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Browse Visuals and Modules
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P Area of a Circle as a Limit
1 Functions and Models
V1.4 Family of Functions
M1.5 Exponential Functions
M1.7A Parametric Curves
M1.7B Families of Cycloids
2 Limits and Derivatives
V2.1 Secant Line and Tangent
V2.6 Tangent Zoom
V2.8 Slope-a-Scope
M2.8 How do Coefficients Affect Graphs?
3 Differentiation Rules
V3.1 Slope-a-Scope (Exponential)
M3.3 The Dynamics of Linear Motion
V3.4 Slope-a-Scope (Trigonometric)
4 Applications of Differentiation
M4.3 Using Derivatives to Sketch f
V4.4 Family of Rational Functions
M4.6 Analyzing Optimization Problems
M4.8 Newton's Method
5 Integrals
V5.1 Area Under a Parabola
M5.2/5.9 Estimating Areas under Curves
V5.2 Integral with Riemann Sums
M5.4 Fundamental Theorem of Calculus
M5.10 Improper Integrals
6 Applications of Integration
V6.2A Approximating the Volume
V6.2B Volumes of Revolution
V6.2C A Solid With Triangular Slices
V6.3 Circumference as Limit of Polygons
7 Differential Equations
M7.2A Direction Fields and Solution Curves
M7.2B Euler's Method
M7.6 Predator Prey
8 Infinite Sequences and Series
M8.2 An Unusual Series and Its Sums
M8.7/8.9 Taylor and MacLaurin Series
9 Vectors and the Geometry of Space
V9.2 Adding Vectors
V9.3A The Dot Product of Two Vectors
V9.3B Vector Projections
V9.4 The Cross Product
M9.6A Traces of a Surface
M9.6B Quadric Surfaces
M9.7 Surfaces in Cyl. and Sph. Coords
10 Vector Functions
V10.1A Vector Functions and Space Curves
V10.1B The Twisted Cubic Curve
V10.1C Visualizing Space Curves
V10.2 Secant and Tangent Vectors
V10.3A The Unit Tangent Vector
V10.3B The TNB Frame
V10.3C Osculating Circle
V10.4 Velocity and Acceleration Vectors
V10.5 Grid Curves on Parametric Surface
M10.5 Families of Parametric Surfaces
11 Partial Derivatives
V11.1A Animated Level Curves
V11.1B Level Curves of a Surface
V11.2 Limit that Does Not Exist
V11.4 The Tangent Plane of a Surface
V11.6A Directional Derivatives
V11.6B Maximizing Directional Derivative
M11.7 Critical Points from Contour Maps
V11.7 Families of Surfaces
V11.8 Lagrange Multipliers
12 Multiple Integrals
V12.2 Fubini's Theorem
V12.7 Regions of Triple Integrals
V12.8 Region in Spherical Coordinates
13 Vector Calculus
V13.1 Vector Fields
V13.6 A Nonorientable Surface
Appendixes
D Precise Definitions of Limits
H Polar Curves
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
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 31
Exercise 39a
Exercise 39d
Exercise 46
Exercise 53
Exercise 65
1.4 Graphing Calculators and Computers
Exercise 8
Exercise 9
Exercise 21
Exercise 25
Exercise 27
Exercise 31
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 25
Exercise 43
Exercise 45
Exercise 49a
Exercise 49b
Exercise 59
1.7 Parametric Curves
Exercise 4
Exercise 7
Exercise 11
Exercise 17
Exercise 27
Exercise 29
Exercise 30
Exercise 37
Exercise 41
Exercise 43
2 Limits and Derivatives
2.1 The Tangent and Velocity Problems
