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Browse Visuals and Modules
1 Functions and Models
V1.4 Family of Functions
M1.5 Exponential Functions
2 Limits and Derivatives
V2.1 Secant Line and Tangent
M2.4/2.6 PreciseDefinitionsOfLimits
V2.7 Tangent Zoom
V2.8 Slope-a-Scope
M2.8 How do Coefficients Affect Graphs?
3 Differentiation Rules
V3.1 Slope-a-Scope (Exponential)
V3.3 Slope-a-Scope (Trigonometric)
M3.7 The Dynamics of Linear Motion
4 Applications of Differentiation
M4.3 Using Derivatives to Sketch f
V4.6 Family of Rational Functions
M4.7 Analyzing Optimization Problems
M4.8 Newton's Method
5 Integrals
V5.1 Area Under a Parabola
M5.2/7.7 Estimating Areas under Curves
V5.2 Integral with Riemann Sums
M5.3 Fundamental Theorem of Calculus
6 Applications of Integration
V6.2A Approximating the Volume
V6.2B Volumes of Revolution
V6.2C A Solid With Triangular Slices
7 Techniques of Integrations
M5.2/7.7 Estimating Areas under Curves
M7.8 Improper Integrals
8 Further Applications of Integration
V8.1 Circumference as Limit of Polygons
9 Differential Equations
M9.2A Direction Fields and Solution Curves
M9.2B Euler's Method
M9.4 Predator Prey
10 Parametric Equations and Polar Coordinates
M10.1A Parametric Curves
M10.1B Families of Cycloids
M10.3 Polar Curves
11 Infinite Sequences and Series
M11.2 An Unusual Series and Its Sums
M11.10/11.11 Taylor and MacLaurin Series
12 Vectors and the Geometry of Space
V12.2 Adding Vectors
V12.3A The Dot Product of Two Vectors
V12.3B Vector Projections
V12.4 The Cross Product
M12.6A Traces of a Surface
M12.6B Quadric Surfaces
13 Vector Functions
V13.1A Vector Functions and Space Curves
V13.1B The Twisted Cubic Curve
V13.1C Visualizing Space Curves
V13.2 Secant and Tangent Vectors
V13.3A The Unit Tangent Vector
V13.3B The TNB Frame
V13.3C Osculating Circle
V13.4 Velocity and Acceleration Vectors
14 Partial Derivatives
V14.1A Animated Level Curves
V14.1B Level Curves of a Surface
V14.2 Limit that Does Not Exist
V14.4 The Tangent Plane of a Surface
V14.6A Directional Derivatives
V14.6B Maximizing Directional Derivative
M14.7 Critical Points from Contour Maps
V14.7 Families of Surfaces
V14.8 Lagrange Multipliers
15 Multiple Integrals
V15.2 Fubini's Theorem
V15.6 Regions of Triple Integrals
M15.8 Surfaces in Cyl. and Sph. Coords
V15.8 Region in Spherical Coordinates
16 Vector Calculus
V16.1 Vector Fields
V16.6 Grid Curves on Parametric Surface
M16.6 Families of Parametric Surfaces
V16.7 A Nonorientable Surface
Browse Homework Hints
1 Functions and Models
1.1 Four Ways to Represent a Function
Exercise 2
Exercise 9
Exercise 11
Exercise 13
Exercise 23
Exercise 31
Exercise 39
Exercise 43
Exercise 47
Exercise 55
Exercise 59
1.2 Mathematical Models: A Catalog of Essential Functions
Exercise 3
Exercise 5
Exercise 15
Exercise 17
1.3 New Functions from Old Functions
Exercise 1a
Exercise 1d
Exercise 1h
Exercise 5a
Exercise 5d
Exercise 7
Exercise 15
Exercise 27a
Exercise 27c
Exercise 29
