃ´pital's Rule. Chapter Review. Challenge Problems. 7. TECHNIQUES OF INTEGRATION. Integration by Parts. Trigonometric Integrals. Trigonometric Substitutions. The Method of Partial Fractions. Integration Using Tables of Integrals and CAS. Improper Integrals. Chapter Review. ProblemSolving Techniques. Challenge Problems. 8. DIFFERENTIAL EQUATIONS. Differential Equations: Separable Equations. Direction Fields and Euler's Method. The Logistic Equation. FirstOrder Linear Differential Equations. Chapter Review. Challenge Problems. 9. INFINITE SEQUENCES AND SERIES. Sequences. Series. The Integral Test. The Comparison Tests. Alternating Series. Absolute Convergence; The Ratio and Root Tests. Power Series. Taylor and Maclaurin Series. Approximation by Taylor Polynomials. Chapter Review. ProblemSolving Techniques. Challenge Problems. 10. CONIC SECTIONS, PARAMETRIC EQUATIONS, AND POLAR COORDINATES. Conic Sections. Plane Curves and Parametric Equations. The Calculus of Parametric Equations. Polar Coordinates. Areas and Arc Lengths in Polar Coordinates. Conic Sections in Polar Coordinates. Chapter Review. Challenge Problems. 11. VECTORS AND THE GEOMETRY OF SPACE. Vectors in the Plane. Coordinate Systems and Vectors in ThreeSpace. The Dot Product. The Cross Product. Lines and Planes in Space. Surfaces in Space. Cylindrical and Spherical Coordinates. Chapter Review. Challenge Problems. 12. VECTORVALUED FUNCTIONS. VectorValued Functions and Space Curves. Differentiation and Integration of Vector Valued. Functions. Arc Length and Curvature. Velocity and Acceleration. Tangential and Normal Components of Acceleration. Chapter Review. Challenge Problems. 13. FUNCTIONS OF SEVERAL VARIABLES. Functions of Two or More Variables. Limits and Continuity. Partial Derivatives. Differentials. The Chain Rule. Directional Derivatives and Gradient Vectors. Tangent Planes and Normal Lines. Extrema of Functions of Two Variables. Lagrange Multipliers. Chapter Review. Challenge Problems. 14. MULTIPLE INTEGRAL.S Double Integrals. Iterated Integrals. Double Integrals in Polar Coordinates. Applications of Double Integrals. Surface Area. Triple Integrals. Triple Integrals in Cylindrical and Spherical Coordinates. Change of Variables in Multiple Integrals. Chapter Review. Challenge Problems. 15. VECTOR ANALYSIS. Vector Fields. Divergence and Curl. Line Integrals. Independence of Path and Conservative Vector Fields. Green's Theorem. Parametric Surfaces. Surface Integrals. The Divergence Theorem. Stoke's Theorem. Chapter Review. Challenge Problems. APPENDICES. A. The Real Number Line, Inequalities, and Absolute Value. B. Proofs of Selected Theorems.