Calculus with Early Transcendentals, 2nd Edition
1.1 Functions and How We Represent Them
- Examples
- Example 1: Finding the Domain and Range of a Relation
- Example 2: Determining Whether a Relation Is a Function
- Example 3: Identifying the Dependent and Independent Variables of a Function Defined by an Equation
- Example 4: Finding the Implied Domain and the Range of a Function
- Example 5: Writing Formulas as Functions
- Example 6: Describing Functions Verbally
- Example 7: Applying the Vertical Line Test
- Example 8: Identifying Intervals of Monotonicity
- Example 9: Identifying Types of Symmetry in Graphs of Relations
1.2 Common Functions
- Examples
- Example 1: Describing Graphs of Polynomial Functions
- Example 2: Applying Position and Velocity Functions
- Example 3: Graphing Rational Functions
- Example 4: Constructing Algebraic Functions from Polynomials
- Example 5: Simplifying Trigonometric Expressions
- Example 6: Using Trigonometric Models
- Example 7: Using an Exponential Function to Model Population Growth
- Example 8: Graphing and Evaluating a Piecewise-Defined Function
1.3 Transforming and Combining Functions
- Examples
- Example 1: Shifting Graphs of Functions Horizontally
- Example 2: Shifting Graphs of Functions Vertically
- Example 3: Reflecting Graphs of Functions with Respect to the Axes
- Example 4: Stretching and Compressing Graphs of Functions Vertically
- Example 5: Graphing a Function Using a Sequence of Function Transformations
- Example 6: Subtracting and Dividing Functions
- Example 7: Adding and Multiplying Functions
- Example 8: Finding the Domain of a Quotient Function
- Example 9: Composing Functions
- Example 10: Finding Formulas and Domains for Composite Functions
- Example 11: Decomposing a Composite Function
1.4 Inverse Functions
- Examples
- Example 1: Finding the Inverse of a Relation and Its Domain and Range
- Example 2: Applying the Horizontal Line Test
- Example 3: Finding the Inverse of a Function
- Example 4: Graphing Logarithmic Functions
- Example 5: Expanding Logarithmic Expressions
- Example 6: Condensing Logarithmic Expressions
- Example 7: Deriving the Arccosine Function
- Example 8: Evaluating Compositions of Trigonometric and Inverse Trigonometric Functions
- Example 9: Evaluating Compositions of Trigonometric and Inverse Trigonometric Functions
- Example 10: Expressing a Composition of Trigonometric and Inverse Trigonometric Functions as an Algebraic Function
1.5 Calculus, Calculators, and Computer Algebra Systems
2.1 Rates of Change and Tangent Lines
- Examples
- Example 1: Using Average Velocities to Estimate Instantaneous Velocity
- Example 2: Using Difference Quotients to Estimate Instantaneous Velocity
- Example 3: Using Difference Quotients to Approximate the Slope of a Tangent Line
- Example 4: Showing That a Tangent Line Does Not Exist
- Example 5: Estimating Area under a Curve and Arc Length
2.2 Limits All around the Plane
- Examples
- Example 1: Using Graphs to Find Limits
- Example 2: Applying Limits to a Paradox
- Example 3: Using One-Sided Limit Notation to Describe Asymptotic Behavior
- Example 4: Finding One-Sided Infinite Limits without a Graph
- Example 6: Using Limits at Infinity to Identify Horizontal Asymptotes
- Example 7: Using Limit Notation to Describe Unbounded Behavior
2.3 The Mathematical Definition of Limit
- Examples
- Example 1: Using Graphs to Estimate δ Given ε
- Example 2: Using Graphs to Find a Limit
- Example 3: Using the ε-δ Definition of Limit to Prove a Limit Exists
- Example 4: Using the ε-δ Definition of Limit to Prove a Limit Exists
- Example 5: Using the ε-δ Definition of Limit to Prove a Limit Exists
- Example 6: Proving a Limit Does Not Exist
- Example 7: Proving a Limit Does Not Exist
2.4 Determining Limits of Functions
- Examples
- Example 1: Using Limit Laws and a Graph to Find Limits
- Example 2: Using Limit Laws to Find Limits
- Example 3: Using Limit Laws to Find Limits
- Example 4: Using Limit Laws to Find Limits
- Example 5: Using Algebraic Techniques to Find a Limit
- Example 6: Using Algebraic Techniques to Find a Limit
- Example 7: Using Algebraic Techniques to Find a Limit
- Example 8: Using the Squeeze Theorem to Prove a Limit Claim
2.5 Continuity
- Examples
- Example 1: Finding Points of Continuity and Discontinuity
- Example 2: Finding Points of Discontinuity
- Example 3: Classifying Removable and Nonremovable Points of Discontinuity
- Example 4: Finding Points of Continuity and Discontinuity