Exercise 35a
Exercise 35d
Exercise 46
Exercise 53
Exercise 65
1.4 Graphing Calculators and Computers
Exercise 8
Exercise 9
Exercise 23
Exercise 27
Exercise 29
Exercise 33
1.5 Exponential Functions
Exercise 5
Exercise 9
Exercise 11
Exercise 13
Exercise 17
Exercise 23
1.6 Inverse Functions and Logarithms
Exercise 3
Exercise 8
Exercise 13
Exercise 17
Exercise 19
Exercise 22
Exercise 23
Exercise 25
Exercise 43
Exercise 45
Exercise 49a
Exercise 49b
Exercise 67
2 Limits and Derivatives
2.1 The Tangent and Velocity Problems
Exercise 3
Exercise 5
Exercise 9
2.2 The Limit of a Function
Exercise 4
Exercise 11
Exercise 15
Exercise 27
Exercise 35
Exercise 42
2.3 Calculating Limits Using the Limit Laws
Exercise 8
Exercise 15
Exercise 19
Exercise 20
Exercise 35
Exercise 39
Exercise 49
Exercise 58
Exercise 61
2.4 The Precise Definition of a Limit
Exercise 3
Exercise 17
Exercise 25
Exercise 29
Exercise 31
Exercise 37
Exercise 43
2.5
Exercise 3
Exercise 7
Exercise 11
Exercise 18
Exercise 27
Exercise 29
Exercise 32
Exercise 39
Exercise 41
Exercise 47
Exercise 53
Exercise 61
2.6 Limits at Infinity; Horizontal Asymptotes
Exercise 2
Exercise 7
Exercise 19
Exercise 25
Exercise 41
Exercise 55
Exercise 57
Exercise 65a
Exercise 65b
2.7 Derivatives and Rates of Change
Exercise 5
Exercise 7
Exercise 9
Exercise 13
Exercise 17
Exercise 18
Exercise 19
Exercise 23
Exercise 27
Exercise 35
Exercise 39
Exercise 43
Exercise 45
Exercise 47
Exercise 51
2.8 The Derivative as a Function
Exercise 3b
Exercise 3c
Exercise 5
Exercise 11
Exercise 17a
Exercise 17b
Exercise 25
Exercise 27
Exercise 33
Exercise 35
Exercise 41
Exercise 47
Exercise 51
Exercise 53
3 Differentiation Rules
3.1 Derivatives of Polynomials and Exponential Functions
Exercise 23
Exercise 31
Exercise 35
Exercise 46
Exercise 49
Exercise 51
Exercise 59
Exercise 73
Exercise 77
3.2 The Product and Quotient Rules
Exercise 11
Exercise 25
Exercise 33
Exercise 36
Exercise 40
Exercise 43
Exercise 45
Exercise 50
Exercise 51
Exercise 55
3.3 Derivatives of Trigonometric Functions
Exercise 9
Exercise 22
Exercise 29
Exercise 33
Exercise 37
Exercise 41
Exercise 45
Exercise 51
3.4 The Chain Rule
Exercise 5
Exercise 19
Exercise 23
Exercise 37
Exercise 55
Exercise 59
Exercise 61
Exercise 65
Exercise 67
Exercise 75
Exercise 77
Exercise 84
Exercise 93
3.5 Implicit Differentiation
Exercise 15
Exercise 27
Exercise 39
Exercise 47
Exercise 61
Exercise 63
Exercise 67a
Exercise 67b
3.6 Derivatives of Logarithmic Functions
Exercise 19
Exercise 27
Exercise 41
Exercise 43
Exercise 50
Exercise 53
3.7 Rates of Change in the Natural and Social Sciences
Exercise 1g
Exercise 1h
Exercise 1i
Exercise 15
Exercise 19
Exercise 21
Exercise 28
Exercise 31
Exercise 35
3.8 Exponential Growth and Decay
Exercise 3
Exercise 5
Exercise 9
Exercise 13
Exercise 19
3.9 Related Rates
Exercise 12
Exercise 15
Exercise 19
Exercise 25
Exercise 27
Exercise 31
Exercise 37a
Exercise 37b
Exercise 43
3.10 Linear Approximations and Differentials
