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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
absolute convergence
if the series n=1|an|n=1|an| converges, the series n=1ann=1an is said to converge absolutely
alternating series
a series of the form n=1(−1)n+1bnn=1(−1)n+1bn or n=1(−1)nbn,n=1(−1)nbn, where bn0,bn0, is called an alternating series
alternating series test
for an alternating series of either form, if bn+1bnbn+1bn for all integers n1n1 and bn0,bn0, then an alternating series converges
arithmetic sequence
a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence
bounded above
a sequence {an}{an} is bounded above if there exists a constant MM such that anManM for all positive integers nn
bounded below
a sequence {an}{an} is bounded below if there exists a constant MM such that ManMan for all positive integers nn
bounded sequence
a sequence {an}{an} is bounded if there exists a constant MM such that |an|M|an|M for all positive integers nn
comparison test
if 0anbn0anbn for all nNnN and n=1bnn=1bn converges, then n=1ann=1an converges; if anbn0anbn0 for all nNnN and n=1bnn=1bn diverges, then n=1ann=1an diverges
conditional convergence
if the series n=1ann=1an converges, but the series n=1|an|n=1|an| diverges, the series n=1ann=1an is said to converge conditionally
convergence of a series
a series converges if the sequence of partial sums for that series converges
convergent sequence
a convergent sequence is a sequence {an}{an} for which there exists a real number LL such that anan is arbitrarily close to LL as long as nn is sufficiently large
divergence of a series
a series diverges if the sequence of partial sums for that series diverges
divergence test
if limnan0,limnan0, then the series n=1ann=1an diverges
divergent sequence
a sequence that is not convergent is divergent
explicit formula
a sequence may be defined by an explicit formula such that an=f(n)an=f(n)
geometric sequence
a sequence {an}{an} in which the ratio an+1/anan+1/an is the same for all positive integers nn is called a geometric sequence
geometric series
a geometric series is a series that can be written in the form
n=1arn1=a+ar+ar2+ar3+n=1arn1=a+ar+ar2+ar3+
harmonic series
the harmonic series takes the form
n=11n=1+12+13+n=11n=1+12+13+
index variable
the subscript used to define the terms in a sequence is called the index
infinite series
an infinite series is an expression of the form
a1+a2+a3+=n=1ana1+a2+a3+=n=1an
integral test
for a series n=1ann=1an with positive terms an,an, if there exists a continuous, decreasing function ff 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
limit comparison test
suppose an,bn0an,bn0 for all n1.n1. If limnan/bnL0,limnan/bnL0, then n=1ann=1an and n=1bnn=1bn both converge or both diverge; if limnan/bn0limnan/bn0 and n=1bnn=1bn converges, then n=1ann=1an converges. If limnan/bn,limnan/bn, and n=1bnn=1bn diverges, then n=1ann=1an diverges
limit of a sequence
the real number LL to which a sequence converges is called the limit of the sequence
monotone sequence
an increasing or decreasing sequence
p-series
a series of the form n=11/npn=11/np
partial sum
the kthkth partial sum of the infinite series n=1ann=1an is the finite sum
Sk=n=1kan=a1+a2+a3++akSk=n=1kan=a1+a2+a3++ak
ratio test
for a series n=1ann=1an with nonzero terms, let ρ=limn|an+1/an|;ρ=limn|an+1/an|; if 0ρ<1,0ρ<1, the series converges absolutely; if ρ>1,ρ>1, the series diverges; if ρ=1,ρ=1, the test is inconclusive
recurrence relation
a recurrence relation is a relationship in which a term anan in a sequence is defined in terms of earlier terms in the sequence
remainder estimate
for a series n=1ann=1an with positive terms anan and a continuous, decreasing function ff such that f(n)=anf(n)=an for all positive integers n,n, the remainder RN=n=1ann=1NanRN=n=1ann=1Nan satisfies the following estimate:
N+1f(x)dx<RN<Nf(x)dxN+1f(x)dx<RN<Nf(x)dx
root test
for a series n=1an,n=1an, let ρ=limn|an|n;ρ=limn|an|n; if 0ρ<1,0ρ<1, the series converges absolutely; if ρ>1,ρ>1, the series diverges; if ρ=1,ρ=1, the test is inconclusive
sequence
an ordered list of numbers of the form a1,a2,a3,…a1,a2,a3,… is a sequence
telescoping series
a telescoping series is one in which most of the terms cancel in each of the partial sums
term
the number anan in the sequence {an}{an} is called the nthnth term of the sequence
unbounded sequence
a sequence that is not bounded is called unbounded
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