- Example 5: Using the ε-δ Definition of Continuity to Prove a Function Is Continuous
- Example 6: Showing That a Rational Function Is Continuous
- Example 7: Using Theorems to Describe the Continuity of Functions
- Example 8: Using the Alternate Formulation of Continuity to Prove a Function Is Continuous
- Example 9: Using Theorems to Describe the Continuity of Functions
- Example 10: Defining the Continuous Extension of a Function
- Example 11: Using the Intermediate Value Theorem
2.6 Rate of Change Revisited: The Derivative
3.1 Differentiation Notation and Consequences
- Examples
- Example 1: Finding the Derivative of a Function
- Example 2: Evaluating the Derivative of a Function at a Point
- Example 3: Finding Velocity and Acceleration Functions
- Example 4: Finding the First, Second, and Third Derivatives of a Function
- Example 5: Proving a Function Is Continuous but Not Differentiable
- Example 6: Finding One-Sided Derivatives
3.2 Derivatives of Polynomials, Exponentials, Products, and Quotients
- Examples
- Example 1: Using the Constant Rule and the Positive Integer Power Rule
- Example 2: Using Elementary Differentiation Rules
- Example 3: Finding the Equation of a Tangent Line
- Example 4: Finding Instantaneous Velocity and Acceleration
- Example 5: Finding the First, Second, and Third Derivatives of a Function
- Example 6: Using the Product Rule
- Example 7: Using the Product Rule
- Example 8: Using the Reciprocal Rule
- Example 9: Using the Quotient Rule
3.3 Derivatives of Trigonometric Functions
- Examples
- Example 1: Finding Limits Involving Trigonometric Functions
- Example 2: Analyzing Slopes of Lines Tangent to the Sine Function
- Example 3: Using Differentiation Rules with Trigonometric Functions
- Example 4: Using Differentiation Rules with Trigonometric Functions
- Example 5: Applying Position, Velocity, and Acceleration Functions
- Example 6: Using Differentiation Rules with Trigonometric Functions
- Example 7: Finding the Second Derivative of a Trigonometric Function
- Example 8: Finding Points Where Tangent Lines Have a Given Slope
3.4 The Chain Rule
- Examples
- Example 1: Using the Chain Rule
- Example 2: Using the Chain Rule with Trigonometric Functions
- Example 3: Repeatedly Applying the Chain Rule
- Example 4: Using the Chain Rule with the Power Rule
- Example 5: Finding the Equation of a Tangent Line
- Example 6: Using the Chain Rule with Trigonometric Functions
- Example 7: Finding the Equation of a Tangent Line and Locations of Horizontal Tangents
3.5 Implicit Differentiation
- Examples
- Example 1: Comparing the Methods of Explicit and Implicit Differentiation
- Example 2: Using Implicit Differentiation to Locate a Vertical Tangent
- Example 3: Finding the Equation of a Normal Line and Locations of Horizontal Tangents
- Example 4: Using Implicit Differentiation
- Example 5: Using Implicit Differentiation to Find a Second Derivative
- Example 6: Using Implicit Differentiation to Find a Second Derivative
3.6 Derivatives of Inverse Functions
- Examples
- Example 1: Using the Derivative Rule for Inverse Functions
- Example 2: Finding Derivatives Involving the Natural Logarithm
- Example 3: Finding the Equation of a Tangent Line
- Example 4: Finding Derivatives of Logarithmic Functions
- Example 5: Using Logarithmic Differentiation
- Example 6: Finding Points of Differentiability and Derivatives of Power Functions
- Example 7: Using the Power Rule
- Example 8: Determining the Derivative of the Arctangent Function
- Example 9: Finding Derivatives of Inverse Trigonometric Functions
3.7 Rates of Change in Use
3.8 Related Rates
3.9 Linearization and Differentials
4.1 Extreme Values of Functions
- Examples
- Example 1: Using a Graph to Identify Absolute Extrema
- Example 2: Using a Graph to Identify Relative and Absolute Extrema
- Example 3: Finding the Absolute Extrema of a Function
- Example 4: Finding the Absolute Extrema of a Function
- Example 5: Finding the Absolute Extrema of a Function
- Example 6: Minimizing Cost
4.2 The Mean Value Theorem
- Examples
- Example 1: Proving an Equation Has Exactly One Solution
- Example 2: Applying Rolle's Theorem to a Position Function
- Example 3: Finding Points That Satisfy the Mean Value Theorem
- Example 4: Using the Mean Value Theorem to Find Velocity
- Example 5: Using the Mean Value Theorem
- Example 6: Using Corollary 2 of the Mean Value Theorem
4.3 The First and Second Derivative Tests
4.4 L'Hôpital's Rule
- Examples
- Example 1: Using L'Hôpital's Rule to Find a Limit of Indeterminate Form 0\0
