Machine derived contents note: 1. FUNCTIONS AND MODELS Four Ways to Represent a Function / New Functions from Old Functions / Graphing Calculators and Computers / Parametric Curves / Laboratory Project: Families of Hypocycloids / Exponential Functions / Inverse Functions and Logarithms / Models and Curve Fitting 2. LIMITS AND DERIVATIVES The Tangent and Velocity Problems / The Limit of a Function / Calculating Limits Using the Limit Laws / Continuity / Limits Involving Infinity / Tangents, Velocities, and Other Rates of Change / Derivatives / Writing Project: Early Methods for Finding Tangents / The Derivative as a Function / Linear Approximations / What does f'''' say about f? 3. DIFFERENTIATION RULES Derivatives of Polynomials and Exponential Functions / The Product and Quotient Rules / Rates of Change in the Natural and Social Sciences / Derivatives of Trigonometric Functions / The Chain Rule / Laboratory Project: Bezier Curves / Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation / Derivatives of Logarithmic Functions / Discovery Project: Hyperbolic Functions / Linear Approximations and Differentials / Laboratory Project: Taylor Polynomials 4. APPLICATIONS OF DIFFERENTIATION Related Rates / Maximum and Minimum Values / Applied Project: The Calculus of Rainbows / Derivatives and the Shapes of Curves / Graphing with Calculus and Calculators / Indeterminate Forms and l''''Hospital''''s Rule / Writing Project: The Origins of l''''Hospital''''s Rule / Optimization Problems / Applied Project: The Shape of a Can / Applications to Economics / Newton''''s Method / Antiderivatives 5. INTEGRALS Areas and Distances / The Definite Integral / Evaluating Definite Integrals / Discovery Project: Area Functions / The Fundamental Theorem of Calculus / Writing Project: Newton, Leibniz, and the Invention of Calculus / The Substitution Rule / Integration by Parts / Integration Using Tables and Computer Algebra Systems / Discovery Project: Patterns in Integrals / Approximate Integration / Improper Integrals 6. APPLICATIONS OF INTEGRATION More about Areas / Volumes / Arc Length / Average Value of a Function / Applied Project: Where to Sit at the Movies / Applications to Physics / Applications to Economics and Biology / Probability 7. DIFFERENTIAL EQUATIONS Modeling with Differential Equations / Direction Fields / Euler''''s Method / Separable Equations / Applied Project: Which is Faster, Going Up or Coming Down? Exponential Growth and Decay / Applied Project: Calculus and Baseball / The Logistic Equation / Predator-Prey Systems / Some Special Second Order Equations 8. INFINITE SEQUENCES AND SERIES Sequences / Laboratory Project: Logistic Sequences / Series / The Integral and Comparison Tests; Estimating Sums / Other Convergence Tests / Power Series / Representation of Functions as Power Series / Taylor and Maclaurin Series / Writing Project: Newton, Gregory, Taylor, and Maclaurin / The Binomial Series / Applications of Taylor Polynomials / Applied Project: Radiation from the Stars / Using Series to Solve Differential Equations 9. 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 / Functions and Surfaces / Cylindrical and Spherical Coordinates / Laboratory Project: Families and Surfaces 10. VECTOR FUNCTIONS Vector Functions and Space Curves / Derivatives and Integrals of Vector Functi.
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