Preface -- To the student -- A preview of calculus -- 1. Functions and models -- 1.1. Four ways to represent a function -- 1.2. Mathematical models : a catalog of essential functions -- 1.3. New functions from old functions -- 1.4. Graphing calculators and computers -- 1.5. Exponential functions -- 1.6. Inverse functions and logarithms -- Review -- Principles of problem solving -- 2. Limits and derivatives -- 2.1. The tangent and velocity problems -- 2.2. The limit of a function -- 2.3. Calculating limits using the limit laws -- 2.4. The precise definition of a limit -- 2.5. Continuity -- 2.6. Limits at infinity ; horizontal asymptotes -- 2.7. Tangents, velocities, and other rates of change -- 2.8. Derivatives -- Writing project : early methods for finding tangents -- 2.9. The derivative as a function -- Review -- Problems plus -- 3. Differentiation rules -- 3.1. Derivatives of polynomials and exponential functions -- 3.2. The product and quotient rules -- 3.3. Rates of change in the natural and social sciences -- 3.4. Derivatives of trigonometric functions -- 3.5. The chain rule -- 3.6. Implicit differentiation -- 3.7. Higher derivatives -- Applied project : where should a pilot start descent? -- Applied project : building a better roller coaster -- 3.8. Derivatives of logarithmic functions -- 3.9. Hyperbolic functions -- 3.10. Related rates -- 3.11. Linear approximations and differentials -- Laboratory project : Taylor polynomials -- Review -- Problems plus -- 4. Applications of differentiation -- 4.1. Maximum and minimum values -- Applied project : the calculus of rainbows -- 4.2. The mean value theorem -- 4.3. How derivatives affect the shape of a graph -- 4.4. Indeterminate forms and L'Hospital's rule -- Writing project : the origins of L'Hospital's rule -- 4.5. Summary of curve sketching -- 4.6. Graphing with calculus and calculators -- 4.7. Optimization problems -- Applied project : the shape of a can -- 4.8. Applications to business and economics -- 4.9. Newton's method -- 4.10. Antiderivatives -- Review -- Problems plus.

5. Integrals -- 5.1. Areas and distances -- 5.2. The definite integral -- Discovery project : area functions -- 5.3. The fundamental theorem of calculus -- 5.4. Indefinite integrals and the net change theorem -- Writing project : Newton, Leibniz, and the invention of calculus -- 5.5. The substitution rule -- 5.6. The logarithm defined as an integral -- Review -- Problems plus -- 6. Applications of integration -- 6.1. Areas between curves -- 6.2. Volumes -- 6.3. Volumes by cylindrical shells -- 6.4. Work -- 6.5. Average value of a function -- Applied projects : where to sit at the movies -- Review -- Problems plus -- 7. Techniques of integration -- 7.1. Integration by parts -- 7.2. Trigonometric integrals -- 7.3. Trigonometric substitution -- 7.4. Integration of rational functions by partial fractions -- 7.5. Strategy for integration -- 7.6. Integration using tables and computer algebra systems -- Discovery project : patterns in integrals -- 7.7. Approximate integration -- 7.8. Improper integrals -- Review -- Problems plus -- 8. Further applications of integration -- 8.1. Arc length -- Discovery project : arc length contest -- 8.2. Area of a surface of revolution -- Discovery project : rotating on a slant -- 8.3. Applications to physics and engineering -- 8.4. Applications to economics and biology -- 8.5. Probability -- Review -- Problems plus -- 9. Differential equations -- 9.1. Modeling with differential equations -- 9.2. Direction fields and Euler's method -- 9.3. Separable equations -- Applied project : how fast does a tank drain? -- Applied project : which is faster, going up or coming down? -- 9.4. Exponential growth and decay -- Applied project : calculus and baseball -- 9.5. The logistic equation -- 9.6. Linear equations -- 9.7. Predator-prey systems -- Review -- Problems plus.

10. Parametric equations and polar coordinates -- 10.1. Curves defined by parametric equations -- Laboratory project : running circles around circles -- 10.2. Calculus with parametric curves -- Laboratory project : Bézier curves -- 10.3. Polar coordinates -- 10.4. Areas and lengths in polar coordinates -- 10.5. Conic sections -- 10.6. Conic sections in polar coordinates -- Review -- Problems plus -- 11. Infinite sequences and series -- 11.1. Sequences -- Laboratory project : logistic sequences -- 11.2. Series -- 11.3. The integral test and estimates of sums -- 11.4. The comparison tests -- 11.5. Alternating series -- 11.6. Absolute convergence and the ratio and root tests -- 11.7. Strategy for testing series -- 11.8. Power series -- 11.9. Representations of functions as power series -- 11.10. Taylor and Maclaurin series -- Laboratory project : an elusive limit -- 11.11. The binomial series -- Writing project : how Newton discovered the binomial series -- 11.12. Applications of Taylor polynomials -- Applied project : radiation from the stars -- Review -- Problems plus -- 12. Vectors and geometry of space -- 12.1. Three-dimensional coordinate systems -- 12.2. Vectors -- 12.3. The dot product -- 12.4. The cross product -- Discovery project : the geometry of a tetrahedron -- 12.5. Equations of lines and planes -- Laboratory project : putting 3D in perspective -- 12.6. Cylinders and quadric surfaces -- 12.7. Cylindrical and spherical coordinates -- Laboratory project : families of surfaces -- Review -- Problems plus -- 13. Vector functions -- 13.1. Vector functions and space curves -- 13.2. Derivatives and integrals of vector functions -- 13.3. Arc length and curvature -- 13.4. Motion in space : velocity and acceleration -- Applied project : Kepler's laws -- Review -- Problems plus.

14. Partial derivatives -- 14.1. Functions of several variables -- 14.2. Limits and continuity -- 14.3. Partial derivatives -- 14.4. Tangent planes and linear approximations -- 14.5. The chain rule -- 14.6. Directional derivatives and the gradient vector -- 14.7. Maximum and minimum values -- Applied project : designing a dumpster -- Discovery project : quadratic approximations and critical points -- 14.8. Lagrange multipliers -- Applied project : rocket science -- Applied project : hydro-turbine optimization -- Review -- Problems plus -- 15. Multiple integrals -- 15.1. Double integrals over rectangles -- 15.2. Iterated integrals -- 15.3. Double integrals over general regions -- 15.4. Double integrals in polar coordinates -- 15.5. Applications of double integrals -- 15.6. Surface area -- 15.7. Triple integrals -- Discovery project : volumes of hyperspheres -- 15.8. Triple integrals in cylindrical coordinates -- Applied project : roller derby -- Discovery project : the intersection of three cylinders -- 15.9. Change of variables in multiple integrals -- Review -- Problems plus -- 16. Vector calculus -- 16.1. Vector fields -- 16.2. Line integrals -- 16.3. The fundamental theorem for line integrals -- 16.4. Green's theorem -- 16.5. Curl and divergence -- 16.6. Parametric surfaces and their areas -- 16.7. Surface integrals -- 16.8. Stokes' theorem -- Writing project : three men and two theorems -- 16.9. The divergence theorem -- 16.10. Summary -- Review -- Problems plus -- 17. Second-order differential equations -- 17.1. Second-order linear equations -- 17.2. Nonhomogeneous linear equations -- 17.3. Applications of second-order differential equations -- 17.4. Series solutions -- Review -- Appendixes -- A. Numbers, inequalities, and absolute values -- B. Coordinate geometry and lines -- C. Graphs of second-degree equations -- D. Trigonometry -- E. Sigma notation -- F. Proofs of theorems -- G. Complex numbers -- H. Answers to odd-numbered exercises -- Index.