Exercise 3
Exercise 5
Exercise 9
2.2 Limit of a Function
Exercise 4
Exercise 7
Exercise 11
Exercise 23
Exercise 28
2.3 Calculating Limits Using the Limit Law
Exercise 6
Exercise 13
Exercise 17
Exercise 27
Exercise 31
Exercise 37
Exercise 43
Exercise 45
2.4 Continuity
Exercise 3
Exercise 7
Exercise 11
Exercise 16
Exercise 21
Exercise 23
Exercise 26
Exercise 31
Exercise 33
Exercise 37
Exercise 43
Exercise 47
2.5 Limits Involving Infinity
Exercise 2
Exercise 7
Exercise 17
Exercise 21
Exercise 25
Exercise 35
Exercise 43
Exercise 45
2.6 Tangents, Velocities, and Other Rates of Change
Exercise 3
Exercise 7
Exercise 9
Exercise 11
Exercise 17
Exercise 21
Exercise 27
2.7 Derivatives
Exercise 3
Exercise 4
Exercise 5
Exercise 9
Exercise 15
Exercise 23
Exercise 27
Exercise 31
Exercise 35
2.8 The Derivative as a Function
Exercise 3b
Exercise 3c
Exercise 5
Exercise 11
Exercise 17a
Exercise 17b
Exercise 23
Exercise 25
Exercise 29
Exercise 31
Exercise 37
Exercise 43
Exercise 47
2.9 What Does f' Say about f'?
Exercise 3
Exercise 5
Exercise 9
Exercise 10
Exercise 11
Exercise 17
Exercise 25
3 Differentiation Rules
3.1 Derivatives of Polynomials and Exponential Functions
Exercise 19
Exercise 23
Exercise 25
Exercise 27
Exercise 38
Exercise 41
Exercise 44
Exercise 45
Exercise 53
Exercise 59
Exercise 61
Exercise 65
3.2 The Product and Quotient Rules
Exercise 9
Exercise 19
Exercise 21
Exercise 24
Exercise 28
Exercise 31
Exercise 33
Exercise 38
Exercise 41
Exercise 43
Exercise 45
3.3 Rates of Change in the Natural and Social Sciences
Exercise 13
Exercise 17
Exercise 19
Exercise 26
Exercise 29
Exercise 33
3.4 Derivatives of Trigonometric Function
Exercise 7
Exercise 18
Exercise 23
Exercise 27
Exercise 29
Exercise 33
Exercise 35
Exercise 37
Exercise 39
Exercise 41
Exercise 43
3.5 The Chain Rule
Exercise 5
Exercise 19
Exercise 29
Exercise 37
Exercise 41
Exercise 45
Exercise 49
Exercise 53
Exercise 57
Exercise 59
Exercise 66
Exercise 73a
Exercise 73b
Exercise 79
3.6 Implicit Differentiation
Exercise 11
Exercise 17
Exercise 25
Exercise 31
Exercise 41
Exercise 45
Exercise 49
Exercise 53a
Exercise 53b
Exercise 55
3.7 Derivatives of Logarithmic Functions
Exercise 17
Exercise 21
Exercise 25
Exercise 31
Exercise 38
Exercise 41
3.8 Linear Approximations and Differentials
Exercise 4
Exercise 7
Exercise 9
Exercise 13
Exercise 19
Exercise 25
Exercise 27
Exercise 31
Exercise 33
4 Applications of Differentiation
4.1 Related Rates
Exercise 9
Exercise 13
Exercise 17
Exercise 25
Exercise 29
Exercise 33a
Exercise 33b
Exercise 37
4.2 Maximum and Minimum Values
Exercise 9
Exercise 11
Exercise 13
Exercise 21
Exercise 31
Exercise 33
Exercise 39
Exercise 53
Exercise 60
4.3 Derivatives and the Shapes of Curves
Exercise 5
Exercise 13
Exercise 23
Exercise 35
Exercise 39
Exercise 49
Exercise 51
Exercise 53
Exercise 55
4.4 Graphing with Calculus and Calculators
Exercise 11
Exercise 21
Exercise 23
Exercise 24
Exercise 28
Exercise 29
4.5 Indeterminate Forms and l'Hospital's Rule
Exercise 1
Exercise 17
