ISBN-10:
0618247505
ISBN-13:
9780618247509
Pub. Date:
07/28/2001
Publisher:
Houghton Mifflin-High School
Calculus From Graphical, Numerical, and Symbolic Points of Views, Volume I / Edition 2

Calculus From Graphical, Numerical, and Symbolic Points of Views, Volume I / Edition 2

by Arnold Ostebee, Paul Zorn

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Product Details

ISBN-13: 9780618247509
Publisher: Houghton Mifflin-High School
Publication date: 07/28/2001
Edition description: New Edition
Pages: 371
Product dimensions: 8.60(w) x 10.40(h) x 0.70(d)

Table of Contents

Contents

Note: Each chapter contains a Summary.

  • 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
    Interlude: Zooming in on Differences
  • 2. Functions and Derivatives: The Symbolic View
    Defining the Derivative
    Derivatives of Power Functions and Polynomials
    Limits
    Using Derivative and Antiderivative Formulas
    Differential Equations; Modeling Motion
    Derivatives of Exponential and Logarithm Functions; Modeling Growth
    Derivatives of Trigonometric Functions: Modeling Oscillation
    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
    Interlude: Vibrations—Simple and Damped
    Interlude: Hyperbolic Functions
  • 4. Using the Derivative
    Slope Fields; More Differential Equation Models
    More on Limits: Limits Involving Infinity and l'Hôpital's Rule
    Optimization
    Parametric Equations, Parametric Curves
    Related Rates
    Newton's Method: Finding Roots
    Building Polynomials to Order; Taylor Polynomials
    Why Continuity Matters
    Why Differentiability Matters: The Mean Value Theorem
    Interlude: Growth withInterest
    Interlude: Logistic Growth
    Interlude: Digging Deeper for Roots
  • 5. The Integral
    Areas and Integrals
    The Area Function
    The Fundamental Theorem of Calculus
    Finding Antiderivatives; The Method of Substitution
    Integral Aids: Tables and Computers
    Approximating Sums: The Integral as a Limit
    Working with Sums
    Interlude: Mean Value Theorems and Integrals
  • 6. Numerical Integration
    Approximating Integrals Numerically
    Error Bounds for approximating Sums
    Euler's Method: Solving DEs Numerically
    Interlude: Simpson's Rule
    Interlude: Gaussian Quadrature: Approximating Integrals Efficiently
  • 7. Using the Integral
    Measurement and the Definite Integral; Arc Length
    Finding Volumes by Integration
    Work
    Separating Variables: Solving DEs Symbolically
    Present Value
    Interlude: Mass and Center of Mass
  • 8. Symbolic Antidifferentiation Techniques
    Integration by Parts
    Partial Fractions
    Trigonometric Antiderivatives
    Miscellaneous Antiderivatives
    Interlude: Beyond Elementary Functions
    Interlude: First-Order Linear Differential Equations
  • 9. Function Approximation
    Taylor Polynomials
    Taylor's Theorem: Accuracy Guarantees for Taylor Polynomials
    Fourier Polynomials: Approximating Periodic Functions
    Interlude: Splines—Connecting the Dots
  • 10. Improper Integrals
    Improper Integrals: Ideas and Definitions
    Detecting Convergence, Estimating Limits
    Improper Integrals and Probability
  • 11. Infinite Series
    Sequences and Their Limits
    Infinite Series, Convergence, and Divergence
    Testing for Convergence; Estimating Limits
    Absolute Convergence; Alternating Series
    Power Series
    Power Series as Functions
    Taylor Series
    Interlude: Fourier Series
  • V. Vectors and Polar Coordinates
    Vectors and Vector-Valued Functions
    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: A First Look
    Multiple Integrals and Approximating Sums
    Calculating Multiple Integrals by Iteration
    Double Integrals in Polor Coordinates
  • 12. Curves and Vectors
    Three-dimensional Space
    Curves and Parametric Equations
    Vectors
    Vector-valued Functions, Derivatives, and Integrals
    Derivatives, Antiderivatives, and Motion
    The Dot Product
    Lines and Planes in Three Dimensions
    The Cross Product
  • 13. Derivatives
    Functions of Several Variables
    Partial Derivatives
    Partial Derivatives and Linear Approximation
    The Gradient and Directional Derivatives
    Local Linearity: Theory of the Derivative
    Higher Order Derivatives and Quadratic Approximation
    Maxima, Minima, and Quadratic Approximation
    The Chain Rule
  • 14. Integrals
    Multiple Integrals and Approximating Sums
    Calculating Integrals by Iteration
    Double Integrals in Polar Coordinates
    More on Triple Integrals; Cylindrical and Spherical Coordinates
    Multiple Integrals Overviewed; Change of Variables
  • 15. Other Topics
    Linear, Circular, and Combined Motion
    Using the Dot Product: More on Curves
    Curvature
    Lagrange Multipliers and Constrained Optimization
  • 16. Vector Calculus
    Line Integrals
    More on Line Integrals; A Fundamental Theorem
    Relating Line and Area Integrals: Green's Theorem
    Surfaces and Their Parametrizations
    Surface Integrals
    Derivatives and Integrals of Vector Fields
    Back to Fundamentals: Stokes' Theorem and the Divergence Theorem
  • Appendices
    A. Machine Graphics
    B. Real Numbers and the Coordinate Plane
    C. Lines and Linear Functions
    D. Polynomials and Rational Functions
    E. Algebra of Exponentials and Logarithms
    F. Trigonometric Functions
    G. Real-World Calculus: From Words to Mathematics
    H. Selected Proofs
    I. A Graphical Glossary of Functions

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