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Calculus Volume 2

Index

Calculus Volume 2Index
  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
A
absolute convergence 5.5 Alternating Series
absolute error 3.6 Numerical Integration
air resistance 4.5 First-order Linear Equations
Airy’s equation 6.4 Working with Taylor Series
algebraic function 3.1 Integration by Parts
alternating series 5.5 Alternating Series
alternating series test 5.5 Alternating Series
angular coordinate 7.3 Polar Coordinates
annuities 6.2 Properties of Power Series
annuity payments 6 Review Exercises
aphelion 1.3 The Fundamental Theorem of Calculus
arc length 2.4 Arc Length of a Curve and Surface Area
Archimedean spiral 7.3 Polar Coordinates
Archimedes 1.1 Approximating Areas
area density 2.5 Physical Applications
area under the curve 1.1 Approximating Areas
arithmetic sequence 5.1 Sequences
asymptotically semi-stable solution 4.2 Direction Fields and Numerical Methods
asymptotically stable solution 4.2 Direction Fields and Numerical Methods
asymptotically unstable solution 4.2 Direction Fields and Numerical Methods
autonomous differential equation 4.3 Separable Equations
average value of the function 1.2 The Definite Integral
B
bald eagle 1.4 Integration Formulas and the Net Change Theorem
binomial series 6.4 Working with Taylor Series
bounded above 5.1 Sequences
bounded below 5.1 Sequences
bounded sequence 5.1 Sequences
C
carbon dating 2.8 Exponential Growth and Decay
cardioid 7.3 Polar Coordinates
carrying capacity 4.4 The Logistic Equation
catenary 2.9 Calculus of the Hyperbolic Functions
center of mass 2.6 Moments and Centers of Mass
centroid 2.6 Moments and Centers of Mass
chambered nautilus Introduction, 7.3 Polar Coordinates
change of variables 1.5 Substitution
cissoid of Diocles 7.4 Area and Arc Length in Polar Coordinates
comparison test 5.4 Comparison Tests
compound interest 2.8 Exponential Growth and Decay
computer algebra systems (CAS) 3.5 Other Strategies for Integration
conditional convergence 5.5 Alternating Series
conic section 7.5 Conic Sections
convergence of a series 5.2 Infinite Series
convergent sequence 5.1 Sequences
coupon collector’s problem 5.3 The Divergence and Integral Tests
cross-section 2.2 Determining Volumes by Slicing
curtate cycloid 7.1 Parametric Equations
cusps 7.1 Parametric Equations
cycloid 7.1 Parametric Equations
D
deceleration 1.4 Integration Formulas and the Net Change Theorem
definite integral 1.2 The Definite Integral
density function 2.5 Physical Applications
differential equation 4.1 Basics of Differential Equations
direction field (slope field) 4.2 Direction Fields and Numerical Methods
directrix 7.5 Conic Sections
discriminant 7.5 Conic Sections
disease epidemics 4.2 Direction Fields and Numerical Methods
disk method 2.2 Determining Volumes by Slicing
displacement 1.2 The Definite Integral, 1.4 Integration Formulas and the Net Change Theorem
divergence of a series 5.2 Infinite Series
divergence test 5.3 The Divergence and Integral Tests
divergent sequence 5.1 Sequences
doubling time 2.8 Exponential Growth and Decay
drugs in the bloodstream 4.3 Separable Equations
dummy variable 1.1 Approximating Areas, 1.2 The Definite Integral
E
Earth’s orbit 7.1 Parametric Equations
eccentricity 7.5 Conic Sections
elliptic integral 6.4 Working with Taylor Series
epitrochoid 7.1 Parametric Equations
equilibrium solution 4.2 Direction Fields and Numerical Methods
Euler transform 5.5 Alternating Series
Euler’s constant 5.2 Infinite Series
Euler’s formula 6 Review Exercises
Euler’s Method 4.2 Direction Fields and Numerical Methods
evaluation theorem 1.3 The Fundamental Theorem of Calculus
even function 1.4 Integration Formulas and the Net Change Theorem
explicit formulas 5.1 Sequences
exponential decay 2.8 Exponential Growth and Decay
exponential growth 2.8 Exponential Growth and Decay
F
fave 1.2 The Definite Integral
federal income tax 1.4 Integration Formulas and the Net Change Theorem
Fibonacci numbers 5.1 Sequences
focal parameter 7.5 Conic Sections