Exercise 29
Exercise 32
Exercise 37
Exercise 45
Exercise 49
Exercise 51
Exercise 55
Exercise 65
4.6 Optimization Problems
Exercise 9
Exercise 11
Exercise 13
Exercise 18
Exercise 19
Exercise 21
Exercise 31
Exercise 33
Exercise 37
Exercise 43
4.7 Applications to Business and Economics
Exercise 1
Exercise 5
Exercise 9
Exercise 15
4.8 Newton's Method
Exercise 4
Exercise 14
Exercise 23
Exercise 27
Exercise 29
Exercise 31
4.9 Antiderivatives
Exercise 7
Exercise 13
Exercise 15
Exercise 25
Exercise 31
Exercise 33
Exercise 39
Exercise 43
Exercise 49
Exercise 53
5 Integrals
5.1 Areas and Distances
Exercise 2
Exercise 5
Exercise 11
Exercise 15
Exercise 19
5.2 The Definite Integral
Exercise 5
Exercise 9
Exercise 19
Exercise 23
Exercise 31
Exercise 35
Exercise 41
Exercise 43
5.3 Evaluating Definite Integrals
Exercise 7
Exercise 13
Exercise 35
Exercise 38
Exercise 41
Exercise 46
Exercise 49
Exercise 55
Exercise 57
Exercise 68
5.4 The Fundamental Theorem of Calculus
Exercise 3
Exercise 9
Exercise 11
Exercise 15
Exercise 19
Exercise 24
Exercise 26
Exercise 27
5.5 The Substitution Rule
Exercise 3
Exercise 11
Exercise 13
Exercise 19
Exercise 23
Exercise 33
Exercise 45
Exercise 52
Exercise 53
Exercise 59
Exercise 63
Exercise 67
5.6 Integration by Parts
Exercise 3
Exercise 13
Exercise 16
Exercise 27
Exercise 37
Exercise 41
Exercise 44
5.7 Additional Techniques of Integration
Exercise 3
Exercise 6
Exercise 9
Exercise 19
Exercise 21
Exercise 31
5.8 Integration Using Tables and Computer Algebra Systems
Exercise 6
Exercise 11
Exercise 13
Exercise 17
Exercise 18
Exercise 21
Exercise 23
5.9 Approximate Integration
Exercise 1
Exercise 3
Exercise 4
Exercise 27
Exercise 29
Exercise 37
5.10 Improper Integrals
Exercise 1
Exercise 7
Exercise 13
Exercise 17
Exercise 25
Exercise 29
Exercise 35
Exercise 43
Exercise 49
Exercise 51
Exercise 57
6 Applications of Integration
6.1 More about Areas
Exercise 3
Exercise 9
Exercise 11
Exercise 22
Exercise 27
Exercise 31
Exercise 37
Exercise 43
6.2 Volumes
Exercise 5
Exercise 7
Exercise 9
Exercise 25
Exercise 27
Exercise 33
Exercise 39
Exercise 43
6.3 Arc Length
Exercise 5
Exercise 7
Exercise 17
Exercise 19
Exercise 23
Exercise 26
6.4 Average Value of a Function
Exercise 5
Exercise 9
Exercise 13
Exercise 19
6.5 Applications to Physics and Engineering
Exercise 5
Exercise 9
Exercise 13
Exercise 15
Exercise 19
Exercise 25
Exercise 27
Exercise 37
Exercise 39
6.6 Applications to Economics and Biology
Exercise 3
Exercise 5
Exercise 10
Exercise 15
6.7 Probability
Exercise 1
Exercise 5
Exercise 6
Exercise 11
7 Differential Equations
7.1 Modeling with Differential Equations
Exercise 3
Exercise 7
Exercise 9
Exercise 11
Exercise 13
7.2 Direction Fields and Euler's Method
Exercise 3
Exercise 11
Exercise 13
Exercise 18
Exercise 19
Exercise 21
Exercise 23
7.3 Separable Equations
Exercise 8
Exercise 9
Exercise 19
Exercise 25
Exercise 29
Exercise 33
Exercise 39
7.4 Exponential Growth and Decay
Exercise 3
Exercise 5
Exercise 9
Exercise 13
Exercise 19
Exercise 22