- Example 2: Using L'Hôpital's Rule to Find a Limit of Indeterminate Form ∞\∞
- Example 3: Using L'Hôpital's Rule to Find a Limit of Indeterminate Form 0\0
- Example 4: Repeatedly Applying L'Hôpital's Rule
- Example 5: Using L'Hôpital's Rule to Find a Limit of Indeterminate Form 0⋅∞
- Example 6: Using L'Hôpital's Rule to Find a Limit of Indeterminate Form ∞−∞
- Example 7: Using L'Hôpital's Rule to Find a Limit of Indeterminate Form 1^∞
- Example 8: Using L'Hôpital's Rule to Find a Limit of Indeterminate Form 0^0
- Example 9: Using L'Hôpital's Rule to Find a Limit of Indeterminate Form ∞^0
4.5 Calculus and Curve Sketching
- Examples
- Example 1: Using the Curve-Sketching Strategy
- Example 2: Using the Curve-Sketching Strategy
- Example 3: Using the Curve-Sketching Strategy
- Example 4: Finding the Relative Extrema of a Rational Function
- Example 5: Using the Curve-Sketching Strategy
- Example 6: Using Newton's Method to Approximate a Root
- Example 7: Using Newton's Method to Approximate a Root
4.6 Optimization Problems
4.7 Antiderivatives
- Examples
- Example 1: Finding Antiderivatives of Functions
- Example 2: Finding the General Antiderivative and a Particular Antiderivative of a Function
- Example 3: Finding General Antiderivatives of Functions
- Example 4: Finding the Maximum Height of a Ball
- Example 5: Solving an Initial Value Problem
- Example 6: Sketching the Graph of a Function Given the Graph of Its Derivative
5.1 Area, Distance, and Riemann Sums
5.2 The Definite Integral
- Examples
- Example 1: Finding the Height of a Stone Given Its Vertical Velocity Function
- Example 2: Evaluating a Definite Integral as a Limit of a Right Riemann Sum
- Example 3: Evaluating a Definite Integral by Using Geometry to Find the Signed Area
- Example 4: Using the Properties of the Definite Integral
- Example 5: Using the Properties of the Definite Integral
- Example 6: Finding the Average Value of a Function on an Interval
5.3 The Fundamental Theorem of Calculus
- Examples
- Example 1: Finding Points Where a Function Takes on Its Average Value
- Example 2: Using the Fundamental Theorem of Calculus, Part I
- Example 3: Using the Fundamental Theorem of Calculus, Part I
- Example 4: Using the Fundamental Theorem of Calculus, Part I
- Example 5: Using the Fundamental Theorem of Calculus, Part II, to Evaluate Definite Integrals
- Example 6: Finding the Function with a Given Derivative That Passes through a Particular Point
- Example 7: Using the Fundamental Theorem of Calculus, Part II, to Evaluate a Definite Integral
5.4 Indefinite Integrals and the Substitution Rule
5.5 The Substitution Rule and Definite Integration
6.1 Finding Volumes Using Slices
- Examples
- Example 1: Using Integration to Derive the Formula for Volume of a Sphere
- Example 2: Using Integration to Derive the Formula for Volume of a Pyramid
- Example 3: Using Integration to Find the Volume of a Curved Wedge
- Example 4: Using the Disk Method to Find the Volume of a Solid of Revolution
- Example 5: Using the Disk Method to Find the Volume of a Solid of Revolution
- Example 6: Using the Washer Method to Find the Volume of a Solid of Revolution
- Example 7: Using the Washer Method to Find the Volume of a Solid of Revolution
- Example 8: Using Integration in Multiple Ways to Derive the Formula for Area of a Circle
- Example 9: Using Integration to Derive the Formula for Volume of a Sphere
6.2 Finding Volumes Using Cylindrical Shells
6.3 Arc Length and Surface Area
6.4 Moments and Centers of Mass
- Examples
- Example 1: Comparing the Centers of Mass of Rods with Different Density Functions
- Example 2: Finding the Center of Mass of a Plate of Constant Density
- Example 3: Finding the Center of Mass of a Plate of Varying Density
- Example 4: Finding the Center of Mass of a Curved Wire
- Example 5: Using Pappus' Theorem to Find the Volume of a Torus
- Example 6: Using Pappus' Theorem to Find the Surface Area of a Torus
- Example 7: Using Pappus' Theorem for Surface Areas to Find Center of Mass
6.5 Force, Work, and Pressure
- Examples
- Example 1: Calculating the Work Done by a Constant Force
- Example 2: Finding the Work Done in Stretching a Spring
- Example 3: Finding the Work Done in Lifting a Bucket on a Rope
- Example 4: Finding the Work Done in Pumping Gasoline Out of a Conical Tank
- Example 5: Finding the Work Done in Pumping Gasoline Out of a Cylindrical Tank
- Example 6: Finding the Fluid Force Exerted on a Vertical Plate
- Example 7: Finding the Fluid Force Exerted on a Vertical Plate