focus 7.5 Conic Sections
Fresnel integrals 6.4 Working with Taylor Series
fruit flies 1.6 Integrals Involving Exponential and Logarithmic Functions
frustum 2.4 Arc Length of a Curve and Surface Area
Fundamental Theorem of Calculus 1.3 The Fundamental Theorem of Calculus
Fundamental Theorem of Calculus, Part 1 1.3 The Fundamental Theorem of Calculus
Fundamental Theorem of Calculus, Part 2 1.3 The Fundamental Theorem of Calculus
G
Gabriel’s Horn 3.7 Improper Integrals
general form 7.5 Conic Sections
general solution 4.1 Basics of Differential Equations
geometric sequence 5.1 Sequences
geometric series 5.2 Infinite Series
golden ratio 5.1 Sequences
Gompertz equation 4.4 The Logistic Equation
growth of bacteria 1.6 Integrals Involving Exponential and Logarithmic Functions
growth rate 4.4 The Logistic Equation
H
half-life 2.8 Exponential Growth and Decay
hanging cables 2.9 Calculus of the Hyperbolic Functions
harmonic series 5.2 Infinite Series
Hooke’s law 2.5 Physical Applications
Hoover Dam 2.5 Physical Applications
hydrostatic pressure 2.5 Physical Applications
hypocycloid 7.1 Parametric Equations
I
iceboat 1.4 Integration Formulas and the Net Change Theorem
improper integral 3.7 Improper Integrals
indefinite integrals 3.1 Integration by Parts
index 1.1 Approximating Areas
index variable 5.1 Sequences
infinite sequence 5.1 Sequences
infinite series 5.2 Infinite Series
initial population 4.4 The Logistic Equation
initial value 4.1 Basics of Differential Equations
initial velocity 4.1 Basics of Differential Equations
initial-value problem 4.1 Basics of Differential Equations
integrable function 1.2 The Definite Integral
integral test 5.3 The Divergence and Integral Tests
integrand 1.2 The Definite Integral
integrating factor 4.5 First-order Linear Equations
integration by parts 3.1 Integration by Parts
integration by substitution 1.5 Substitution
integration tables 3.5 Other Strategies for Integration
interval of convergence 6.1 Power Series and Functions
J
joule 2.5 Physical Applications
K
Koch’s snowflake 5.2 Infinite Series
L
lamina 2.6 Moments and Centers of Mass
Laplace transform 3.7 Improper Integrals, 3.7 Improper Integrals
left-endpoint approximation 1.1 Approximating Areas
Leibniz 1.2 The Definite Integral
limaçon 7.3 Polar Coordinates
limit comparison test 5.4 Comparison Tests
limit of the sequence 5.1 Sequences
limits of integration 1.2 The Definite Integral
linear 4.5 First-order Linear Equations
logarithmic function 3.1 Integration by Parts
logistic differential equation 4.4 The Logistic Equation
lower sum 1.1 Approximating Areas
M
Maclaurin polynomials 6.3 Taylor and Maclaurin Series
Maclaurin series 6.3 Taylor and Maclaurin Series
major axis 7.5 Conic Sections
Mean Value Theorem for Integrals 1.3 The Fundamental Theorem of Calculus
method of cylindrical shells. 2.3 Volumes of Revolution: Cylindrical Shells
method of equating coefficients 3.4 Partial Fractions
method of exhaustion 1.1 Approximating Areas
method of strategic substitution 3.4 Partial Fractions
midpoint rule 3.6 Numerical Integration
minor axis 7.5 Conic Sections
moment 2.6 Moments and Centers of Mass
monotone sequence 5.1 Sequences
N
nappes 7.5 Conic Sections
net change theorem 1.4 Integration Formulas and the Net Change Theorem
net signed area 1.2 The Definite Integral
Newton 1.3 The Fundamental Theorem of Calculus
Newton’s law of cooling 2.8 Exponential Growth and Decay, 4.3 Separable Equations
Newton’s second law of motion 4.1 Basics of Differential Equations
nonelementary integral 6.4 Working with Taylor Series
numerical integration 3.6 Numerical Integration
O
odd function 1.4 Integration Formulas and the Net Change Theorem
order of a differential equation 4.1 Basics of Differential Equations
orientation 7.1 Parametric Equations
P
p-series 5.3 The Divergence and Integral Tests
parameter 7.1 Parametric Equations
parameterization of a curve 7.1 Parametric Equations
parametric curve 7.1 Parametric Equations
parametric equations 7.1 Parametric Equations
partial fraction decomposition 3.4 Partial Fractions
