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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

5.1 Sequences

  • To determine the convergence of a sequence given by an explicit formula an=f(n),an=f(n), we use the properties of limits for functions.
  • If {an}{an} and {bn}{bn} are convergent sequences that converge to AA and B,B, respectively, and cc is any real number, then the sequence {can}{can} converges to c·A,c·A, the sequences {an±bn}{an±bn} converge to A±B,A±B, the sequence {an·bn}{an·bn} converges to A·B,A·B, and the sequence {an/bn}{an/bn} converges to A/B,A/B, provided B0.B0.
  • If a sequence is bounded and monotone, then it converges, but not all convergent sequences are monotone.
  • If a sequence is unbounded, it diverges, but not all divergent sequences are unbounded.
  • The geometric sequence {rn}{rn} converges if and only if |r|<1|r|<1 or r=1.r=1.

5.2 Infinite Series

  • Given the infinite series
    n=1an=a1+a2+a3+n=1an=a1+a2+a3+

    and the corresponding sequence of partial sums {Sk}{Sk} where
    Sk=n=1kan=a1+a2+a3++ak,Sk=n=1kan=a1+a2+a3++ak,

    the series converges if and only if the sequence {Sk}{Sk} converges.
  • The geometric series n=1arn1n=1arn1 converges if |r|<1|r|<1 and diverges if |r|1.|r|1. For |r|<1,|r|<1,
    n=1arn1=a1r.n=1arn1=a1r.
  • The harmonic series
    n=11n=1+12+13+n=11n=1+12+13+

    diverges.
  • A series of the form n=1[bnbn+1]=[b1b2]+[b2b3]+[b3b4]++[bnbn+1]+n=1[bnbn+1]=[b1b2]+[b2b3]+[b3b4]++[bnbn+1]+
    is a telescoping series. The kthkth partial sum of this series is given by Sk=b1bk+1.Sk=b1bk+1. The series will converge if and only if limkbk+1limkbk+1 exists. In that case,
    n=1[bnbn+1]=b1limk(bk+1).n=1[bnbn+1]=b1limk(bk+1).

5.3 The Divergence and Integral Tests

  • If limnan0,limnan0, then the series n=1ann=1an diverges.
  • If limnan=0,limnan=0, the series n=1ann=1an may converge or diverge.
  • If n=1ann=1an is a series with positive terms anan and ff is a continuous, decreasing function such that f(n)=anf(n)=an for all positive integers n,n, then
    n=1anand1f(x)dxn=1anand1f(x)dx

    either both converge or both diverge. Furthermore, if n=1ann=1an converges, then the NthNth partial sum approximation SNSN is accurate up to an error RNRN where N+1f(x)dx<RN<Nf(x)dx.N+1f(x)dx<RN<Nf(x)dx.
  • The p-series n=11/npn=11/np converges if p>1p>1 and diverges if p1.p1.

5.4 Comparison Tests

  • The comparison tests are used to determine convergence or divergence of series with positive terms.
  • When using the comparison tests, a series n=1ann=1an is often compared to a geometric or p-series.

5.5 Alternating Series

  • For an alternating series n=1(−1)n+1bn,n=1(−1)n+1bn, if bk+1bkbk+1bk for all kk and bk0bk0 as k,k, the alternating series converges.
  • If n=1|an|n=1|an| converges, then n=1ann=1an converges.

5.6 Ratio and Root Tests

  • For the ratio test, we consider
    ρ=limn|an+1an|.ρ=limn|an+1an|.

    If ρ<1,ρ<1, the series n=1ann=1an converges absolutely. If ρ>1,ρ>1, the series diverges. If ρ=1,ρ=1, the test does not provide any information. This test is useful for series whose terms involve factorials.
  • For the root test, we consider
    ρ=limn|an|n.ρ=limn|an|n.

    If ρ<1,ρ<1, the series n=1ann=1an converges absolutely. If ρ>1,ρ>1, the series diverges. If ρ=1,ρ=1, the test does not provide any information. The root test is useful for series whose terms involve powers.
  • For a series that is similar to a geometric series or pseries,pseries, consider one of the comparison tests.
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