- Example 8: Using the Centroid and Area of a Vertical Plate to Find the Fluid Force Exerted on It
6.6 Hyperbolic Functions
7.1 Integration by Parts
- Examples
- Example 1: Using Integration by Parts to Evaluate an Indefinite Integral
- Example 2: Using Integration by Parts to Evaluate an Indefinite Integral
- Example 3: Repeatedly Applying Integration by Parts
- Example 4: Repeatedly Applying Integration by Parts
- Example 5: Using Integration by Parts to Derive a Reduction Formula
- Example 6: Using a Reduction Formula to Evaluate an Indefinite Integral
- Example 7: Using Integration by Parts to Evaluate a Definite Integral
- Example 8: Using Integration by Parts and u-Substitution to Find Area under a Curve
7.2 The Partial Fractions Method
- Examples
- Example 1: Using Partial Fractions to Integrate a Rational Function
- Example 2: Using Partial Fraction Decomposition to Rewrite a Rational Function
- Example 3: Using Partial Fractions to Integrate a Rational Function
- Example 4: Using Partial Fractions to Integrate a Rational Function
- Example 5: Using Differentiation and Substitution to Determine Partial Fraction Constants
- Example 6: Using the Heaviside Cover-Up Method to Determine Partial Fraction Constants
- Example 7: Using Partial Fractions to Integrate a Rational Function
7.3 Trigonometric Integrals
- Examples
- Example 1: Integrating a Product of Powers of Sine and Cosine
- Example 2: Integrating a Power of Cosine
- Example 3: Integrating a Product of Powers of Sine and Cosine
- Example 4: Using a Product-to-Sum Identity to Evaluate a Trigonometric Integral
- Example 5: Integrating the First and Second Powers of Tangent and Secant
- Example 6: Integrating a Product of Powers of Tangent and Secant
- Example 7: Integrating a Power of Tangent
- Example 8: Integrating a Power of Secant
7.4 Trigonometric Substitutions
- Examples
- Example 1: Using a Trigonometric Substitution to Evaluate an Indefinite Integral
- Example 2: Using a Trigonometric Substitution to Evaluate an Indefinite Integral
- Example 3: Using Integration with Trigonometric Substitution to Derive the Formula for Area of an Ellipse
- Example 4: Using a Trigonometric Substitution to Evaluate an Indefinite Integral
- Example 5: Using a Hyperbolic Substitution to Evaluate an Indefinite Integral
- Example 6: Using Two Changes of Variable to Evaluate a Definite Integral
7.5 Integration Summary and Integration Using Computer Algebra Systems
7.6 Numerical Integration
- Examples
- Example 1: Using the Trapezoidal Rule to Approximate a Definite Integral
- Example 2: Using the Trapezoidal Rule to Estimate Average Humidity
- Example 3: Using the Error Estimate for the Trapezoidal Rule
- Example 4: Using Simpson's Rule to Approximate a Definite Integral
- Example 5: Using Simpson's Rule and Estimating the Error in the Approximation
- Example 6: Using Simpson's Rule to Estimate Volume of a Pond
7.7 Improper Integrals
- Examples
- Example 1: Finding the Area under a Curve over an Unbounded Interval
- Example 2: Evaluating an Improper Integral of Type I
- Example 3: Evaluating an Improper Integral of Type I
- Example 4: Finding the Volume of an Infinitely Long Horn
- Example 5: Evaluating an Improper Integral of Type II
- Example 6: Evaluating an Improper Integral of Type II
- Example 7: Evaluating an Improper Integral of Type II
- Example 8: Using the Direct Comparison Test
- Example 9: Using the Direct Comparison Test
8.1 Separable Differential Equations
8.2 First-Order Linear Differential Equations
8.3 Autonomous Differential Equations and Slope Fields
- Examples
- Example 1: Solving an Autonomous Differential Equation and Graphing Its Slope Field
- Example 2: Using an Autonomous Differential Equation to Model Newton’s Law of Cooling
- Example 3: Using a Logistic Differential Equation to Model Population Growth
- Example 4: Using an Autonomous Differential Equation to Find Terminal Velocity
- Example 5: Using Euler's Method
8.4 Second-Order Linear Differential Equations
- Examples
- Example 1: Solving a Homogeneous Second-Order Linear Differential Equation
- Example 2: Solving a Homogeneous Second-Order Linear Differential Equation
- Example 3: Solving a Homogeneous Second-Order Linear Differential Equation
- Example 4: Solving a Second-Order Initial Value Problem Involving Simple Harmonic Motion
- Example 5: Solving a Second-Order Initial Value Problem Involving Damped Harmonic Motion
- Example 6: Solving Second-Order Boundary Value Problems
9.1 Parametric Equations
9.2 Calculus and Parametric Equations
