Calculus: Single and Multivariable

Cover
Title Page
Copyright
Preface
Acknowledgements
Contents
1 FOUNDATION FOR CALCULUS: FUNCTIONS AND LIMITS
	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 LIMITS AND CONTINUITY
	1.8 EXTENDING THE IDEA OF A LIMIT
	1.9 FURTHER LIMIT CALCULATIONS USING ALGEBRA
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
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
4 USING THE DERIVATIVE
	4.1 USING FIRST AND SECOND DERIVATIVES
	4.2 OPTIMIZATION
	4.3 OPTIMIZATION AND MODELING
	4.4 FAMILIES OF FUNCTIONS 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
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
6 CONSTRUCTING ANTIDERIVATIVES
	6.1 ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY
	6.2 CONSTRUCTING ANTIDERIVATIVES ANALYTICALLY
	6.3 DIFFERENTIAL EQUATIONS AND MOTION
	6.4 SECOND FUNDAMENTAL THEOREM OF CALCULUS
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 NUMERICAL METHODS FOR DEFINITE INTEGRALS
	7.6 IMPROPER INTEGRALS
	7.7 COMPARISON OF IMPROPER INTEGRALS
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
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
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
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
12 FUNCTIONS OF SEVERAL VARIABLES
	12.1 FUNCTIONS OF TWO VARIABLES
	12.2 GRAPHS AND SURFACES
	12.3 CONTOUR DIAGRAMS
	12.4 LINEAR FUNCTIONS
	12.5 FUNCTIONS OF THREE VARIABLES
	12.6 LIMITS AND CONTINUITY
13 A FUNDAMENTAL TOOL: VECTORS
	13.1 DISPLACEMENT VECTORS
	13.2 VECTORS IN GENERAL
	13.3 THE DOT PRODUCT
	13.4 THE CROSS PRODUCT
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
15 OPTIMIZATION: LOCAL AND GLOBAL EXTREMA
	15.1 CRITICAL POINTS: LOCAL EXTREMA AND SADDLE POINTS
	15.2 OPTIMIZATION
	15.3 CONSTRAINED OPTIMIZATION: LAGRANGE MULTIPLIERS
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
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
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
19 FLUX INTEGRALS AND DIVERGENCE
	19.1 THE IDEA OF A FLUX INTEGRAL
	19.2 FLUX INTEGRALS FOR GRAPHS, CYLINDERS, AND SPHERES
	19.3 THE DIVERGENCE OF A VECTOR FIELD
	19.4 THE DIVERGENCE THEOREM
20 THE CURL AND STOKES’ THEOREM
	20.1 THE CURL OF A VECTOR FIELD
	20.2 STOKES’ THEOREM
	20.3 THE THREE FUNDAMENTAL THEOREMS
21 PARAMETERS, COORDINATES, AND INTEGRALS
	21.1 COORDINATES AND PARAMETERIZED SURFACES
	21.2 CHANGE OF COORDINATES IN A MULTIPLE INTEGRAL
	21.3 FLUX INTEGRALS OVER PARAMETERIZED SURFACES
READY REFERENCE
ANSWERS TO ODD-NUMBERED PROBLEMS
INDEX
EULA