Exercise 3
Exercise 5
Exercise 9
Exercise 13
Exercise 15
Exercise 29
Exercise 33
Exercise 40
Exercise 43
3.11 Hyperbolic Functions
Exercise 9
Exercise 15
Exercise 17
Exercise 35
Exercise 45
Exercise 51
Exercise 53
4 Applications of Differentiation
4.1 Maximum and Minimum Values
Exercise 9
Exercise 11
Exercise 13
Exercise 25
Exercise 39
Exercise 41
Exercise 49
Exercise 67
Exercise 78
4.2 The Mean Value Theorem
Exercise 5
Exercise 11
Exercise 19
Exercise 23
Exercise 25
Exercise 35
4.3 How Derivatives Affect the Shape of a Graph
Exercise 5a
Exercise 5b
Exercise 7
Exercise 11a
Exercise 11b
Exercise 11c
Exercise 17
Exercise 25
Exercise 31
Exercise 39
Exercise 41a
Exercise 41b
Exercise 41c
Exercise 51
Exercise 51d
Exercise 55
Exercise 63
Exercise 64
Exercise 67
Exercise 75
4.4 Indeterminate Forms and L'Hospital's Rule
Exercise 1
Exercise 21
Exercise 29
Exercise 43
Exercise 48
Exercise 55
Exercise 69
Exercise 81
4.5 Summary of Curve Sketching
Exercise 5
Exercise 9
Exercise 17
Exercise 19
Exercise 33
Exercise 41
Exercise 57
4.6 Graphing with Calculus and Calculators
Exercise 11
Exercise 13
Exercise 23
Exercise 30
Exercise 31
Exercise 34
4.7 Optimization Problems
Exercise 13
Exercise 16
Exercise 17
Exercise 19
Exercise 22
Exercise 30
Exercise 33
Exercise 49
Exercise 50
Exercise 53
Exercise 55
Exercise 63
Exercise 69
4.8 Newton's Method
Exercise 4
Exercise 24
Exercise 31
Exercise 37
Exercise 41
4.9 Antiderivatives
Exercise 13
Exercise 21
Exercise 23
Exercise 39
Exercise 49
Exercise 53
Exercise 57
Exercise 65
Exercise 73
5 Integrals
5.1 Areas and Distances
Exercise 2
Exercise 5
Exercise 11
Exercise 15
Exercise 21
5.2 The Definite Integral
Exercise 5
Exercise 9
Exercise 19
Exercise 23
Exercise 33
Exercise 37
Exercise 47
Exercise 49
Exercise 53
5.3 The Fundamental Theorem of Calculus
Exercise 3
Exercise 9
Exercise 13
Exercise 17
Exercise 51
Exercise 53
Exercise 62
Exercise 63
Exercise 72
Exercise 73
Exercise 74
5.4 Indefinite Integrals and the Net Change Theorem
Exercise 2
Exercise 9
Exercise 23
Exercise 31
Exercise 43
Exercise 48
Exercise 51
Exercise 57
Exercise 59
5.5 The Substitution Rule
Exercise 3
Exercise 13
Exercise 19
Exercise 21
Exercise 25
Exercise 33
Exercise 43
Exercise 59
Exercise 64
Exercise 67
Exercise 75
Exercise 81
Exercise 85
6 Applications of Integration
6.1 Areas between Curves
Exercise 3
Exercise 9
Exercise 13
Exercise 21
Exercise 29
Exercise 45
Exercise 51
Exercise 53
6.2 Volumes
Exercise 7
Exercise 9
Exercise 11
Exercise 43
Exercise 49
Exercise 51
Exercise 57
Exercise 63
Exercise 67
6.3 Volumes by Cylindrical Shells
Exercise 5
Exercise 13
Exercise 17
Exercise 25
Exercise 29
Exercise 41
Exercise 45
6.4 Work
Exercise 7
Exercise 9
Exercise 13
Exercise 17
Exercise 19
Exercise 27
6.5 Average Value of a Function
Exercise 7
Exercise 9
Exercise 13
Exercise 17
Exercise 23
7 Techniques of Integrations
7.1 Integration by Parts
Exercise 3
Exercise 15
Exercise 17
Exercise 20
Exercise 35
Exercise 47
Exercise 57