- Examples
- Example 1: Identifying Points of Differentiability and Smoothness of a Parametrization
- Example 2: Finding Equations of Lines Tangent to a Parametric Curve
- Example 3: Finding the Inflection Points of a Parametric Curve
- Example 4: Finding the Area of the Region Bounded by a Parametric Curve
- Example 5: Using Parametric Arc Length to Derive the Formula for Circumference of a Circle
- Example 6: Finding the Arc Length of a Spiral
- Example 7: Finding the Arc Length of One Arch of a Cycloid
- Example 8: Finding the Surface Area of a Sphere Formed by Revolving a Parametric Curve
9.3 Polar Coordinates
- Examples
- Example 1: Plotting Points in Polar Coordinates
- Example 2: Converting from Polar to Cartesian Coordinates
- Example 3: Converting from Cartesian to Polar Coordinates
- Example 4: Rewriting a Rectangular Equation in Polar Form
- Example 5: Rewriting a Polar Equation in Rectangular Form
- Example 6: Graphing Polar Equations and Converting to Rectangular Coordinates
- Example 7: Graphing a Polar Equation
- Example 8: Graphing Common Polar Equations Using Symmetry
- Example 9: Graphing Common Polar Equations Using Symmetry
9.4 Calculus in Polar Coordinates
9.5 Conic Sections in Cartesian Coordinates
- Examples
- Example 1: Graphing an Ellipse and Finding Its Foci
- Example 2: Estimating the Closest Approach of Earth to the Sun
- Example 3: Finding the Focus and Directrix of a Parabola
- Example 4: Finding the Focus of a Parabolic Mirror
- Example 5: Finding the Foci, Vertices, and Asymptotes of a Hyperbola
- Example 6: Using Rotation Relations to Find x'y'-Coordinates
- Example 7: Using Rotation Relations to Convert an Equation to x'y'-Coordinates
- Example 8: Graphing a Rotated Conic Section
9.6 Conic Sections in Polar Coordinates
10.1 Sequences
- Examples
- Example 1: Using Sequence Notation
- Example 2: Defining Sequences with Explicit and Recursive Formulas
- Example 3: Using Newton's Method to Generate a Recursively Defined Sequence
- Example 4: Examining a Recursively Generated Sequence of Complex Numbers
- Example 5: Proving a Sequence Converges to a Limit or Diverges
- Example 6: Finding the Limit of a Sequence
- Example 7: Proving a Sequence Is Strictly Increasing
- Example 8: Using the Bounded Monotonic Sequence Theorem to Find the Limit of a Sequence
10.2 Infinite Series
- Examples
- Example 1: Determining Whether a Series Converges or Diverges
- Example 2: Finding the Total Vertical Distance Traveled by a Bouncing Ball
- Example 3: Finding the Total Time a Bouncing Ball Is in Motion
- Example 4: Expressing a Repeating Decimal as a Ratio of Two Integers
- Example 5: Showing That a Series Converges and Finding Its Sum
- Example 6: Showing That a Series Diverges
- Example 7: Determining Whether a Series Converges or Diverges
- Example 8: Finding the Sum of a Series
10.3 The Integral Test
10.4 Comparison Tests
10.5 The Ratio and Root Tests
10.6 Absolute and Conditional Convergence
10.7 Power Series
- Examples
- Example 1: Finding Values of x for Which a Power Series Converges
- Example 2: Finding Values of x for Which a Series Converges and the Corresponding Limiting Function
- Example 3: Finding the Radius of Convergence and Interval of Convergence of a Series
- Example 4: Differentiating a Power Series and Expressing the Result in Closed Form
- Example 5: Finding the Closed Form of a Series
- Example 6: Using the Multiplication of Power Series Theorem
10.8 Taylor and Maclaurin Series
- Examples
- Example 1: Finding the Taylor Series Expansion of a Function and Where It Converges
- Example 2: Finding the Maclaurin Polynomials Generated by a Function
- Example 3: Finding the Maclaurin Series Generated by a Function and Where It Converges
- Example 4: Finding Values of x for Which a Maclaurin Series Converges to Its Generating Function
- Example 5: Exploring a Maclaurin Series That Converges to the Desired Function at Only One Point
- Example 6: Finding the Maclaurin Series Expansion of Cosine
- Example 7: Using a Known Maclaurin Series to Find the Maclaurin Series Expansion of a Related Function
- Example 8: Approximating a Nonelementary Definite Integral
- Example 9: Finding a Taylor Series Expansion of the Natural Logarithm
- Example 10: Using Known Maclaurin and Taylor Series Expansions to Find Series Expansions of Related Functions
10.9 Further Applications of Series
- Examples
- Example 1: Using Euler's Formula
- Example 2: Using the Binomial Series to Find Series Expansions of Functions
