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Calculus: Single and Multivariable, 5th Edition International Student Version

Description

CALCULUS 5/e brings together the best of both new and traditional curricula in an effort to meet the needs of even more instructors teaching calculus. The author team's extensive experience teaching from both traditional and innovative books and their expertise in developing innovative problems put them in an unique position to make this new curriculum meaningful to students going into mathematics and those going into the sciences and engineering.  The authors believe this edition will work well for those departments who are looking for a calculus book that offers a middle ground for their calculus instructors.

CALCULUS 5/e exhibits the same strengths from earlier editions including the Rule of Four, an emphasis on modeling, exposition that students can read and understand and a flexible approach to technology. The conceptual and modeling problems, praised for their creativity and variety, continue to motivate and challenge students.

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1 A LIBRARY OF FUNCTIONS.

1.1 Functions and Change.

1.2 Exponential Functions.

1.3 New Functions from Old.

1.4 Logarithmic Functions.

1.5 Trigonometric Functions.

1.6 Powers, Polynomials, and Rational Functions.

1.7 Introduction to Continuity.

1.8 Limits.

Review Problems.

Check Your Understanding.

Projects: Matching Functions to Data, Which Way Is the Wind Blowing?

2 KEY CONCEPT: THE DERIVATIVE.

2.1 How Do We Measure Speed?

2.2 The Derivative at a Point.

2.3 The Derivative Function.

2.4 Interpretations of the Derivative.

2.5 The Second Derivative.

2.6 Differentiability.

Review Problems.

Check Your Understanding.

Projects: Hours of Daylight as a Function of Latitude, US Population.

3 SHORT-CUTS TO DIFFERENTIATION.

3.1 Powers and Polynomials.

3.2 The Exponential Function.

3.3 The Product and Quotient Rules.

3.4 The Chain Rule.

3.5 The Trigonometric Functions.

3.6 The Chain Rule and Inverse Functions.

3.7 Implicit Functions.

3.8 Hyperbolic Functions.

3.9 Linear Approximation and the Derivative.

3.10 Theorems about Differentiable Functions.

Review Problems.

Check Your Understanding.

Projects: Rule of 70, Newton’s Method.

4 USING THE DERIVATIVE.

4.1 Using First and Second Derivatives.

4.2 Optimization.

4.3 Families of Functions.

4.4 Optimization, Geometry, and Modeling.

4.5 Applications to Marginality.

4.6 Rates and Related Rates.

4.7 L'hopital's Rule, Growth, and Dominance.

4.8 Parametric Equations.

Review Problems.

Check Your Understanding.

Projects: Building a Greenhouse, Fitting a Line to Data, Firebreaks.

5 KEY CONCEPT: THE DEFINITE INTEGRAL.

5.1 How Do We Measure Distance Traveled?

5.2 The Definite Integral.

5.3 The Fundamental Theorem and Interpretations.

5.4 Theorems about Definite Integrals.

Review Problems.

Check Your Understanding.

Projects: The Car and the Truck, An Orbiting Satellite.

6 CONSTRUCTING ANTIDERIVATIVES.

6.1 Antiderivatives Graphically and Numerically.

6.2 Constructing Antiderivatives Analytically.

6.3 Differential Equations.

6.4 Second Fundamental Theorem of Calculus.

6.5 The Equations of Motion.

Review Problems.

Check Your Understanding.

Projects: Distribution of Resources, Yield from an Apple Orchard, Slope Fields.

7 INTEGRATION.

7.1 Integration by Substitution.

7.2 Integration by Parts.

7.3 Tables of Integrals.

7.4 Algebraic Identities and Trigonometric Substitutions.

7.5 Approximating Definite Integrals.

7.6 Approximation Errors and Simpson's Rule.

7.7 Improper Integrals.

7.8 Comparison of Improper Integrals.

Review Problems.

Check Your Understanding.

Projects: Taylor Polynomial Inequalities.

8 USING THE DEFINITE INTEGRAL.

8.1 Areas and Volumes.

8.2 Applications to Geometry.

8.3 Area and Arc Length in Polar Coordinates.

8.4 Density and Center of Mass.

8.5 Applications to Physics.

8.6 Applications to Economics.

8.7 Distribution Functions.

8.8 Probability, Mean, and Median.

Review Problems.

Check Your Understanding.

Projects: Volume Enclosed by Two Cylinders, Length of a Hanging Cable, Surface Area of an Unpaintable Can of Paint, Maxwell's Distribution of Molecular Velocities.

9 SEQUENCES AND SERIES.

9.1 Sequences.

9.2 Geometric Series.

9.3 Convergence of Series.

9.4 Tests for Convergence.

9.5 Power Series and Interval of Convergence.

Review Problems.

Check Your Understanding.

Projects: A Definition of e, Probability of Winning in Sports, Prednisone.

