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Calculus Volume 2

Key Terms

Calculus Volume 2Key Terms
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Table of contents
  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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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

Key Terms

binomial series
the Maclaurin series for f(x)=(1+x)r;f(x)=(1+x)r; it is given by
 (1+x)r=∑n=0∞(rn)xn=1+rx+r(r−1)2!x2+⋯+r(r−1)⋯(r−n+1)n!xn+⋯(1+x)r=∑n=0∞(rn)xn=1+rx+r(r−1)2!x2+⋯+r(r−1)⋯(r−n+1)n!xn+⋯ for |x|<1|x|<1
interval of convergence
the set of real numbers x for which a power series converges
Maclaurin polynomial
a Taylor polynomial centered at 0; the nth Taylor polynomial for ff at 0 is the nth Maclaurin polynomial for ff
Maclaurin series
a Taylor series for a function ff at x=0x=0 is known as a Maclaurin series for ff
nonelementary integral
an integral for which the antiderivative of the integrand cannot be expressed as an elementary function
power series
a series of the form ∑n=0∞cnxn∑n=0∞cnxn is a power series centered at x=0;x=0; a series of the form ∑n=0∞cn(x−a)n∑n=0∞cn(x−a)n is a power series centered at x=ax=a
radius of convergence
if there exists a real number R>0R>0 such that a power series centered at x=ax=a converges for |x−a|<R|x−a|<R and diverges for |x−a|>R,|x−a|>R, then R is the radius of convergence; if the power series only converges at x=a,x=a, the radius of convergence is R=0;R=0; if the power series converges for all real numbers x, the radius of convergence is R=∞R=∞
Taylor polynomials
the nth Taylor polynomial for ff at x=ax=a is pn(x)=f(a)+f′(a)(x−a)+f″(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)npn(x)=f(a)+f′(a)(x−a)+f″(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)n
Taylor series
a power series at a that converges to a function ff on some open interval containing a
Taylor’s theorem with remainder
for a function ff and the nth Taylor polynomial for ff at x=a,x=a, the remainder Rn(x)=f(x)−pn(x)Rn(x)=f(x)−pn(x) satisfies Rn(x)=f(n+1)(c)(n+1)!(x−a)n+1Rn(x)=f(n+1)(c)(n+1)!(x−a)n+1
 for some c between x and a; if there exists an interval I containing a and a real number M such that |f(n+1)(x)|≤M|f(n+1)(x)|≤M for all x in I, then |Rn(x)|≤M(n+1)!|x−a|n+1|Rn(x)|≤M(n+1)!|x−a|n+1
term-by-term differentiation of a power series
a technique for evaluating the derivative of a power series ∑n=0∞cn(x−a)n∑n=0∞cn(x−a)n by evaluating the derivative of each term separately to create the new power series ∑n=1∞ncn(x−a)n−1∑n=1∞ncn(x−a)n−1
term-by-term integration of a power series
a technique for integrating a power series ∑n=0∞cn(x−a)n∑n=0∞cn(x−a)n by integrating each term separately to create the new power series C+∑n=0∞cn(x−a)n+1n+1C+∑n=0∞cn(x−a)n+1n+1
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