| |
| |
| Introduction: Calculus: A New Horizon from Ancient Roots | |
| |
| |
| |
| Functions | |
| |
| |
| |
| Functions and the Analysis of Graphical Information | |
| |
| |
| |
| Properties of Functions | |
| |
| |
| |
| Graphing Functions on Calculators and Computers; Computer Algebra Systems | |
| |
| |
| |
| New Functions from Old | |
| |
| |
| |
| Lines | |
| |
| |
| |
| Families of Functions | |
| |
| |
| |
| Mathematical Models | |
| |
| |
| |
| Parametric Equations | |
| |
| |
| Horizon Module: Iteration and Dynamical Systems | |
| |
| |
| |
| Limits and Continuity | |
| |
| |
| |
| Limits (An Intuitive Approach) | |
| |
| |
| |
| Computing Limits | |
| |
| |
| |
| Computing Limits: End Behavior | |
| |
| |
| |
| Limits (Discussed More Rigorously) | |
| |
| |
| |
| Continuity | |
| |
| |
| |
| Limits and Continuity of Trigonometric Functions | |
| |
| |
| |
| The Derivative | |
| |
| |
| |
| Slopes and Rates of Change | |
| |
| |
| |
| The Derivative | |
| |
| |
| |
| Techniques of Differentiation | |
| |
| |
| |
| Derivatives of Trigonometric Functions | |
| |
| |
| |
| The Chain Rule | |
| |
| |
| |
| Implicit Differentiation | |
| |
| |
| |
| Related Rates | |
| |
| |
| |
| Local Linear Approximation; Differentials | |
| |
| |
| Horizon Module: Robotics | |
| |
| |
| |
| The Derivative in Graphing and Applications | |
| |
| |
| |
| Analysis of Functions I: Increase, Decrease, and Concavity | |
| |
| |
| |
| Analysis of Functions II: Relative Extrema; First and Second Derivative Tests | |
| |
| |
| |
| Analysis of Functions III: Applying Technology and the Tools of Calculus | |
| |
| |
| |
| Rectilinear Motion (Motion Along a Line) | |
| |
| |
| |
| Absolute Maxima and Minima | |
| |
| |
| |
| Applied Maximum and Minimum Problems | |
| |
| |
| |
| Newton's Method | |
| |
| |
| |
| Rolle's Theorem; Mean-Value Theorem | |
| |
| |
| |
| Integration | |
| |
| |
| |
| An Overview of the Area Problem | |
| |
| |
| |
| The Indefinite Integral; Integral Curves and Direction Fields | |
| |
| |
| |
| Integration by Substitution | |
| |
| |
| |
| Sigma Notation; Area as a Limit | |
| |
| |
| |
| The Definite Integral | |
| |
| |
| |
| The Fundamental Theorem of Calculus | |
| |
| |
| |
| Rectilinear Motion Revisited; Average Value | |
| |
| |
| |
| Evaluating Definite Integrals by Substitution | |
| |
| |
| Horizon Module: Blammo the Human Cannonball | |
| |
| |
| |
| Applications of the Definite Integral in Geometry, Science, and Engineering | |
| |
| |
| |
| Area Between Two Curves | |
| |
| |
| |
| Volumes by Slicing; Disks and Washers | |
| |
| |
| |
| Volumes by Cylindrical Shells | |
| |
| |
| |
| Length of a Plane Curve | |
| |
| |
| |
| Area of a Surface of Revolution | |
| |
| |
| |
| Work | |
| |
| |
| |
| Fluid Pressure and Force | |
| |
| |
| |
| Exponential, Logarithmic, and Inverse Trigonometric Functions | |
| |
| |
| |
| Inverse Functions | |
| |
| |
| |
| Exponential and Logarithmic Functions | |
| |
| |
| |
| Derivatives and Integrals Involving Logarithmic and Exponential Functions | |
| |
| |
| |
| Graphs and Applications Involving Logarithmic and Exponential Functions | |
| |
| |
| |
| Logarithmic Functions from the Integral Point of View | |
| |
| |
| |
| Derivatives and Integrals Involving Inverse Trigonometric Functions | |
| |
| |
| |
| L'Hopital's Rule; Indeterminate Forms | |
| |
| |
| |
| Hyperbolic Functions and Hanging Cables | |
| |
| |
| |
| Principles of Integral Evaluation | |
| |
| |
| |
| An Overview of Integration Methods | |
| |
| |
| |
| Integration by Parts | |
| |
| |
| |
| Trigonometric Integrals | |
| |
| |
| |
| Trigonometric Substitutions | |
| |
| |
| |
| Integrating Rational Functions by Partial Fractions | |
| |
| |
| |
| Using Tables of Integrals and Computer Algebra Systems | |
| |
| |
| |
| Numerical Integration; Simpson's Rule | |
| |
| |
| |
| Improper Integrals | |
| |
| |
| Horizon Module: Railroad Design | |
| |
| |
| |
| Mathematical Modeling with Differential Equations | |
| |
| |
| |
| First-Order Differential Equations and Applications | |
| |
| |
| |
| Direction Fields; Euler's Method | |
| |
| |
| |
| Modeling with First-Order Differential Equations | |
| |
| |
| |
| Second-Order Linear Homogeneous Differential Equations; The Vibrating Spring | |
| |
| |
| |
| Infinite Series | |
| |
| |
| |
| Maclaurin and Taylor Polynomial Approximations | |
| |
| |
| |
| Sequences | |
| |
| |
| |
| Monotone Sequences | |
| |
| |
| |
| Infinite Series | |
| |
| |
| |
| Convergence Tests | |
| |
| |
| |
| The Comparison, Ratio, and Root Tests | |
| |
| |
| |
| Alternating Series; Conditional Convergence | |
| |
| |
| |
| Maclaurin and Taylor Series; Power Series | |
| |
| |
| |
| Convergence of Taylor Series; Computational Methods | |
| |
| |
| |
| Differentiating and Integrating Power Series; Modeling with Taylor Series | |
| |
| |
| |
| Analytic Geometry in Calculus | |
| |
| |
| |
| Polar Coordinates | |
| |
| |
| |
| Tangent Lines and Arc Length for Parametric and Polar Curves | |
| |
| |
| |
| Area in Polar Coordinates | |
| |
| |
| |
| Conic Sections in Calculus | |
| |
| |
| |
| Rotation of Axes; Second-Degree Equations | |
| |
| |
| |
| Conic Sections in Polar Coordinates | |
| |
| |
| Horizon Module: Comet Collision | |
| |
| |
| |
| Real Numbers, Intervals, and Inequalities | |
| |
| |
| |
| Absolute Value | |
| |
| |
| |
| Coordinate Planes and Lines | |
| |
| |
| |
| Distance, Circles, and Quadratic Equations | |
| |
| |
| |
| Trigonometry Review | |
| |
| |
| |
| Solving Polynomial Equations | |
| |
| |
| |
| Selected Proofs | |
| |
| |
| Answers | |
| |
| |
| Photo Credits | |
| |
| |
| Index | |