Preface
To the Student
Diagnostic Tests.
A Preview of Calculus.
1. FUNCTIONS AND MODELS
Function. Exponential Function and Logarithmic Functions. Inverse Functions of Trigonometric Functions. Principles of Problem Solving.
2. LIMITS AND DERIVATIVES. The Tangent and Velocity Problems. The Limit of a Function. Calculating Limits Using the Limit Laws. The Precise Definition of a Limit. Continuity. Limits at Infinity; Horizontal Asymptotes. Derivatives and Rates of Change. Writing Project: Early Methods for Finding Tangents. The Derivative as a Function. Review. Problems Plus.
3. DIFFERENTIATION RULES. Derivatives of Polynomials and Exponential Functions. Applied Project: Building a Better Roller Coaster. The Product and Quotient Rules. Derivatives of Trigonometric Functions. The Chain Rule. Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation. Laboratory Project. Families of Implicit Curves.잻erivatives of Logarithmic Functions. . Related Rates. Linear Approximations and Differentials. Laboratory Project: Taylor Polynomials. Hyperbolic Functions. Review. Problems Plus.
4. APPLICATIONS OF DIFFERENTIATION. Maximum and Minimum Values. Applied Project: The Calculus of Rainbows. The Mean Value Theorem. How Derivatives Affect the Shape of a Graph. Indeterminate Forms and L'Hospital's Rule. Writing Project: The Origins of l'Hospital's Rule. Summary of Curve Sketching.쟏ptimization Problems. Applied Project: The Shape of a Can. Newton's Metho.쟓eview. Problems Plus.
5. INTEGRALS. Areas and Distances. The Definite Integral. Discovery Project: Area Functions. The Fundamental Theorem of Calculus. Indefinite Integrals and the Net Change Theorem. Writing Project: Newton, Leibniz, and the Invention of Calculus.쟓eview. Problems Plus.
6. TECHNIQUES OF INTEGRATION. The Substitution Rule. Integration by Parts. Trigonometric Integrals. Trigonometric Substitution. Integration of Rational Functions by Partial Fractions.잸pproximate Integration. Improper Integrals. Review. Problems Plus.
7. APPLICATIONS OF INTEGRATION. Areas between Curves. Applied Project. The Gini Index. Volume. Volumes by Cylindrical Shells. Are Length. Discovery Project. Arc Length Contest. Area of쟞 Surface of Revolution Discovery Project. Rotating on a Slant. Pappus Theorem.잻iscovery Project. Complementary Coffee잺ups Review. Problems Plus.
8. PARAMETRIC EQUATIONS AND POLAR COORDINATES. Curves Defined by Parametric Equations. Laboratory Project. Running Circles around Circles.잺alculus with Parametric Curves. Laboratory Project: Bezier Curves. Polar Coordinates. Laboratory Project. Families of Polar Curves. Areas and Lengths in Polar Coordinates. Conic Sections. Conic Sections in Polar Coordinates. Review. Problems Plus.
9. INFINITE SEQUENCES AND SERIES. Sequences. Laboratory Project: Logistic Sequences. Series. The Integral Test and Estimates of Sums. The Comparison Tests. Alternating Series. Absolute Convergence and the Ratio and Root Tests. Strategy for Testing Series. Power Series. Representations of Functions as Power Series. Taylor and Maclaurin Series. Laboratory Project: An Elusive Limit. Writing Project: How Newton Discovered the Binomial Series.쟓eview. Problems Plus.
10. VECTORS AND THE GEOMETRY OF SPACE. Three-Dimensional Coordinate Systems. Vectors. The Dot Product. The Cross Product. Discovery Project: The Geometry of a Tetrahedron. Equations of Lines and Planes. Laboratory Project. Putting 3D in Perspective. Cylinders and Quadric쟔urfaces. Review. Problems Plus.
11. VECTOR FUNCTIONS. Vector Functions and Space Curves. Derivatives and Integrals of Vector Functions. Arc Length and Curvature. Motion in Space: Velocity and Acceleration.쟓eview. Problems Plus.
12. PARTIAL DERIVATIVES. Functions of Several Variables. Limits and Continuity. Partial Derivatives. Tangent Planes and Linear Approximations. The Chain Rule. Directional Derivatives and the Gradient Vector. Maximum and Minimum Values. Applied Project: Designing a Dumpster. Discovery Project: Quadratic Approximations and Critical Points. Lagrange Multipliers. Applied Project: Rocket Science. Applied Project: Hydro-Turbine Optimization. Review. Problems Plus
13. MULTIPLE INTEGRALS. Double Integrals over Rectangles. Iterated Integrals. Double Integrals over General Regions. Double Integrals in Polar Coordinates. Applications of Double Integrals.쟕riple Integrals. Discovery Project: Volumes of Hyperspheres. Triple Integrals in Cylindrical잺oordinates. Laboratory Project. The Intersection of Three Cylinders. Triple Integrals in쟔pherical Coordinates.잸pplied Project: Roller Derby.잺hange of Variables in Multiple Integrals. Review. Problems Plus.
14. VECTOR CALCULUS. Vector Fields. Line Integrals. The Fundamental Theorem for Line Integrals. Green's Theorem. Curl and Divergence. Parametric Surfaces and Their Areas. Surface Integrals. Stokes' Theorem. Writing Project: Three Men and Two Theorems. The Divergence Theorem. Summary. Review. Problems Plus.
APPENDIXES. A Numbers, Inequalities, and Absolute Values. B Trigonometry C Proofs of Theorems D The Logarithm Defined as an Integral. E Complex Numbers. F잸nswers of Exercises
