by Arnold Ostebee and Paul Zorn
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1. FUNCTIONS AND DERIVATIVES: THE GRAPHICAL VIEW.
Functions, Calculus Style.
Graphs.
A Field Guide to Elementary Functions.
Amount Functions and Rate Functions: The Idea of the Derivative.
Estimating Derivatives: A Closer Look.
The Geometry of Derivatives.
The Geometry of Higher-Order Derivatives.
Chapter Summary.
Interlude: Zooming in on Differences.
2. FUNCTIONS AND DERIVATIVES: THE SYMBOLIC VIEW.
Defining the Derivative.
Derivatives of Power Functions ad Polynomials.
Limits.
Derivatives, Antiderivatives, and Their Uses.
Differential Equations; Modeling Motion.
Derivatives of Exponential and Logarithm Functions; Modeling Growth.
Derivatives of Trigonometric Functions; Modeling Oscillation.
Chapter Summary.
Interlude: Tangent Lines in History.
Interlude: Limit--the Formal Definition.
3. NEW DERIVATIVES FROM OLD.
Algebraic Combinations: The Product and Quotient Rules.
Composition and the Chain Rule.
Implicit Functions and Implicit Differentiation.
Inverse Functions and their Derivatives; Inverse Trigonometric Functions.
Miscellaneous Derivatives and Antiderivatives.
Chapter Summary.
Interlude: Vibrations--Simple and Damped.
Interlude: Hyperbolic Functions.
4. USING THE DERIVATIVE.
Direction Fields; More on Growth and Motion.
Limits Involving Infinity; l'Hôspital's Rule: Comparing Rates.
More on Optimization.
Parametric Equations.
Related Rates.
Newton's Method.
Linear Approximation and Taylor Polynomials.
Continuity.
The Mean Value Theorem.
Chapter Summary.
Interlude: Growth with Interest.
Interlude: Logistic Growth.
Interlude: Digging Deeper for Roots (More on Newton's Method).
5. THE INTEGRAL.
Areas and Integrals.
The Area Function.
The Fundamental Theorem of Calculus.
Finding Antiderivatives by Substitution.
Finding Antiderivatives Using Tables and Computers.
Approximating Sums: The Integral as a Limit.
Working with Approximating Sums.
Chapter Summary.
Interlude: Mean Value Theorems and Integrals.
6. NUMERICAL INTEGRATION.
Approximating Integrals: The Idea and Basic Methods.
Euler's Method for Differential Equations.
Error Bounds.
7.USING THE INTEGRAL.
Interpretations of the Integral.
Areas and Volumes.
Motion; Work.
Probability.
Separable Differential Equations.
8. SYMBOLIC ANTIDIFFERENTIATION TECHNIQUES.
Integration by Parts.
Partial Fractions.
Trigonometric Antiderivatives.
Miscellaneous Antidifferentiation Exercises.
9. FUNCTION APPROXIMATION.
Taylor Polynomials.
Taylor's Theorem.
Fourier Polynomials.
10. IMPROPER INTEGRALS.
When Is an Integral Improper? Detecting Convergence; Estimating Limits.
11. INFINITE SERIES.
Sequences and Their Limits; Infinite Series, Convergence, and Divergence.
Testing for Convergence; Estimating Limits.
Absolute Convergence; Alternating Series.
Power Series; Taylor Series.
Algebra and Calculus with Power Series.
V.VECTORS AND POLAR COORDINATES.
Vectors.
Polar Coordinates and Polar Curves.
Calculus in Polar Coordinates.
M. MULTIVARIABLE CALCULUS: A FIRST LOOK.
Three-dimensional Space.
Functions of Several Variables.
Partial Derivatives.
Optimization and Partial Derivatives.
Multiple Integrals and Approximating Sums.
Calculating Multiple Integrals by Iteration.
Summary and Review.
APPENDICES.
Machine Graphics.
Real Numbers and the Coordinate Plane.
Lines and Linear Functions.
Polynomials and Rational Functions.
Exponential and Logarithm Functions.
Trigonometric Functions.
Real-World Calculus: From Words to Mathematics.
Selected Proofs.
A Graphical Glossary of Functions.
Complex Numbers.
Matrices and Matrix Algebra: A Crash Course.
Formulas, Identities, and Table of Integrals.