- Example 3: Using the Binomial Series to Expand an Equation
- Example 4: Finding a Power Series Solution to a Differential Equation
- Example 5: Approximating a Sawtooth Function with Its Fourier Series
11.1 Three-Dimensional Cartesian Space
- Examples
- Example 1: Graphing in Cartesian Three-Dimensional Space
- Example 2: Determining Projections of Points and Objects onto Planes
- Example 3: Graphing an Equation in Cartesian Three-Dimensional Space
- Example 4: Finding the Distance between Points in Cartesian Three-Dimensional Space
- Example 5: Using the Distance Formula to Construct an Equation of a Sphere
11.2 Vectors and Vector Algebra
- Examples
- Example 1: Finding the Component Form and Length of a Vector
- Example 2: Using Vectors with Horizontal and Vertical Velocity
- Example 3: Performing Vector Algebra Using Vector Components
- Example 4: Graphing Vector Sums and Scalar Multiples of Vectors
- Example 5: Using Vector Algebra to Find Coordinates of Points
- Example 6: Using Standard Basis and Unit Vectors
- Example 7: Determining Tension Force Vectors Using Trigonometry
11.3 The Dot Product
- Examples
- Example 1: Evaluating Dot Products
- Example 2: Finding the Angle between Two Vectors
- Example 3: Finding an Unknown Component of Orthogonal Vectors
- Example 4: Describing a Set of Orthogonal Vectors
- Example 5: Finding the Direction Angles of a Vector
- Example 6: Decomposing a Vector into a Sum of Orthogonal Vectors
- Example 7: Using a Projection of a Vector to Calculate Force
- Example 8: Using Vectors to Find the Work Done in Pulling a Wagon
11.4 The Cross Product
- Examples
- Example 2: Using the Cross Product Determinant Formula
- Example 3: Constructing a Vector Normal to a Plane Defined by Three Points
- Example 4: Using the Cross Product Formula for Area of a Parallelogram
- Example 5: Using a Cross Product to Find the Area of a Triangle
- Example 6: Using the Triple Scalar Product Formula for Volume of a Parallelepiped
- Example 7: Using a Cross Product to Find the Torque Generated by a Wrench
11.5 Describing Lines and Planes
- Examples
- Example 1: Constructing Position Vectors to Describe a Line
- Example 2: Finding the Vector Form and Parametric Equations of a Line
- Example 3: Finding the Points of Intersection of a Line with the Coordinate Planes
- Example 4: Finding the Point of Intersection of Two Lines
- Example 5: Finding Equations for a Plane Given a Point and a Normal Vector
- Example 6: Describing a Plane as a Two-Parameter Set of Points
- Example 7: Finding an Equation for a Plane Given Three Points
- Example 8: Finding the Point of Intersection of a Line and a Plane
- Example 9: Finding Parametric Equations for a Line of Intersection of Two Planes
- Example 10: Finding the Shortest Distance between a Point and a Plane
- Example 11: Finding the Shortest Distance between Two Skew Lines
11.6 Cylinders and Quadric Surfaces
12.1 Vector-Valued Functions
- Examples
- Example 1: Understanding the Parameters of a Vector Function
- Example 2: Finding the Limit of a Vector Function
- Example 3: Describing a Space Curve in Terms of a Vector Function
- Example 4: Finding the Derivative of a Vector Function
- Example 5: Finding the Vector Form of an Equation for a Tangent Line
- Example 6: Proving a Constant-Magnitude Vector Function and Its Tangent Are Orthogonal
- Example 7: Using Integration to Find Velocity and Position Vector Functions
12.2 Arc Length and the Unit Tangent Vector
- Examples
- Example 1: Finding the Arc Length of a Space Curve
- Example 2: Parametrizing a Space Curve with Respect to Arc Length
- Example 3: Finding the Unit Tangent Vector for a Curve
- Example 4: Finding the Unit Tangent Vector for a Curve
- Example 5: Finding the Maximum Height of a Projectile
- Example 6: Finding the Distance Traveled by a Projectile and Its Speed at Impact
12.3 The Unit Normal and Binormal Vectors, Curvature, and Torsion
- Examples
- Example 1: Finding the Unit Tangent, Normal, and Binormal Vectors for a Curve
- Example 2: Determining Equations for the Osculating, Normal, and Rectifying Planes of a Curve
- Example 3: Finding Parametric Descriptions of a Ribbon and a Tube
- Example 4: Determining the Curvature of a Line and a Circle
- Example 5: Calculating the Curvature and Torsion Functions of a Helix
- Example 6: Finding the Tangential and Normal Components of Acceleration of an Object
12.4 Planetary Motion and Kepler's Laws
13.1 Functions of Several Variables
- Examples
- Example 1: Finding the Domain and Range of a Function of Two Variables