7.5 The Logistic Equation
Exercise 1
Exercise 3
Exercise 7
Exercise 9
Exercise 11
Exercise 13
7.6 Predator-Prey Systems
Exercise 1
Exercise 3
Exercise 5
Exercise 7
8 Infinite Sequences and Series
8.1 Sequences
Exercise 5
Exercise 9
Exercise 21
Exercise 25
Exercise 37
Exercise 41
Exercise 45
Exercise 47
8.2 Series
Exercise 3
Exercise 9
Exercise 15
Exercise 25
Exercise 27
Exercise 31
Exercise 35
Exercise 41
Exercise 45
Exercise 49
Exercise 53
Exercise 55
8.3 The Integral and Comparison Tests; Estimating Sums
Exercise 3
Exercise 11
Exercise 13
Exercise 15
Exercise 16
Exercise 18
Exercise 21
Exercise 25
Exercise 29
Exercise 35
Exercise 37
8.4 Other Convergence Tests
Exercise 3
Exercise 7
Exercise 11
Exercise 13
Exercise 19
Exercise 24
Exercise 25
Exercise 31
Exercise 33
Exercise 35
8.5 Power Series
Exercise 3
Exercise 7
Exercise 8
Exercise 13
Exercise 17
Exercise 18
Exercise 19
Exercise 25
8.6 Representations of Functions as Power Series
Exercise 6
Exercise 7
Exercise 11a
Exercise 11b
Exercise 13
Exercise 19
Exercise 21
Exercise 33
Exercise 35
8.7 Taylor and Maclaurin Series
Exercise 5
Exercise 11
Exercise 24
Exercise 29
Exercise 43
Exercise 45
Exercise 49
8.8 The Binomial Series
Exercise 3
Exercise 5
Exercise 9
Exercise 13
8.9 Applications of Taylor Polynomials
Exercise 3
Exercise 7
Exercise 15
Exercise 16
Exercise 21
Exercise 25
Exercise 27
9 Vectors and the Geometry of Space
9.1 Three-Dimensional Coordinate Systems
Exercise 5
Exercise 11
Exercise 17
Exercise 23
Exercise 27
Exercise 31
Exercise 35
9.2 Vectors
Exercise 3
Exercise 9
Exercise 19
Exercise 21
Exercise 23
Exercise 29
Exercise 33
Exercise 37
9.3 The Dot Product
Exercise 9
Exercise 15
Exercise 21
Exercise 27
Exercise 29
Exercise 35
Exercise 37
Exercise 43
9.4 The Cross Product
Exercise 1
Exercise 4
Exercise 9
Exercise 13
Exercise 17
Exercise 27
Exercise 29
Exercise 33
9.5 Equations of Lines and Planes
Exercise 5
Exercise 7
Exercise 11
Exercise 17
Exercise 25
Exercise 39
Exercise 51
9.6 Functions and Surfaces
Exercise 1
Exercise 5
Exercise 15
Exercise 25
9.7 Cylindrical and Spherical Coordinates
Exercise 5
Exercise 7
Exercise 13
Exercise 17
Exercise 23
Exercise 33
10 Vector Functions
10.1 Vector Functions and Space Curves
Exercise 11
Exercise 17
Exercise 19
Exercise 23
Exercise 33
Exercise 35
10.2 Derivatives and Integrals of Vector Functions
Exercise 1
Exercise 3
Exercise 13
Exercise 15
Exercise 21
Exercise 25
Exercise 45
10.3 Arc Length and Curvature
Exercise 3
Exercise 11
Exercise 25
Exercise 27
Exercise 31
Exercise 37
Exercise 43
Exercise 45
10.4 Motion in Space: Velocity and Acceleration
Exercise 11
Exercise 17
Exercise 20
Exercise 23
Exercise 33
10.5 Parametric Surfaces
Exercise 1
Exercise 13
Exercise 17
Exercise 21
Exercise 24
Exercise 29
11 Partial Derivatives
11.1 Functions of Several Variables
Exercise 1
Exercise 5
Exercise 11
Exercise 19
Exercise 31
Exercise 37
Exercise 41
11.2 Limits and Continuity
Exercise 7
Exercise 11
Exercise 17
Exercise 21
Exercise 24
Exercise 31
Exercise 33
