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(Practice Exercises, Additional Exercises, and Questions to Guide Your Review appear at the end of each chapter.)
Functions and Graphs.
Inverse Functions and Logarithms.
Trigonometric Functions and Their Inverses.
1. Limits and Continuity.
Rates of Change and Limits.
Finding Limits and One-Sided Limits.
Limits Involving Infinity.
The Derivative as a Function.
The Derivative as a Rate of Change.
Derivatives of Products, Quotients, and Negative Powers.
Derivatives of Trigonometric Functions.
The Chain Rule and Parametric Equations.
Derivatives of Inverse Trigonometric Functions.
Derivatives of Exponential and Logarithmic Functions.
3. Applications of Derivatives.
Extreme Values of Functions.
The Mean Value Theorem and Differential Equations.
The Shape of a Graph.
Graphical Solutions of Autonomous Differential Equations.
Modeling and Optimization.
Linearization and Differentials.
Indefinite Integrals, Differential Equations, and Modeling.
Integral Rules; Integration by Substitution.
Estimating with Finite Sums.
Riemann Sums and Definite Integrals.
The Mean Value and Fundamental Theorems.
Substitution in Definite Integrals.
5. Applications of Integrals.
Volumes by Slicing andRotation About an Axis.
Modeling Volume Using Cylindrical Shells.
Lengths of Plane Curves.
First-Order Separable Differential Equations.
Springs, Pumping, and Lifting.
Moments and Centers of Mass.
6. Transcendental Functions and Differential Equations.
Linear First-Order Differential Equations.
Euler's Method; Population Models.
7. Integration Techniques, L'Hôpital's Rule, and Improper Integrals.
Basic Integration Formulas.
Integration by Parts.
Integral Tables, Computer Algebra Systems, and Monte Carlo Integration.
8. Infinite Series.
Limits of Sequences of Numbers.
Subsequences, Bounded Sequences, and Picard's Method.
Series of Nonnegative Terms.
Alternating Series, Absolute and Conditional Convergence.
Taylor and Maclaurin Series.
Applications of Power Series.
Fourier Cosine and Sine Series.
9. Vectors in the Plane and Polar Functions.
Vectors in the Plane.
Modeling Projectile Motion.
Polar Coordinates and Graphs.
Calculus of Polar Curves.
10. Vectors and Motion in Space.
Cartesian (Rectangular) Coordinates and Vectors in Space.
Dot and Cross Products.
Lines and Planes in Space.
Cylinders and Quadratic Surfaces.
Vector-Valued Functions and Space Curves.
Arc Length and the Unit Tangent Vector T.
The TNB Frame; Tangential and Normal Components of Acceleration.
Planetary Motion and Satellites.
11. Multivariable Functions and Their Derivatives.
Functions of Several Variables.
Limits and Continuity in Higher Dimensions.
The Chain Rule.
Directional Derivatives, Gradient Vectors, and Tangent Planes.
Linearization and Differentials.
Extreme Values and Saddle Points.
Partial Derivatives with Constrained Variables.
Taylor's Formula for Two Variables.
12. Multiple Integrals.
Areas, Moments, and Centers of Mass.
Double Integrals in Polar Form.
Triple Integrals in Rectangular Coordinates.
Masses and Moments in Three Dimensions.
Triple Integrals in Cylindrical and Spherical Coordinates.
Substitutions in Multiple Integrals.
13. Integration in Vector Fields.
Vector Fields, Work, Circulation, and Flux.
Path Independence, Potential Functions, and Conservative Fields.
Green's Theorem in the Plane.
Surface Area and Surface Integrals.
Divergence Theorem and a Unified Theory.
Proofs of Limit Theorems in Section 1.2.
Proof of the Chain Rule.
Simpson's One-Third Rule.
Cauchy's Mean Value Theorem and the Stronger Form of L'Hôpital's Rule.
Limits That Arise Frequently.
Proof of Taylor's Theorem.
The Distributive Law for Vector Cross Products.
Determinants and Cramer's Rule.
The Mixed Derivative Theorem and the Increment Theorem.
The Area of a Parallelogram's Projection on a Plane.