by Robert T. Smith and Roland B. Minton
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This modern calculus textbook places a strong emphasis on developing students' conceptual understanding and on building connections between key calculus topics and their relevance for the real world. It is written for the average student -- one who is mostly unfamiliar with the subject and who requires significant motivation. It follows a relatively standard order of presentation, with early coverage of transcendentals, and integrates thought-provoking applications, examples and exercises throughout. The text also provides balanced guidance on the appropriate role of technology in problem-solving, including its benefits and its potential pitfalls. Wherever practical, concepts are developed from graphical, numerical, algebraic and verbal perspectives (the "Rule of Four") to give students a complete understanding of calculus.
New Features
0 Preliminaries
0.1 Polynomial and Rational Functions
0.2 Graphing Calculators and Computer Algebra Systems
0.3 Inverse Functions
0.4 Trigonometric and Inverse Trigonometric Functions
0.5 Exponential and Logarithmic Functions
0.6 Transformations of Functions
0.7 Parametric Equations and Polar Coordinates
1 Limits and Continuity
1.1 A Brief Preview of Calculus
1.2 The Concept of Limit
1.3 Computation of Limits
1.4 Continuity and its Consequences
1.5 Limits Involving Infinity
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 The Chain Rule
2.6 Derivatives of Trigonometric and Inverse Trigonometric Functions
2.7 Derivatives of Exponential and Logarithmic Functions
2.8 Implicit Differentiation
2.9 The Mean Value Theorem
3 Applications of Differentiation
3.1 Linear Approximations and Newton's Method
3.2 Indeterminate Forms and L'Hopital's Rule
3.3 Maximum and Minimum Values
3.4 Increasing and Decreasing Functions
3.5 Concavity and Overview of Curve Sketching
3.6 Optimization
3.7 Rates of Change in Economics and the Sciences
3.8 Related Rates and Parametric Equations
4 Integration
4.1 Area Under a Curve
4.2 The Definite Integral
4.3 Antiderivatives
4.4 The Fundamental Theorem of Calculus
4.5 Integration by Substitution
4.6 Integration by Parts
4.7 Other Techniques of Integration
4.8 Integration Tables and Computer Algebra Systems
4.9 Numerical Integration
4.10 Improper Integrals
5 Applications of the Definite Integral
5.1 Area Between Curves
5.2 Volume
5.3 Arc Length and Surface Area
5.4 Projectile Motion
5.5 Applications of Integration to Physics and Engineering
5.6 Probability
6 Differential Equations
6.1 Growth and Decay Problems
6.2 Separable Differential Equations
6.3 Euler's Method
6.4 Second Order Equations with Constant Coefficients
6.5 Nonhomogeneous Equations: Undetermined Coefficients
6.6 Applications of Differential Equations
7 Infinite Series
7.1 Sequences of Real Numbers
7.2 Infinite Series
7.3 The Integral Test and Comparison Tests
7.4 Alternating Series
7.5 Absolute Convergence and the Ratio Test
7.6 Power Series
7.7 Taylor Series
7.8 Applications of Taylor Series
7.9 Fourier Series
7.10 Power Series Solutions of Differential Equations
8 Vectors and the Geometry of Space
8.1 Vectors in the Plane
8.2 Vectors in Space
8.3 The Dot Product
8.4 The Cross Product
8.5 Lines and Planes in Space
8.6 Surfaces in Space
9 Vector-Valued Functions
9.1 Vector-Valued Functions
9.2 Parametric Surfaces
9.3 The Calculus of Vector-Valued Functions
9.4 Motion in Space
9.5 Curvature
9.6 Tangent and Normal Vectors
10 Functions of Several Variables and Differentiation
10.1 Functions of Several Variables
10.2 Limits and Continuity
10.3 Partial Derivatives
10.4 Tangent Planes and Linear Approximations
10.5 The Chain Rule
10.6 The Gradient and Directional Derivatives
10.7 Extrema of Functions of Several Variables
10.8 Constrained Optimization and Lagrange Multipliers
11 Multiple Integrals
11.1 Double Integrals
11.2 Area, Volume and Center of Mass
11.3 Double Integrals in Polar Coordinates
11.4 Surface Area
11.5 Triple Integrals
11.6 Cylindrical Coordinates
11.7 Spherical Coordinates
11.8 Change of Variables in Multiple Integrals
12 Vector Calculus
12.1 Vector Fields
12.2 Curl and Divergence
12.3 Line Integrals
12.4 Independence of Path and Conservative Vector Fields
12.5 Green's Theorem
12.6 Surface Integrals
12.7 The Divergence Theorem
12.8 Stokes' Theorem
12.9 Applications of Vector Calculus
Appendix A Graphs of Additional Polar Equations
Appendix B Formal Definition of Limit
Appendix C Complete Derivation of Derivatives of sin x and cos x
Appendix D Natural Logarithm Defined as an Integral; Exponential Defined as the Inverse of the Natural Logarithm
Appendix E Conic Sections in Polar Coordinates
Appendix F Proofs of Selected Theorems
Robert T. Smith and Roland B. Minton
ISBN13: 978-0073309293This modern calculus textbook places a strong emphasis on developing students' conceptual understanding and on building connections between key calculus topics and their relevance for the real world. It is written for the average student -- one who is mostly unfamiliar with the subject and who requires significant motivation. It follows a relatively standard order of presentation, with early coverage of transcendentals, and integrates thought-provoking applications, examples and exercises throughout. The text also provides balanced guidance on the appropriate role of technology in problem-solving, including its benefits and its potential pitfalls. Wherever practical, concepts are developed from graphical, numerical, algebraic and verbal perspectives (the "Rule of Four") to give students a complete understanding of calculus.