Exercise 63
Exercise 66
7.2 Trigonometric Integrals
Exercise 3
Exercise 7
Exercise 13
Exercise 23
Exercise 29
Exercise 43
Exercise 55
Exercise 61
7.3 Trigonometric Substitution
Exercise 3
Exercise 7
Exercise 13
Exercise 17
Exercise 22
Exercise 31a
Exercise 31b
7.4 Integration of Rational Functions by Partial Fractions
Exercise 5
Exercise 11
Exercise 17
Exercise 25
Exercise 29
Exercise 31
Exercise 43
Exercise 47
Exercise 55
7.5 Strategy for Integration
Exercise 7
Exercise 17
Exercise 23
Exercise 31
Exercise 41
Exercise 45
Exercise 49
Exercise 57
Exercise 61
Exercise 69
7.6 Integration Using Tables and Computer Algebra Systems
Exercise 10
Exercise 17
Exercise 19
Exercise 26
Exercise 27
Exercise 29
Exercise 33
7.7 Approximate Integration
Exercise 1
Exercise 3
Exercise 4
Exercise 33
Exercise 35
Exercise 45
7.8 Improper Integrals
Exercise 1
Exercise 7
Exercise 13
Exercise 21
Exercise 29
Exercise 31
Exercise 43
Exercise 49
Exercise 57
Exercise 61
Exercise 69
8 Further Applications of Integration
8.1 Arc Length
Exercise 7
Exercise 11
Exercise 13
Exercise 31
Exercise 33
Exercise 39
Exercise 42
8.2 Area of a Surface of Revolution
Exercise 1a
Exercise 1b
Exercise 5
Exercise 11
Exercise 15
Exercise 25
Exercise 31
8.3 Applications to Physics and Engineering
Exercise 7
Exercise 13
Exercise 27
Exercise 31
Exercise 41
8.4 Applications to Economics and Biology
Exercise 3
Exercise 5
Exercise 10
Exercise 17
8.5 Probability
Exercise 1
Exercise 7
Exercise 8
Exercise 13
9 Differential Equations
9.1 Modeling with Differential Equations
Exercise 3
Exercise 7
Exercise 9
Exercise 11
Exercise 13
9.2 Direction Fields and Eulerís Method
Exercise 3
Exercise 11
Exercise 13
Exercise 18
Exercise 19
Exercise 21
Exercise 23
9.3 Separable Equations
Exercise 10
Exercise 15
Exercise 25
Exercise 31
Exercise 35
Exercise 39
Exercise 45
9.4 Models for Population Growth
Exercise 1
Exercise 3
Exercise 7
Exercise 9
Exercise 14
Exercise 15
Exercise 17
9.5 Linear Equations
Exercise 5
Exercise 9
Exercise 19
Exercise 25
Exercise 31
Exercise 33
9.6 Predator-Prey Systems
Exercise 1
Exercise 3
Exercise 5
Exercise 7
10 Parametric Equations and Polar Coordinates
10.1 Curves Defined by Parametric Equations
Exercise 4
Exercise 9
Exercise 13
Exercise 21
Exercise 31
Exercise 33
Exercise 34
Exercise 41
Exercise 47
Exercise 49
10.2 Calculus with Parametric Curves
Exercise 5
Exercise 11
Exercise 23
Exercise 25
Exercise 31
Exercise 41
Exercise 45
Exercise 61
Exercise 65
10.3 Polar Coordinates
Exercise 11
Exercise 17
Exercise 25
Exercise 35
Exercise 39
Exercise 49
Exercise 55
Exercise 59
Exercise 63
Exercise 69
10.4 Areas and Lengths in Polar Coordinates
Exercise 7
Exercise 11
Exercise 21
Exercise 27
Exercise 31
Exercise 41
Exercise 47
10.5 Conic Sections
Exercise 5
Exercise 15
Exercise 19
Exercise 27
Exercise 33
Exercise 37
Exercise 47
10.6 Conic Sections in Polar Coordinates
Exercise 3
Exercise 13
Exercise 13b
Exercise 13c
Exercise 13d
Exercise 21
Exercise 27
