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Calculus Volume 1

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Calculus Volume 1Index
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Table of contents
  1. Preface
  2. 1 Functions and Graphs
    1. Introduction
    2. 1.1 Review of Functions
    3. 1.2 Basic Classes of Functions
    4. 1.3 Trigonometric Functions
    5. 1.4 Inverse Functions
    6. 1.5 Exponential and Logarithmic Functions
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Limits
    1. Introduction
    2. 2.1 A Preview of Calculus
    3. 2.2 The Limit of a Function
    4. 2.3 The Limit Laws
    5. 2.4 Continuity
    6. 2.5 The Precise Definition of a Limit
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Derivatives
    1. Introduction
    2. 3.1 Defining the Derivative
    3. 3.2 The Derivative as a Function
    4. 3.3 Differentiation Rules
    5. 3.4 Derivatives as Rates of Change
    6. 3.5 Derivatives of Trigonometric Functions
    7. 3.6 The Chain Rule
    8. 3.7 Derivatives of Inverse Functions
    9. 3.8 Implicit Differentiation
    10. 3.9 Derivatives of Exponential and Logarithmic Functions
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Applications of Derivatives
    1. Introduction
    2. 4.1 Related Rates
    3. 4.2 Linear Approximations and Differentials
    4. 4.3 Maxima and Minima
    5. 4.4 The Mean Value Theorem
    6. 4.5 Derivatives and the Shape of a Graph
    7. 4.6 Limits at Infinity and Asymptotes
    8. 4.7 Applied Optimization Problems
    9. 4.8 L’Hôpital’s Rule
    10. 4.9 Newton’s Method
    11. 4.10 Antiderivatives
    12. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Integration
    1. Introduction
    2. 5.1 Approximating Areas
    3. 5.2 The Definite Integral
    4. 5.3 The Fundamental Theorem of Calculus
    5. 5.4 Integration Formulas and the Net Change Theorem
    6. 5.5 Substitution
    7. 5.6 Integrals Involving Exponential and Logarithmic Functions
    8. 5.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Applications of Integration
    1. Introduction
    2. 6.1 Areas between Curves
    3. 6.2 Determining Volumes by Slicing
    4. 6.3 Volumes of Revolution: Cylindrical Shells
    5. 6.4 Arc Length of a Curve and Surface Area
    6. 6.5 Physical Applications
    7. 6.6 Moments and Centers of Mass
    8. 6.7 Integrals, Exponential Functions, and Logarithms
    9. 6.8 Exponential Growth and Decay
    10. 6.9 Calculus of the Hyperbolic Functions
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. A | Table of Integrals
  9. B | Table of Derivatives
  10. C | Review of Pre-Calculus
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
  12. Index
A
absolute extremum 4.3 Maxima and Minima
absolute maximum 4.3 Maxima and Minima
absolute minimum 4.3 Maxima and Minima
absolute value function 1.1 Review of Functions
acceleration 3.1 Defining the Derivative, 3.4 Derivatives as Rates of Change
algebraic function 1.2 Basic Classes of Functions
amount of change 3.4 Derivatives as Rates of Change
antiderivative 4.10 Antiderivatives
aphelion 5.3 The Fundamental Theorem of Calculus
arc length 6.4 Arc Length of a Curve and Surface Area
Archimedes 2.3 The Limit Laws, 5.1 Approximating Areas
area density 6.5 Physical Applications
Area Problem 2.1 A Preview of Calculus
area under the curve 5.1 Approximating Areas
average rate of change 3.4 Derivatives as Rates of Change
average value of the function 5.2 The Definite Integral
average velocity 2.1 A Preview of Calculus, 3.1 Defining the Derivative
B
bald eagle 5.4 Integration Formulas and the Net Change Theorem
base 1.5 Exponential and Logarithmic Functions
C
carbon dating 6.8 Exponential Growth and Decay
catenary 6.9 Calculus of the Hyperbolic Functions
center of mass 6.6 Moments and Centers of Mass
centroid 6.6 Moments and Centers of Mass
chain rule 3.6 The Chain Rule
change of variables 5.5 Substitution
chaos 4.9 Newton’s Method
common logarithm 1.5 Exponential and Logarithmic Functions
composite function 1.1 Review of Functions
compound interest 6.8 Exponential Growth and Decay
compounding interest 1.5 Exponential and Logarithmic Functions
concave down 4.5 Derivatives and the Shape of a Graph
concave up 4.5 Derivatives and the Shape of a Graph
concavity 4.5 Derivatives and the Shape of a Graph
concavity test 4.5 Derivatives and the Shape of a Graph