Table of Contents
1. FUNCTIONS AND DERIVATIVES: THE GRAPHICAL VIEW.
Functions, Calculus Style.
Graphs.
A Field Guide to Elementary Functions.
Amount Functions and Rate Functions: The Idea of the Derivative.
Estimating Derivatives: A Closer Look.
The Geometry of Derivatives.
The Geometry of Higher-Order Derivatives.
Chapter Summary.
Interlude: Zooming in on Differences.
2. FUNCTIONS AND DERIVATIVES: THE SYMBOLIC VIEW.
Defining the Derivative.
Derivatives of Power Functions ad Polynomials.
Limits.
Derivatives, Antiderivatives, and Their Uses.
Differential Equations; Modeling Motion.
Derivatives of Exponential and Logarithm Functions; Modeling Growth.
Derivatives of Trigonometric Functions; Modeling Oscillation.
Chapter Summary.
Interlude: Tangent Lines in History.
Interlude: Limit--the Formal Definition.
3. NEW DERIVATIVES FROM OLD.
Algebraic Combinations: The Product and Quotient Rules.
Composition and the Chain Rule.
Implicit Functions and Implicit Differentiation.
Inverse Functions and their Derivatives; Inverse Trigonometric Functions.
Miscellaneous Derivatives and Antiderivatives.
Chapter Summary.
Interlude: Vibrations--Simple and Damped.
Interlude: Hyperbolic Functions.
4. USING THE DERIVATIVE.
Direction Fields; More on Growth and Motion.
Limits Involving Infinity; l'Hôspital's Rule: Comparing Rates.
More on Optimization.
Parametric Equations.
Related Rates.
Newton's Method.
Linear Approximation and Taylor Polynomials.
Continuity.
The Mean Value Theorem.
Chapter Summary.
Interlude: Growth with Interest.
Interlude: Logistic Growth.
Interlude: Digging Deeper for Roots (More on Newton's Method).
5. THE INTEGRAL.
Areas and Integrals.
The Area Function.
The Fundamental Theorem of Calculus.
Finding Antiderivatives by Substitution.
Finding Antiderivatives Using Tables and Computers.
Approximating Sums: The Integral as a Limit.
Working with Approximating Sums.
Chapter Summary.
Interlude: Mean Value Theorems and Integrals.
6. NUMERICAL INTEGRATION.
Approximating Integrals: The Idea and Basic Methods.
Euler's Method for Differential Equations.
Error Bounds.
7.USING THE INTEGRAL.
Interpretations of the Integral.
Areas and Volumes.
Motion; Work.
Probability.
Separable Differential Equations.
8. SYMBOLIC ANTIDIFFERENTIATION TECHNIQUES.
Integration by Parts.
Partial Fractions.
Trigonometric Antiderivatives.
Miscellaneous Antidifferentiation Exercises.
9. FUNCTION APPROXIMATION.
Taylor Polynomials.
Taylor's Theorem.
Fourier Polynomials.
10. IMPROPER INTEGRALS.
When Is an Integral Improper? Detecting Convergence; Estimating Limits.
11. INFINITE SERIES.
Sequences and Their Limits; Infinite Series, Convergence, and Divergence.
Testing for Convergence; Estimating Limits.
Absolute Convergence; Alternating Series.
Power Series; Taylor Series.
Algebra and Calculus with Power Series.
V.VECTORS AND POLAR COORDINATES.
Vectors.
Polar Coordinates and Polar Curves.
Calculus in Polar Coordinates.
M. MULTIVARIABLE CALCULUS: A FIRST LOOK.
Three-dimensional Space.
Functions of Several Variables.
Partial Derivatives.
Optimization and Partial Derivatives.
Multiple Integrals and Approximating Sums.
Calculating Multiple Integrals by Iteration.
Summary and Review.
APPENDICES.
Machine Graphics.
Real Numbers and the Coordinate Plane.
Lines and Linear Functions.
Polynomials and Rational Functions.
Exponential and Logarithm Functions.
Trigonometric Functions.
Real-World Calculus: From Words to Mathematics.
Selected Proofs.
A Graphical Glossary of Functions.
Complex Numbers.
Matrices and Matrix Algebra: A Crash Course.
Formulas, Identities, and Table of Integrals.