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Summary
The wide-ranging debate brought about by the calculus reform movement has had a significant impact on calculus textbooks. In response to many of the questions and concerns surrounding this debate, the authors have written a modern calculus textbook, intended for students majoring in mathematics, physics, chemistry, engineering and related fields. The text is written for the average student -- one who does not already know the subject, whose background is somewhat weak in spots, and who requires a significant motivation to study calculus.The authors follow a relatively standard order of presentation, while integrating technology and thought-provoking exercises throughout the text. Some minor changes have been made in the order of topics to reflect shifts in the importance of certain applications in engineering and science. This text also gives an early introduction to logarithms, exponentials and the trigonometric functions. Wherever practical, concepts are developed from graphical, numerical, and algebraic perspectives (the "Rule of Three") to give students a full understanding of calculus. This text places a significant emphasis on problem solving and presents realistic applications, as well as open-ended problems.
Table of Contents
0 Preliminaries
0.1 The Real Numbers and the Cartesian Plane
0.2 Lines and Functions
0.3 Graphing Calculators and Computer Algebra Systems
0.4 Solving Equations
0.5 Trigonometric Functions
0.6 Exponential and Logarithmic Functions
0.7 Transformations of Functions
0.8 Preview of Calculus
1 Limits and Continuity
1.1 The Concept of Limit
1.2 Computation of Limits
1.3 Continuity and its Consequences
1.4 Limits Involving Infinity
1.5 Formal Definition of the Limit
1.6 Limits and Loss-of-Significance Errors
2 Differentiation
2.1 Tangent Lines and Velocity
2.2 The Derivative
2.3 Computation of Derivatives: The Power Rule
2.4 The Product and Quotient Rules
2.5 Derivatives of Trigonometric Functions
2.6 Derivatives of Exponential and Logarithmic Functions
2.7 The Chain Rule
2.8 Implicit Differentiation and Related Rates
2.9 The Mean Value Theorem
3 Applications of Differentiation
3.1 Linear Approximations adn L'Hopital's Rule
3.2 Newton's Method
3.3 Maximum and Minimum Values
3.4 Increasing and Decreasing Functions
3.5 Concavity
3.6 Overview of Curve Sketching
3.7 Optimization
3.8 Rates of Change in Applications
4 Integration
4.1 Antiderivatives
4.2 Sums and Sigma Notation
4.3 Area
4.4 The Definite Integral
4.5 The Fundamental Theorem of Calculus
4.6 Integration by Substitution
4.7 Numerical Integration
5 Applications of the Definite Integral
5.1 Area Between Curves
5.2 Volume
5.3 Volumes by Cylindrical Shells
5.4 Arc Length and Surface Area
5.5 Projectile Motion
5.6 Work, Moments, and Hydrostatic Force
5.7 Probability
6 Exponentials, Logarithms, and Other Transcendental Functions
6.1 The Natural Logarithm Revisited
6.2 Inverse Functions
6.3 The Exponential Function Revisited
6.4 Growth and Decay Problems
6.5 Separable Differential Equations
6.6 Euler's Method
6.7 The Inverse Trigonometric Functions
6.8 The Calculus of the Inverse Trigonometric Functions
6.9 The Hyperbolic Functions
7 Integration Techniques
7.1 Review of Formulas and Techniques
7.2 Integration by Parts
7.3 Trigonometric Techniques of Integration
7.4 Integration of Rational Functions using Partial Fractions
7.5 Integration Tables and Computer Algebra Systems
7.6 Indeterminate Forms and L'Hopital's Rule
7.7 Improper Integrals
8 Infinite Series
8.1 Sequences of Real Numbers
8.2 Infinite Series
8.3 The Integral Test and Comparison Tests
8.4 Alternating Series
8.5 Absolute Convergence and the Ratio Test
8.6 Power Series
8.7 Taylor Series
8.8 Fourier Series
9 Parametric Equations and Polar Coordinates
9.1 Plane Curves and Parametric Equations
9.2 Calculus and Parametric Equations
9.3 Arc Length and Surface Area in Parametric Equations
9.4 Polar Coordinates
9.5 Calculus and Polar Coordinates
9.6 Conic Sections
9.7 Conic Sections in Polar Coordinates
10 Vectors and the Geometry of Space
10.1 Vectors in the Plane
10.2 Vectors in Space
10.3 The Dot Product
10.4 The Cross Product
10.5 Lines and Planes in Space
10.6 Surfaces in Space
11 Vector-Valued Functions
11.1 Vector-Valued Functions
11.2 The Calculus of Vector-Valued Functions
11.3 Motion in Space
11.4 Curvature
11.5 Tangent and Normal Vectors
12 Functions of Several Variables and Differentiation
12.1 Functions of Several Variables
12.2 Limits and Continuity
12.3 Partial Derivatives
12.4 Tangent Planes and Linear Approximations
12.5 The Chain Rule
12.6 The Gradient and Directional Derivatives
12.7 Extrema of Functions of Several Variables
12.8 Constrained Optimization and Lagrange Multipliers
13 Multiple Integrals
13.1 Double Integrals
13.2 Area, Volume and Center of Mass
13.3 Double Integrals in Polar Coordinates
13.4 Surface Area
13.5 Triple Integrals
13.6 Cylindrical Coordinates
13.7 Spherical Coordinates
13.8 Change of Variables in Multiple Integrals
14 Vector Calculus
14.1 Vector Fields
14.2 Line Integrals
14.3 Independence of Path and Conservative Vector Fields