11 Infinite Sequences and Series
11.1 Sequences
Exercise 13
Exercise 19
Exercise 35
Exercise 36
Exercise 43
Exercise 54
Exercise 59
Exercise 61
Exercise 69
11.2 Series
Exercise 3
Exercise 9
Exercise 17
Exercise 31
Exercise 35
Exercise 41
Exercise 47
Exercise 55
Exercise 59
Exercise 65
Exercise 71
Exercise 73
11.3 The Integral Test and Estimates of Sums
Exercise 7
Exercise 11
Exercise 17
Exercise 21
Exercise 33
Exercise 39
11.4 The Comparison Tests
Exercise 1
Exercise 8
Exercise 10
Exercise 17
Exercise 31
Exercise 37
Exercise 41
11.5 Alternating Series
Exercise 3
Exercise 7
Exercise 11
Exercise 13
Exercise 17
Exercise 23
Exercise 32
11.6 Absolute Convergence and the Ratio and Root Tests
Exercise 4
Exercise 13
Exercise 19
Exercise 21
Exercise 29
Exercise 31
Exercise 33
11.8 Power Series
Exercise 3
Exercise 7
Exercise 15
Exercise 23
Exercise 24
Exercise 29
Exercise 37
11.9 Representations of Functions as Power Series
Exercise 5
Exercise 8
Exercise 13a
Exercise 13b
Exercise 15
Exercise 21
Exercise 23
Exercise 35
Exercise 37
11.10 Taylor and Maclaurin Series
Exercise 5
Exercise 15
Exercise 27
Exercise 33
Exercise 35
Exercise 39
Exercise 45
Exercise 57
Exercise 59
Exercise 63
11.11 Applications of Taylor Polynomials
Exercise 5
Exercise 9
Exercise 18
Exercise 19
Exercise 25
Exercise 31
Exercise 33
12 Vectors and the Geometry of Space
12.1 Three-Dimensional Coordinate Systems
Exercise 5
Exercise 13
Exercise 21
Exercise 27
Exercise 31
Exercise 35
Exercise 39
12.2 Vectors
Exercise 3
Exercise 11
Exercise 23
Exercise 25
Exercise 39
Exercise 41
Exercise 45
12.3 The Dot Product
Exercise 19
Exercise 27
Exercise 41
Exercise 43
Exercise 49
Exercise 51
Exercise 57
12.4 The Cross Product
Exercise 7
Exercise 13
Exercise 16
Exercise 19
Exercise 29
Exercise 31
Exercise 43
Exercise 45
Exercise 49
12.5 Equations of Lines and Planes
Exercise 5
Exercise 9
Exercise 13
Exercise 19
Exercise 31
Exercise 49
Exercise 61
Exercise 73
12.6 Cylinders and Quadric Surfaces
Exercise 9
Exercise 19
13 Vector Functions
13.1 Vector Functions and Space Curves
Exercise 13
Exercise 19
Exercise 25
Exercise 39
13.2 Derivatives and Integrals of Vector Functions
Exercise 1
Exercise 3
Exercise 15
Exercise 19
Exercise 25
Exercise 49
13.3 Arc Length and Curvature
Exercise 3
Exercise 5
Exercise 17
Exercise 31
Exercise 33
Exercise 43
Exercise 49
Exercise 51
13.4 Motion in Space: Velocity and Acceleration
Exercise 11
Exercise 19
Exercise 22
Exercise 25
Exercise 35
14 Partial Derivatives
14.1 Functions of Several Variables
Exercise 1
Exercise 5
Exercise 13
Exercise 17
Exercise 23
Exercise 33
Exercise 43
Exercise 55
Exercise 61
Exercise 65
14.2 Limits and Continuity
Exercise 9
Exercise 13
Exercise 21
Exercise 25
Exercise 28
Exercise 37
Exercise 39
14.3 Partial Derivatives
Exercise 1
Exercise 5a
Exercise 5b
Exercise 21
Exercise 31
Exercise 50
Exercise 69
Exercise 81
Exercise 86
Exercise 87
14.4 Tangent Planes and Linear Approximations
Exercise 11
Exercise 21
Exercise 31