- Example 2: Identifying Bounded and Unbounded Regions
- Example 3: Writing an Equation of a Surface as a Function of x and y
- Example 4: Examining a Quadric Surface and Its Contour Map
- Example 5: Examining a Surface and Its Contour Map
- Example 6: Writing an Equation of a Surface as a Function of Two Variables Other than x and y
- Example 7: Visualizing the Level Surfaces of a Function of Three Variables
13.2 Limits and Continuity of Multivariable Functions
- Examples
- Example 1: Determining Whether a Limit Exists for a Function of Two Variables
- Example 2: Finding Limits of Functions of Two Variables
- Example 3: Finding a Limit of a Function of Two Variables
- Example 4: Determining Where Multivariable Functions Are Continuous
- Example 5: Defining the Continuous Extension of a Function of Two Variables
- Example 6: Examining the Continuity of Compositions of Multivariable Functions
13.3 Partial Derivatives
- Examples
- Example 1: Using Partial Derivatives to Find Equations of Lines Tangent to a Surface
- Example 2: Finding Partial Derivatives of a Function of Three Variables
- Example 3: Using Implicit Differentiation to Find Partial Derivatives
- Example 4: Using Partial Derivatives to Estimate Propagated Error and Relative Error
- Example 5: Finding Second-Order Partial Derivatives of a Function of Two Variables
- Example 6: Applying Clairaut‘s Theorem to Find a Mixed Partial Derivative
- Example 7: Finding Higher-Order Mixed Partial Derivatives
- Example 8: Verifying a Solution of a Partial Differential Equation
- Example 9: Examining Partial Derivatives and Continuity of a Function of Two Variables
13.4 The Chain Rule for Multivariable Functions
- Examples
- Example 1: Using the Chain Rule (Case I)
- Example 2: Finding the Rate of Change of a Multivariable Function along a Parametric Curve
- Example 3: Using a Tree Diagram to Apply the Chain Rule (Case II)
- Example 4: Using a Tree Diagram to Apply the Chain Rule
- Example 5: Using the Chain Rule to Find Second-Order Partial Derivatives
- Example 6: Using the Chain Rule for Implicit Differentiation
- Example 7: Finding Partial Derivatives of an Implicit Function
13.5 Directional Derivatives and Gradient Vectors
- Examples
- Example 1: Finding a Directional Derivative of a Function of Two Variables
- Example 2: Finding a Directional Derivative of a Function of Two Variables
- Example 3: Exploring the Properties of the Directional Derivative
- Example 4: Finding the Equation of a Line Tangent to an Implicitly Defined Curve
- Example 5: Using the Properties of the Gradient
- Example 6: Applying the Properties of the Directional Derivative to a Function of Three Variables
13.6 Tangent Planes and Differentials
- Examples
- Example 1: Determining the Tangent Plane and Normal Line at a Point on a Surface
- Example 2: Determining the Tangent Plane at a Point on a Surface
- Example 3: Finding Parametric Equations for a Line Tangent to a Smooth Curve on a Surface
- Example 4: Finding the Differential and Linearization of a Function of Two Variables
- Example 5: Finding an Upper Bound on Relative Error
- Example 6: Finding an Upper Bound on Relative Error
- Example 7: Linearizing a Function of Four Variables
13.7 Extreme Values of Functions of Two Variables
- Examples
- Example 1: Finding the Relative Extrema of a Function of Two Variables
- Example 2: Finding the Relative Extrema of a Function of Two Variables
- Example 3: Using the Second Partial Derivative Test
- Example 4: Using the Second Partial Derivative Test
- Example 5: Finding the Absolute Extrema of a Function of Two Variables
- Example 6: Maximizing the Volume of a Box
13.8 Lagrange Multipliers
- Examples
- Example 1: Using Lagrange Multipliers to Find Absolute Extreme Values
- Example 2: Using the Method of Lagrange Multipliers to Maximize the Volume of a Box
- Example 3: Finding the Location on a Plane Closest to a Point
- Example 4: Finding the Points on a Curve of Relative Minimum and Maximum Distance from the Origin
- Example 5: Using the Method of Lagrange Multipliers with Two Constraints
14.1 Double Integrals
- Examples
- Example 1: Using Riemann Sums to Estimate Volume under a Surface
- Example 2: Using Fubini's Theorem to Evaluate a Double Integral
- Example 3: Using Fubini's Theorem to Evaluate a Double Integral
- Example 4: Evaluating a Double Integral
- Example 5: Evaluating a Double Integral
- Example 6: Finding the Volume of a Tetrahedron
14.2 Applications of Double Integrals
- Examples