partial sum 5.2 Infinite Series
particular solution 4.1 Basics of Differential Equations
partition 1.1 Approximating Areas
Pascal’s principle 2.5 Physical Applications
pascals 2.5 Physical Applications
perihelion 1.3 The Fundamental Theorem of Calculus
phase line 4.4 The Logistic Equation
polar axis 7.3 Polar Coordinates
polar coordinate system 7.3 Polar Coordinates
polar equations 7.3 Polar Coordinates
pole 7.3 Polar Coordinates
Population growth 2.8 Exponential Growth and Decay
power reduction formulas 3.2 Trigonometric Integrals
power series 6.1 Power Series and Functions
power-reducing identities 3.2 Trigonometric Integrals
present value 6.2 Properties of Power Series
price–demand function 1.6 Integrals Involving Exponential and Logarithmic Functions
probability 3.7 Improper Integrals
probability density function 3.7 Improper Integrals
prolate cycloid 7.1 Parametric Equations
R
radial coordinate 7.3 Polar Coordinates
radial density 2.5 Physical Applications
radius of convergence 6.1 Power Series and Functions
Ramanujan 5.6 Ratio and Root Tests
rate of change 1.4 Integration Formulas and the Net Change Theorem
ratio test 5.6 Ratio and Root Tests
rational functions 3.4 Partial Fractions
RC circuit 4.5 First-order Linear Equations
recurrence relation 5.1 Sequences, 5.1 Sequences
regular partition 1.1 Approximating Areas
relative error 3.6 Numerical Integration
remainder estimate 5.3 The Divergence and Integral Tests
Riemann sum 1.1 Approximating Areas
Riemann sums 3.6 Numerical Integration
right-endpoint approximation 1.1 Approximating Areas
root test 5.6 Ratio and Root Tests
rose 7.3 Polar Coordinates
S
separable differential equation 4.3 Separable Equations
separation of variables 4.3 Separable Equations
Sierpinski triangle 5.2 Infinite Series
sigma notation 1.1 Approximating Areas
simple interest 2.8 Exponential Growth and Decay
Simpson’s rule 3.6 Numerical Integration
skydiver 1.3 The Fundamental Theorem of Calculus
slicing method 2.2 Determining Volumes by Slicing
smooth 2.4 Arc Length of a Curve and Surface Area
solid of revolution 2.2 Determining Volumes by Slicing
solution concentrations 4.3 Separable Equations
solution curve 4.2 Direction Fields and Numerical Methods
solution to a differential equation 4.1 Basics of Differential Equations
space-filling curve 7.3 Polar Coordinates
space-filling curves 7.1 Parametric Equations
spring constant 2.5 Physical Applications
standard form 4.5 First-order Linear Equations, 7.5 Conic Sections
step size 4.2 Direction Fields and Numerical Methods
summation notation 1.1 Approximating Areas
sums and powers of integers 1.1 Approximating Areas
Surface area 2.4 Arc Length of a Curve and Surface Area
symmetry 7.3 Polar Coordinates
symmetry principle 2.6 Moments and Centers of Mass
T
Taylor polynomials 6.3 Taylor and Maclaurin Series
Taylor series 6.3 Taylor and Maclaurin Series
Taylor’s theorem with remainder 6.3 Taylor and Maclaurin Series
telescoping series 5.2 Infinite Series
term 5.1 Sequences
term-by-term differentiation of a power series 6.2 Properties of Power Series
term-by-term integration of a power series 6.2 Properties of Power Series
theorem of Pappus for volume 2.6 Moments and Centers of Mass
threshold population 4.4 The Logistic Equation
total area 1.2 The Definite Integral
Tour de France 1.4 Integration Formulas and the Net Change Theorem
traffic accidents 3.7 Improper Integrals
trapezoidal rule 3.6 Numerical Integration
trigonometric integrals 3.2 Trigonometric Integrals
trigonometric substitution 3.3 Trigonometric Substitution
U
unbounded sequence 5.1 Sequences
upper sum 1.1 Approximating Areas
V
variable of integration 1.2 The Definite Integral
velocity 1.4 Integration Formulas and the Net Change Theorem
vertex 7.5 Conic Sections
von Bertalanffy growth 5.2 Infinite Series
W
washer method 2.2 Determining Volumes by Slicing
wingsuits 1.3 The Fundamental Theorem of Calculus
witch of Agnesi 7.1 Parametric Equations
work 2.5 Physical Applications
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