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This book focuses on the requirements of a specific group of readers, structuring the book so that calculus is presented as a single subject rather than a collection of topics. With a user-friendly approach that keeps the reader in mind, the material is organized so that vector calculus is thoroughly covered. Approaches the theoretical aspects of calculus with the belief that, at the introductory level, it is important to understand the geometric basis for theorems and develop an intuitive understanding for the statements of the theorems and their implications. Emphasizes the power of calculus as a tool for modeling complex physical problems in order to present the methods of differentiation and integration as necessary skills needed to solve problems that arise from mathematical models. Excellent as a refresher for those in fields requiring a strong mathematical background.
1. N-Dimensional Space.
An Introduction to R n. Graphs in R 2 and R 3 and Their Equations. Algebra in R n. The Dot Product. Determinants, Areas and Volumes. Equations of Lines and Planes.
Functions. Functions and Graphing Technology. Functions from R into R n. The Wrapping Function and Other Functions. Sketching Parametrized Curves. Compositions of Functions. Building New Functions.
3. Limits, Continuity, and Derivatives.
Average Velocity and Average Rate of Change. Limits: An Intuitive Approach. Instantaneous Rate of Change: The Derivative. Linear Approximations of Functions and Newton's Method. More on Limits. Limits: A Formal Approach.
4. Differentiation Rules.
The Sum and Product Rules and Higher Derivatives. The Quotient Rule. The Chain Rule. Implicit Differentiation. Higher Taylor Polynomials.
5. The Geometry of Functions and Curves.
Horizontal and Vertical Asymptotes. Increasing and Decreasing Functions. Increasing and Decreasing Curves in the Plane. Concavity. Tangential and Normal Components of Acceleration. Circular Motion and Curvature. Applications of Maxima and Minima. The Remainder Theorem for Taylor Polynomials.
Antiderivatives and the Integral. The Chain Rule in Reverse. Acceleration, Velocity, and Position. Antiderivatives and Area. Area and Riemann Sums. The Definite Integral. Volumes. The Fundamental Theorem of Calculus.
7. Some Transcendental Functions.
Antiderivatives Revisited. Numerical Methods. The ln Function. The Function e x. Exponents and Logarithms. Euler's Formula (Optional). Inverse Trigonometric Functions. Derivatives of Inverse Trigonometric Functions. Tables of Integrals.
8. Applications of Separation of Variables.
Separation of Variables and Exponential Growth. Equations of the Form y 1 = ky + b. The Logistic Equation.
9. L'Hospital's Rule, Improper Integrals, and Series.
L'Hopital's Rule (0/0 and …à/…à) Improper Integrals. Series. Alternating Series and Absolute Convergence. The Ration Test and Power Series. Power Series of Functions. Radius of Convergence for Rational Functions.
10. Techniques of Integration.
Integration by Parts. Trigonometric Substitutions. Rational Functions. Integration Factors.
11. Work, Energy, and The Line Integral.
Work. The Work-Energy Theorem. Fundamental Curves. Line Integrals of Type I and Arc Length. Center of Mass and Moment of Inertia. Vector Fields. Line Integrals of Type II and Work. Partial Derivatives. Potential Functions and the Gradient. The Gradient and Directional Derivatives. The Curl and Iterated Partial Derivatives.
12. Optimization of Functions From Rn Into R.
Tests for Local Extrema. Extrema on Closed and Bounded Domains. Lagrange Multipliers.
13. Change of Coordinate Systems.
Translations and Linear Transformations. Other Transformations. The Derivative. Arc Length for Curves in Other Coordinate Systems. Change of Area with Linear Transformations. The Jacobian.
14. Multiple Integrals.
The Integral Over a Rectangle. Simple Surfaces. An Introduction to Surface Integrals. Some Applications of Surface Integrals. Change of Variables. Simple Solids. Triple Integrals. More on Triple Integrals.
15. Divergence and Stokes' Theorem.
Oriented Surfaces and Surface Integrals. Gaussian Surfaces. Divergence. Surfaces and Their Boundaries. Stokes' Theorem. Stokes' Theorem and Conservative Fields.
A. Mathematical Induction.
B. Continuity for Functions From Rn into Rm.
C. Conic Sections in Rn.
Parabolas, Ellipses, and Hyperbolas. Some 3-Dimensional Graphs. Translation in Rn.
D. Geometric Formulas.
E. Answers to Odd Exercises.