conditional statement 2.5 The Precise Definition of a Limit
constant function 1.2 Basic Classes of Functions
Constant multiple law for limits 2.3 The Limit Laws
Constant Multiple Rule 3.3 Differentiation Rules
constant rule 3.3 Differentiation Rules
continuity over an interval 2.4 Continuity
continuous at a point 2.4 Continuity
continuous from the left 2.4 Continuity
continuous from the right 2.4 Continuity
critical number 4.3 Maxima and Minima
cross-section 6.2 Determining Volumes by Slicing
cubic function 1.2 Basic Classes of Functions
D
deceleration 5.4 Integration Formulas and the Net Change Theorem
decreasing on the interval I I 1.1 Review of Functions
definite integral 5.2 The Definite Integral
degree 1.2 Basic Classes of Functions
density function 6.5 Physical Applications
dependent variable 1.1 Review of Functions
derivative 3.1 Defining the Derivative
derivative function 3.2 The Derivative as a Function
Difference law for limits 2.3 The Limit Laws
difference quotient 3.1 Defining the Derivative
Difference Rule 3.3 Differentiation Rules
differentiable at a a 3.2 The Derivative as a Function
differentiable function 3.2 The Derivative as a Function
differentiable on S S 3.2 The Derivative as a Function
Differential calculus 2.1 A Preview of Calculus
differential form 4.2 Linear Approximations and Differentials
differentials 4.2 Linear Approximations and Differentials
differentiation 3.1 Defining the Derivative
discontinuous at a point 2.4 Continuity
disk method 6.2 Determining Volumes by Slicing
displacement 5.2 The Definite Integral, 5.4 Integration Formulas and the Net Change Theorem
domain 1.1 Review of Functions
doubling time 6.8 Exponential Growth and Decay
dummy variable 5.1 Approximating Areas, 5.2 The Definite Integral
E
earthquake 1.5 Exponential and Logarithmic Functions
end behavior 1.2 Basic Classes of Functions, 4.6 Limits at Infinity and Asymptotes
endpoints 1.1 Review of Functions
epsilon-delta definition of the limit 2.5 The Precise Definition of a Limit
evaluation theorem 5.3 The Fundamental Theorem of Calculus
even function 1.1 Review of Functions, 5.4 Integration Formulas and the Net Change Theorem
existential quantifier 2.5 The Precise Definition of a Limit
exponent 1.5 Exponential and Logarithmic Functions
exponential decay 6.8 Exponential Growth and Decay
exponential growth 6.8 Exponential Growth and Decay
Extreme Value Theorem 4.3 Maxima and Minima
F
fave 5.2 The Definite Integral
federal income tax 5.4 Integration Formulas and the Net Change Theorem
Fermat’s theorem 4.3 Maxima and Minima
first derivative test 4.5 Derivatives and the Shape of a Graph
folium of Descartes 3.8 Implicit Differentiation
fruit flies 5.6 Integrals Involving Exponential and Logarithmic Functions
frustum 6.4 Arc Length of a Curve and Surface Area
function 1.1 Review of Functions
Fundamental Theorem of Calculus 5.3 The Fundamental Theorem of Calculus
Fundamental Theorem of Calculus, Part 1 5.3 The Fundamental Theorem of Calculus
Fundamental Theorem of Calculus, Part 2 5.3 The Fundamental Theorem of Calculus
G
graph of a function 1.1 Review of Functions
growth of bacteria 5.6 Integrals Involving Exponential and Logarithmic Functions
H
half-life 6.8 Exponential Growth and Decay
hanging cables 6.9 Calculus of the Hyperbolic Functions
higher-order derivatives 3.2 The Derivative as a Function
Holling type I equation 3.4 Derivatives as Rates of Change
Hooke’s law 6.5 Physical Applications
Hoover Dam 6.5 Physical Applications
horizontal asymptote 4.6 Limits at Infinity and Asymptotes
horizontal line test 1.4 Inverse Functions
hydrostatic pressure 6.5 Physical Applications
hyperbolic functions 1.5 Exponential and Logarithmic Functions
I
iceboat 5.4 Integration Formulas and the Net Change Theorem
implicit differentiation 3.8 Implicit Differentiation
increasing on the interval I I 1.1 Review of Functions
indefinite integral 4.10 Antiderivatives
independent variable 1.1 Review of Functions
indeterminate forms 4.8 L’Hôpital’s Rule
index 5.1 Approximating Areas
infinite discontinuity 2.4 Continuity
infinite limit at infinity 4.6 Limits at Infinity and Asymptotes
infinite limits 2.2 The Limit of a Function
inflection point 4.5 Derivatives and the Shape of a Graph
initial-value problem 4.10 Antiderivatives
input 1.1 Review of Functions
