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Calculus Volume 2

Key Concepts

Calculus Volume 2Key Concepts
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  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

6.1 Power Series and Functions

  • For a power series centered at x=a,x=a, one of the following three properties hold:
    1. The power series converges only at x=a.x=a. In this case, we say that the radius of convergence is R=0.R=0.
    2. The power series converges for all real numbers x. In this case, we say that the radius of convergence is R=.R=.
    3. There is a real number R such that the series converges for |xa|<R|xa|<R and diverges for |xa|>R.|xa|>R. In this case, the radius of convergence is R.
  • If a power series converges on a finite interval, the series may or may not converge at the endpoints.
  • The ratio test may often be used to determine the radius of convergence.
  • The geometric series n=0xn=11xn=0xn=11x for |x|<1|x|<1 allows us to represent certain functions using geometric series.

6.2 Properties of Power Series

  • Given two power series n=0cnxnn=0cnxn and n=0dnxnn=0dnxn that converge to functions f and g on a common interval I, the sum and difference of the two series converge to f±g,f±g, respectively, on I. In addition, for any real number b and integer m0,m0, the series n=0bxmcnxnn=0bxmcnxn converges to bxmf(x)bxmf(x) and the series n=0cn(bxm)nn=0cn(bxm)n converges to f(bxm)f(bxm) whenever bxm is in the interval I.
  • Given two power series that converge on an interval (R,R),(R,R), the Cauchy product of the two power series converges on the interval (R,R).(R,R).
  • Given a power series that converges to a function f on an interval (R,R),(R,R), the series can be differentiated term-by-term and the resulting series converges to ff on (R,R).(R,R). The series can also be integrated term-by-term and the resulting series converges to f(x)dxf(x)dx on (R,R).(R,R).

6.3 Taylor and Maclaurin Series

  • Taylor polynomials are used to approximate functions near a value x=a.x=a. Maclaurin polynomials are Taylor polynomials at x=0.x=0.
  • The nth degree Taylor polynomials for a function ff are the partial sums of the Taylor series for f.f.
  • If a function ff has a power series representation at x=a,x=a, then it is given by its Taylor series at x=a.x=a.
  • A Taylor series for ff converges to ff if and only if limnRn(x)=0limnRn(x)=0 where Rn(x)=f(x)pn(x).Rn(x)=f(x)pn(x).
  • The Taylor series for ex, sinx,sinx, and cosxcosx converge to the respective functions for all real x.

6.4 Working with Taylor Series

  • The binomial series is the Maclaurin series for f(x)=(1+x)r.f(x)=(1+x)r. It converges for |x|<1.|x|<1.
  • Taylor series for functions can often be derived by algebraic operations with a known Taylor series or by differentiating or integrating a known Taylor series.
  • Power series can be used to solve differential equations.
  • Taylor series can be used to help approximate integrals that cannot be evaluated by other means.
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