New Features
Table of Contents
0 Preliminaries
0.1 Polynomial and Rational Functions
0.2 Graphing Calculators and Computer Algebra Systems
0.3 Inverse Functions
0.4 Trigonometric and Inverse Trigonometric Functions
0.5 Exponential and Logarithmic Functions
0.6 Transformations of Functions
0.7 Parametric Equations and Polar Coordinates
1 Limits and Continuity
1.1 A Brief Preview of Calculus
1.2 The Concept of Limit
1.3 Computation of Limits
1.4 Continuity and its Consequences
1.5 Limits Involving Infinity
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 The Chain Rule
2.6 Derivatives of Trigonometric and Inverse Trigonometric Functions
2.7 Derivatives of Exponential and Logarithmic Functions
2.8 Implicit Differentiation
2.9 The Mean Value Theorem
3 Applications of Differentiation
3.1 Linear Approximations and Newton's Method
3.2 Indeterminate Forms and L'Hopital's Rule
3.3 Maximum and Minimum Values
3.4 Increasing and Decreasing Functions
3.5 Concavity and Overview of Curve Sketching
3.6 Optimization
3.7 Rates of Change in Economics and the Sciences
3.8 Related Rates and Parametric Equations
4 Integration
4.1 Area Under a Curve
4.2 The Definite Integral
4.3 Antiderivatives
4.4 The Fundamental Theorem of Calculus
4.5 Integration by Substitution
4.6 Integration by Parts
4.7 Other Techniques of Integration
4.8 Integration Tables and Computer Algebra Systems
4.9 Numerical Integration
4.10 Improper Integrals
5 Applications of the Definite Integral
5.1 Area Between Curves
5.2 Volume
5.3 Arc Length and Surface Area
5.4 Projectile Motion
5.5 Applications of Integration to Physics and Engineering
5.6 Probability
6 Differential Equations
6.1 Growth and Decay Problems
6.2 Separable Differential Equations
6.3 Euler's Method
6.4 Second Order Equations with Constant Coefficients
6.5 Nonhomogeneous Equations: Undetermined Coefficients
6.6 Applications of Differential Equations
7 Infinite Series
7.1 Sequences of Real Numbers
7.2 Infinite Series
7.3 The Integral Test and Comparison Tests
7.4 Alternating Series
7.5 Absolute Convergence and the Ratio Test
7.6 Power Series
7.7 Taylor Series
7.8 Applications of Taylor Series
7.9 Fourier Series
7.10 Power Series Solutions of Differential Equations
8 Vectors and the Geometry of Space
8.1 Vectors in the Plane
8.2 Vectors in Space
8.3 The Dot Product
8.4 The Cross Product
8.5 Lines and Planes in Space
8.6 Surfaces in Space
9 Vector-Valued Functions
9.1 Vector-Valued Functions
9.2 Parametric Surfaces
9.3 The Calculus of Vector-Valued Functions
9.4 Motion in Space
9.5 Curvature
9.6 Tangent and Normal Vectors
10 Functions of Several Variables and Differentiation
10.1 Functions of Several Variables
10.2 Limits and Continuity
10.3 Partial Derivatives
10.4 Tangent Planes and Linear Approximations
10.5 The Chain Rule
10.6 The Gradient and Directional Derivatives
10.7 Extrema of Functions of Several Variables
10.8 Constrained Optimization and Lagrange Multipliers
11 Multiple Integrals
11.1 Double Integrals
11.2 Area, Volume and Center of Mass
11.3 Double Integrals in Polar Coordinates
11.4 Surface Area
11.5 Triple Integrals
11.6 Cylindrical Coordinates
11.7 Spherical Coordinates
11.8 Change of Variables in Multiple Integrals
12 Vector Calculus
12.1 Vector Fields
12.2 Curl and Divergence
12.3 Line Integrals
12.4 Independence of Path and Conservative Vector Fields
12.5 Green's Theorem
12.6 Surface Integrals
12.7 The Divergence Theorem
12.8 Stokes' Theorem
12.9 Applications of Vector Calculus
Appendix A Graphs of Additional Polar Equations
Appendix B Formal Definition of Limit
Appendix C Complete Derivation of Derivatives of sin x and cos x
Appendix D Natural Logarithm Defined as an Integral; Exponential Defined as the Inverse of the Natural Logarithm
Appendix E Conic Sections in Polar Coordinates
Appendix F Proofs of Selected Theorems