Exercise 35
Exercise 37
Exercise 43
Exercise 45
14.5 The Chain Rule
Exercise 11
Exercise 17
Exercise 32
Exercise 35
Exercise 39
Exercise 45
Exercise 47
14.6 Directional Derivatives and the Gradient Vector
Exercise 11
Exercise 19
Exercise 23
Exercise 27
Exercise 29
Exercise 33
Exercise 43
Exercise 57
Exercise 63
14.7 Maximum and Minimum Values
Exercise 1
Exercise 3
Exercise 13
Exercise 31
Exercise 41
Exercise 43
Exercise 51
14.8 Lagrange Multipliers
Exercise 1
Exercise 3
Exercise 11
Exercise 19
Exercise 25
Exercise 35
Exercise 45
15 Multiple Integrals
15.1 Double Integrals over Rectangles
Exercise 1
Exercise 7
Exercise 9a
Exercise 9b
Exercise 13
Exercise 17
15.2 Iterated Integrals
Exercise 3
Exercise 9
Exercise 17
Exercise 19
Exercise 23
Exercise 27
Exercise 35
15.3 Double Integrals over General Regions
Exercise 5
Exercise 13
Exercise 17
Exercise 21
Exercise 43
Exercise 45
Exercise 51
Exercise 58
15.4 Double Integrals in Polar Coordinates
Exercise 1
Exercise 11
Exercise 13
Exercise 15
Exercise 25
Exercise 35
15.5 Applications of Double Integrals
Exercise 1
Exercise 5
Exercise 15
Exercise 27
Exercise 29
15.6 Triple Integrals
Exercise 11
Exercise 19
Exercise 23
Exercise 27
Exercise 35
Exercise 39
15.7 Triple Integrals in Cylindrical Coordinates
Exercise 3
Exercise 9
Exercise 17
Exercise 21
15.8 Triple Integrals in Spherical Coordinates
Exercise 1
Exercise 5
Exercise 17
Exercise 21
Exercise 30
Exercise 35
15.9 Change of Variables in Multiple Integrals
Exercise 7
Exercise 13
Exercise 21
16 Vector Calculus
16.1 Vector Fields
Exercise 5
Exercise 11
Exercise 17
Exercise 23
Exercise 29
Exercise 35
16.2 Line Integrals
Exercise 3
Exercise 7
Exercise 11
Exercise 17
Exercise 21
Exercise 33
Exercise 39
Exercise 43
16.3 The Fundamental Theorem for Line Integrals
Exercise 7
Exercise 11
Exercise 15
Exercise 23
Exercise 27
Exercise 29
Exercise 33
16.4 Green's Theorem
Exercise 3
Exercise 7
Exercise 9
Exercise 17
Exercise 21
Exercise 27
16.5 Curl and Divergence
Exercise 1
Exercise 11
Exercise 15
Exercise 19
Exercise 21
Exercise 31
16.6 Parametric Surfaces and Their Areas
Exercise 3
Exercise 13
Exercise 19
Exercise 23
Exercise 26
Exercise 33
Exercise 37
Exercise 41
Exercise 47
Exercise 55
Exercise 57
16.7 Surface Integrals
Exercise 4
Exercise 5
Exercise 15
Exercise 19
Exercise 25
Exercise 37
Exercise 45
16.8 Stokes' Theorem
Exercise 5
Exercise 7
Exercise 15
Exercise 19
16.9 The Divergence Theorem
Exercise 1
Exercise 7
Exercise 9
Exercise 19
Exercise 25
17 Second-Order Differential Equations
17.1 Second-Order Linear Equations
Exercise 1
Exercise 9
Exercise 11
Exercise 17
Exercise 21
Exercise 23
Exercise 30
17.2 Nonhomogeneous Linear Equations
Exercise 5
Exercise 9
Exercise 16
Exercise 18
Exercise 21a
Exercise 21b
Exercise 25
17.3 Applications of Second-Order Differential Equations
Exercise 3
Exercise 9
Exercise 13
Exercise 17
17.4 Series Solutions
Exercise 3
Exercise 9
Appendixes
Appendix G Appendix G
Exercise 1
Exercise 3
Exercise 5