10 APPROXIMATING FUNCTIONS USING SERIES.

10.1 Taylor Polynomials.

10.2 Taylor Series.

10.3 Finding and Using Taylor Series.

10.4 The Error in Taylor Polynomial Approximations.

10.5 Fourier Series.

Review Problems.

Check Your Understanding.

Projects: Shape of Planets, Machin's Formula and the Value of pi, Approximation the Derivative.

11 DIFFERENTIAL EQUATIONS.

11.1 What Is a Differential Equation?

11.2 Slope Fields.

11.3 Euler's Method.

11.4 Separation of Variables.

11.5 Growth and Decay.

11.6 Applications and Modeling.

11.7 The Logistic Model.

11.8 Systems of Differential Equations.

11.9 Analyzing the Phase Plane.

11.10 Second-Order Differential Equations: Oscillations.

11.11 Linear Second-Order Differential Equations.

Review Problems.

Check Your Understanding.

Projects: SARS Predictions for Hong Kong, A S-I-R Model for SARS, Pareto’s Law, Vibrations in a Molecule.

12 FUNCTIONS OF SEVERAL VARIABLES.

12.1 Functions of Two Variables.

12.2 Graphs of Functions of Two Variables.

12.3 Contour Diagrams.

12.4 Linear Functions.

12.5 Functions of Three Variables.

12.6 Limits and Continuity.

Review Problems.

Check Your Understanding.

Projects: A Heater in a Room, Light in a Wave-Guide.

13 A FUNDAMENTAL TOOL: VECTORS.

13.1 Displacement Vectors.

13.2 Vectors in General.

13.3 The Dot Product.

13.4 The Cross Product.

Review Problems.

Check Your Understanding.

Projects: Cross Product of Vectors in the Plane, The Dot Product in Genetics, A Warren Truss.

14 DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES.

14.1 The Partial Derivative.

14.2 Computing Partial Derivatives Algebraically.

14.3 Local Linearity and the Differential.

14.4 Gradients and Directional Derivatives in the Plane.

14.5 Gradients and Directional Derivatives in Space.

14.6 The Chain Rule.

14.7 Second-Order Partial Derivatives.

14.8 Differentiability.

Review Problems.

Check Your Understanding.

15 OPTIMIZATION: LOCAL AND GLOBAL EXTREMA.

15.1 Local Extrema.

15.2 Optimization.

15.3 Constrained Optimization: Lagrange Multipliers.

Review Problems.

Check Your Understanding.

Projects: Optimization in Manufacturing, Fitting a Line to Data Using Least Squares, Hockey and Entropy.

16 INTEGRATING FUNCTIONS OF SEVERAL VARIABLES.

16.1 The Definite Integral of a Function of Two Variables.

16.2 Iterated Integrals.

16.3 Triple Integrals.

16.4 Double Integrals in Polar Coordinates.

16.5 Integrals in Cylindrical and Spherical Coordinates.

16.6 Applications of Integration to Probability.

16.7 Change of Variables in a Multiple Integral.

Review Problems.

Check Your Understanding.

Projects: A Connection Between e and pi, Average Distance Walked to an Airport Gate.

17 PARAMETERIZATION AND VECTOR FIELDS.

17.1 Parameterized Curves.

17.2 Motion, Velocity, and Acceleration.

17.3 Vector Fields.

17.4 The Flow of a Vector Field.

17.5 Parameterized Surfaces.

Review Problems.

Check Your Understanding.

Projects: Shooting a Basketball, Kepler’s Second Law, Flux Diagrams.

18 LINE INTEGRALS.

18.1 The Idea of a Line Integral.

18.2 Computing Line Integrals Over Parameterized Curves.

18.3 Gradient Fields and Path-Independent Fields.

18.4 Path-Dependent Vector Fields and Green's Theorem.

Review Problems.

Check Your Understanding.

Projects: Conservation of Energy, Planimeters, Ampre's Law.

19 FLUX INTEGRALS.

19.1 The Idea of a Flux Integral.

19.2 Flux Integrals for Graphs, Cylinders, and Spheres.

19.3 Flux Integrals Over Parameterized Surfaces.

Review Problems.

Check Your Understanding.

Projects: Gauss's Law Applied to a Charged Wire and a Charged Sheet, Flux across a Cylinder Due to a Point Charge: Obtaining Gauss's Law from Coulomb's Law .

20 CALCULUS OF VECTOR FIELDS.

20.1 The Divergence of a Vector Field.

20.2 The Divergence Theorem.

20.3 The Curl of a Vector Field.

20.4 Stokes Theorem.

20.5 The Three Fundamental Theorems.

Review Problems.

Check Your Understanding.

Projects: Divergence of Spherically Symmetric Vector Fields, Divergence of Cylindrically Symmetric Vector Fields.

Appendix.

A Roots, Accuracy, and Bounds.

B Complex Numbers.

C Newtons's Method.

D Vectors in the Plane.

E Determinants.