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Calculus
Textbook answers
Questions
Review
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1. Review of Functions
1.1
Functions and Their Graphs
1.2
Trigonometric Functions
1.3
Other Special Functions
1.4
Inverse Functions
2. Limits
2.1
Rates of Change and Tangent Lines
2.2
The Definition of a Limit
2.3
Computing Limits with the Limits Laws
2.4
Continuity
2.5
One-Sided Limits
2.6
Limits Involving Infinity
3. Differentiation
3.1
The Derivative as Rate of Change
3.2
The Derivative at a Point
3.3
The Derivative as a Function
3.4
The Basic Rules of Differentiation
3.5
The Product and Quotient Rules
3.6
The Chain Rule
3.7
Derivatives of the Trigonometric Functions
3.8
Implicit Differentiation
3.9
Derivatives of Exponential and Logarithmic Functions
3.10
Derivatives of the Inverse Trigonometric Functions
4. Applications of the Derivative
4.1
Extrema for a Function
4.2
The Mean Value Theorem
4.3
First Derivatives and Increasing/Decreasing Functions
4.4
Second Derivatives and Concavity
4.5
Optimization Problems
4.6
Linear Approximation and Differentials
4.7
Newton's Method
4.8
Related Rates
4.9
L'Hopital's Rule
4.10
Antiderivatives
5. Integration
5.1
Estimating the Area under a Curve
5.2
The Definite Integral
5.3
The Indefinite Integral
5.4
The Fundamental Theorem of Calculus
6. Integration Techniques
6.1
The Basic Rules of Integration
6.2
Integration by Substitution
6.3
Integration by Parts
6.4
Integration of Trigonometric Functions
6.5
Integration by Trigonometric Substitution
6.6
Partial Fraction Decomposition
6.7
Integration Tables and Other Strategies
6.8
Numerical Integration and CAS Systems
6.9
Improper Integrals
7. Applications of the Integral
7.1
The Area Between Two Curves
7.2
Volumes: The Disk Method
7.3
Volumes: The Shell Method
7.4
Arc Length
7.5
Surface Area and Areas of Revolution
7.6
Net Change
7.7
Average Value
7.8
Applications in Science
7.9
Introduction to Probability
8. Ordinary Differential Equations
8.1
Introduction to Differential Equations: Slope Fields and Euler's Method
8.2
Exponential Growth and Decay
8.3
Separable Differential Equations
8.4
The Logistic Equation
8.5
First Order Linear Differential Equations
9. Sequences and Series
9.1
Sequences
9.2
Infinite Series and the nth Term Test
9.3
The Integral and p-Series Tests
9.4
The Comparison Tests
9.5
The Root and Ratio Tests
9.6
Alternating Series and Absolute Convergence
9.7
Polynomial Approximations of Functions
9.8
Power Series
9.9
Taylor Series
9.10
Convergence of Taylor Series
10. Parametrized Functions and Polar Coordinates
10.1
Parametric Equations of Plane Curves
10.2
Tangents, Arc Length, and Surface Area of Parametrized Regions
10.3
The Polar Coordinate System
10.4
Arc Length and Area in Polar Coordinates
10.5
Conic Sections
11. Vectors and the Geometry of Space
11.1
Vectors in the Plane
11.2
Vectors in 3 Dimensional Space
11.3
Spherical and Cylindrical Coordinates in 3D
11.4
The Dot Product
11.5
The Cross Product
11.6
Lines, Curves, and Planes
11.7
Surfaces
12. Vector Functions
12.1
Vector Functions
12.2
Differentiation and Integration of Vector Functions
12.3
Particle Motion in Space
12.4
Arc Length
12.5
Curvature
13. Differentiation of Functions of Several Variables
13.1
Functions of Several Variables
13.2
Limits and Continuity in Higher Dimensions
13.3
Partial Derivatives
13.4
Differentials and the Tangent Plane
13.5
The Chain Rule for Functions of Several Variables
13.6
Directional Derivatives and the Gradient
13.7
Extrema on a Surface
13.8
Lagrange Multipliers
14. Multiple Integration
14.1
Double Integrals over Rectangular Regions
14.2
Double Integrals over Arbitrary Regions
14.3
Double Integrals in Polar Coordinates
14.4
Surface Area
14.5
Triple Integrals
14.6
Triple Integrals in Spherical and Cylindrical Coordinates
14.7
Centroids and Moments of Inertia
14.8
Change of Variables in Multiple Integrals
15. Vector Analysis
15.1
Vector Fields
15.2
Line Integrals
15.3
Conservative Vector Fields and the Fundamental Theorem for Line Integrals
15.4
Green's Theorem
15.5
Parametrized Surfaces
15.6
Surface Integrals
15.7
Divergence and Curl
15.8
Stokes' Theorem
15.9
The Divergence Theorem
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