- Example 1: Using a Double Integral to Find the Area between Two Parabolas
- Example 2: Finding the Average Value of a Function over a Region
- Example 3: Finding the Center of Mass of a Plate of Varying Density
- Example 4: Finding Second Moments of a Disk
- Example 5: Finding the Moment of Inertia about the Origin of an Annulus
- Example 6: Finding Second Moments and Radii of Gyration of a Plate of Constant Density
14.3 Double Integrals in Polar Coordinates
- Examples
- Example 1: Finding the Area Bounded between the Graphs of Two Polar Equations
- Example 2: Using Polar Coordinates to Find the Moment of Inertia about the Origin of an Annulus
- Example 3: Evaluating a Double Integral by Converting from Cartesian to Polar Coordinates
- Example 4: Using a Double Integral in Polar Coordinates to Find Volume of a Solid
14.4 Triple Integrals
- Examples
- Example 1: Using Fubini's Theorem to Evaluate a Triple Integral
- Example 2: Finding the Volume of a Solid Bounded by Two Surfaces
- Example 3: Changing the Order of Integration in a Triple Integral
- Example 4: Finding the First and Second Moments, Center of Mass, and Radii of Gyration of a Tetrahedron
- Example 5: Finding the Centroid, Second Moment, and Radius of Gyration of a Solid
14.5 Triple Integrals in Cylindrical and Spherical Coordinates
- Examples
- Example 1: Using a Triple Integral in Cylindrical Coordinates to Find the Volume of a Solid
- Example 2: Using Cylindrical Coordinates to Find the Mass and Center of Mass of a Solid of Varying Density
- Example 3: Using Cylindrical Coordinates to Find the Mass of a Solid of Varying Density
- Example 4: Converting an Equation from Cartesian to Spherical Coordinates
- Example 5: Setting Up a Triple Integral in Spherical Coordinates
- Example 6: Using a Triple Integral in Spherical Coordinates to Find the Volume of a Solid
14.6 Substitutions and Multiple Integrals
- Examples
- Example 1: Using a Jacobian to Derive a Double Integral in Polar Coordinates
- Example 2: Using a Change of Variables to Evaluate a Double Integral
- Example 3: Using a Jacobian to Derive a Triple Integral in Spherical Coordinates
- Example 4: Using a Change of Variables to Evaluate a Triple Integral
15.1 Vector Fields
- Examples
- Example 1: Describing and Sketching a Vector Field
- Example 2: Describing and Sketching a Vector Field
- Example 3: Expressing Newton‘s Law of Universal Gravitation as a Gravitational Field
- Example 4: Expressing Coulomb‘s Law as an Electric Field
- Example 5: Finding the Gradient Vector Field of a Scalar Field
- Example 6: Showing That a Gravitational Field Is Conservative
15.2 Line Integrals
15.3 The Fundamental Theorem for Line Integrals
- Examples
- Example 1: Using the Fundamental Theorem for Line Integrals
- Example 2: Determining a Fluid‘s Flow along a Curve
- Example 3: Showing That a Vector Field Is Not Conservative
- Example 4: Showing That a Vector Field Is Not Conservative
- Example 5: Finding a Potential Function for a Vector Field
- Example 6: Using a Potential Function to Evaluate a Line Integral
- Example 7: Using a Potential Function to Evaluate a Line Integral
15.4 Green's Theorem
- Examples
- Example 1: Finding and Interpreting the Divergence of a Vector Field at a Point
- Example 2: Finding the Curl of a Vector Field
- Example 3: Using Green's Theorem to Evaluate a Line Integral
- Example 4: Using Green's Theorem (Tangential-Curl Form) to Evaluate a Line Integral
- Example 5: Evaluating the Outward Flux of a Vector Field across a Closed Path
- Example 6: Showing That Two Circulations Are Equivalent
15.5 Parametric Surfaces and Surface Area
- Examples
- Example 1: Describing a Surface with a Vector Function of Two Parameters
- Example 2: Describing a Sphere with a Vector Function of Two Parameters
- Example 3: Examining Grid Curves of a Parametric Surface
- Example 4: Describing a Plane with a Vector Function of Two Parameters
- Example 5: Constructing an Equation for a Plane Tangent to a Parametric Surface
- Example 6: Finding the Area of a Parametric Surface
- Example 7: Deriving the Formula for Surface Area of a Sphere Defined as a Parametric Surface
15.6 Surface Integrals
- Examples
- Example 1: Finding the Mass of a Surface
- Example 2: Finding the Center of Mass of a Surface
- Example 3: Evaluating the Integral of a Function over a Surface
- Example 4: Determining the Positively Oriented Field of Unit Normal Vectors
- Example 5: Evaluating the Flux of a Vector Field across a Surface
- Example 6: Finding the Flow of Heat across a Surface