11.3 Partial Derivatives
Exercise 1
Exercise 5
Exercise 7
Exercise 17
Exercise 27
Exercise 46
Exercise 61
Exercise 71
Exercise 76
Exercise 77
11.4 Tangent Planes and Linear Approximations
Exercise 9
Exercise 15
Exercise 23
Exercise 27
Exercise 33
Exercise 39
Exercise 41
11.5 The Chain Rule
Exercise 3
Exercise 7
Exercise 13
Exercise 26
Exercise 29
Exercise 33
Exercise 37
Exercise 39
11.6 Directional Derivatives and the Gradient Vector
Exercise 1
Exercise 11
Exercise 17
Exercise 23
Exercise 25
Exercise 29
Exercise 37
Exercise 47
Exercise 53
11.7 Maximum and Minimum Values
Exercise 1
Exercise 3
Exercise 11
Exercise 27
Exercise 35
Exercise 37
Exercise 45
11.8 Lagrange Multipliers
Exercise 1
Exercise 3
Exercise 11
Exercise 19
Exercise 23
Exercise 33
Exercise 43
12 Multiple Integrals
12.1 Double Integrals over Rectangles
Exercise 1
Exercise 7
Exercise 9a
Exercise 13
Exercise 17
12.2 Iterated Integrals
Exercise 3
Exercise 9
Exercise 13
Exercise 15
Exercise 19
Exercise 23
Exercise 31
12.3 Double Integrals over General Regions
Exercise 5
Exercise 11
Exercise 15
Exercise 19
Exercise 37
Exercise 39
Exercise 45
Exercise 50
12.4 Double Integrals in Polar Coordinates
Exercise 5
Exercise 13
Exercise 19
Exercise 23
Exercise 31
12.5 Applications of Double Integrals
Exercise 1
Exercise 5
Exercise 13
Exercise 21
Exercise 23
12.6 Surface Area
Exercise 3
Exercise 7
Exercise 11
Exercise 21
Exercise 23
12.7 Triple Integrals
Exercise 9
Exercise 17
Exercise 21
Exercise 25
Exercise 33
Exercise 37
Exercise 47
12.8 Triple Integrals in Cylindrical and Spherical Coordinates
Exercise 3
Exercise 7
Exercise 11
Exercise 17
Exercise 22
Exercise 27
12.9 Change of Variables in Multiple Integrals
Exercise 2
Exercise 5
Exercise 7
Exercise 13
Exercise 21
13 Vector Calculus
13.1 Vector Fields
Exercise 5
Exercise 11
Exercise 17
Exercise 23
Exercise 31
Exercise 35
13.2 Line Integrals
Exercise 3
Exercise 5
Exercise 9
Exercise 15
Exercise 19
Exercise 27
Exercise 33
Exercise 37
13.3 The Fundamental Theorem for Line Integrals
Exercise 9
Exercise 11
Exercise 15
Exercise 23
Exercise 27
Exercise 29
Exercise 33
13.4 Green's Theorem
Exercise 3
Exercise 9
Exercise 11
Exercise 17
Exercise 21
Exercise 27
13.5 Curl and Divergence
Exercise 1
Exercise 9
Exercise 13
Exercise 17
Exercise 19
Exercise 29
13.6 Surface Integrals
Exercise 4
Exercise 7
Exercise 15
Exercise 19
Exercise 25
Exercise 33
Exercise 41
13.7 Stokes' Theorem
Exercise 1
Exercise 5
Exercise 7
Exercise 15
Exercise 19
13.8 The Divergence Theorem
Exercise 1
Exercise 7
Exercise 9
Exercise 19
Exercise 25
Appendixes
Appendix D Precise Definitions of Limits
Exercise 15a
Appendix G Integration of Rational Functions by Partial Fractions
Exercise 5
Exercise 11
Exercise 17
Exercise 23
Exercise 27
Exercise 37
Appendix H.1 Polar Coordinates
Exercise 11
Exercise 13
Exercise 19
Exercise 29
Exercise 33
Exercise 41
Exercise 45
Exercise 49
Exercise 51
Exercise 55
Appendix H.2 Polar Coordinates
Exercise 7
Exercise 9
Exercise 17
Exercise 21
Exercise 25
Exercise 31
Exercise 37