instantaneous rate of change 3.1 Defining the Derivative
instantaneous velocity 2.1 A Preview of Calculus, 3.1 Defining the Derivative
integrable function 5.2 The Definite Integral
Integral calculus 2.1 A Preview of Calculus
integrand 5.2 The Definite Integral
integration by substitution 5.5 Substitution
interior points 3.2 The Derivative as a Function
Intermediate Value Theorem 2.4 Continuity
interval notation 1.1 Review of Functions
intuitive definition of the limit 2.2 The Limit of a Function
inverse function 1.4 Inverse Functions
inverse hyperbolic functions 1.5 Exponential and Logarithmic Functions
inverse trigonometric functions 1.4 Inverse Functions
iterative process 4.9 Newton’s Method
J
joule 6.5 Physical Applications
jump discontinuity 2.4 Continuity
L
L’Hôpital’s rule 4.8 L’Hôpital’s Rule
lamina 6.6 Moments and Centers of Mass
leading coefficient 1.2 Basic Classes of Functions
left-endpoint approximation 5.1 Approximating Areas
Leibniz 3.1 Defining the Derivative, 5.2 The Definite Integral
limit 2.1 A Preview of Calculus
limit at infinity 4.6 Limits at Infinity and Asymptotes, 4.6 Limits at Infinity and Asymptotes
limit laws 2.3 The Limit Laws
limits of integration 5.2 The Definite Integral
linear approximation 4.2 Linear Approximations and Differentials
linear function 1.2 Basic Classes of Functions
linearization 4.2 Linear Approximations and Differentials
local extremum 4.3 Maxima and Minima
local maximum 4.3 Maxima and Minima
local minimum 4.3 Maxima and Minima
logarithmic differentiation 3.9 Derivatives of Exponential and Logarithmic Functions
logarithmic function 1.2 Basic Classes of Functions
lower sum 5.1 Approximating Areas
M
Mandelbrot set 4.9 Newton’s Method
marginal cost 3.4 Derivatives as Rates of Change
marginal profit 3.4 Derivatives as Rates of Change
mathematical models 1.2 Basic Classes of Functions
maximizing revenue 4.7 Applied Optimization Problems
Mean Value Theorem 4.4 The Mean Value Theorem
Mean Value Theorem for Integrals 5.3 The Fundamental Theorem of Calculus
method of cylindrical shells. 6.3 Volumes of Revolution: Cylindrical Shells
method of exhaustion 5.1 Approximating Areas
moment 6.6 Moments and Centers of Mass
multivariable calculus 2.1 A Preview of Calculus
N
natural exponential function 1.5 Exponential and Logarithmic Functions, 3.9 Derivatives of Exponential and Logarithmic Functions
natural logarithm 1.5 Exponential and Logarithmic Functions
natural logarithmic function 3.9 Derivatives of Exponential and Logarithmic Functions
net change theorem 5.4 Integration Formulas and the Net Change Theorem
net signed area 5.2 The Definite Integral
Newton 3.1 Defining the Derivative, 5.3 The Fundamental Theorem of Calculus
Newton’s law of cooling 6.8 Exponential Growth and Decay
Newton’s method 4.9 Newton’s Method
number e e 1.5 Exponential and Logarithmic Functions
O
oblique asymptote 4.6 Limits at Infinity and Asymptotes
odd function 1.1 Review of Functions, 5.4 Integration Formulas and the Net Change Theorem
one-sided limit 2.2 The Limit of a Function
one-to-one function 1.4 Inverse Functions
optimization problems 4.7 Applied Optimization Problems
output 1.1 Review of Functions
P
partition 5.1 Approximating Areas
Pascal’s principle 6.5 Physical Applications
pascals 6.5 Physical Applications
percentage error 4.2 Linear Approximations and Differentials
perihelion 5.3 The Fundamental Theorem of Calculus
periodic functions. 1.3 Trigonometric Functions
piecewise-defined function 1.2 Basic Classes of Functions
piecewise-defined functions 1.1 Review of Functions
point-slope equation 1.2 Basic Classes of Functions
polynomial function 1.2 Basic Classes of Functions
population growth 1.5 Exponential and Logarithmic Functions, 6.8 Exponential Growth and Decay
population growth rates 3.4 Derivatives as Rates of Change
power function 1.2 Basic Classes of Functions
Power law for limits 2.3 The Limit Laws
power rule 3.3 Differentiation Rules
price–demand function 5.6 Integrals Involving Exponential and Logarithmic Functions
Product law for limits 2.3 The Limit Laws
product rule 3.3 Differentiation Rules
propagated error 4.2 Linear Approximations and Differentials
Pythagorean theorem 4.1 Related Rates
Q
quadratic function 1.2 Basic Classes of Functions
Quotient law for limits 2.3 The Limit Laws
quotient rule 3.3 Differentiation Rules
R
radial density 6.5 Physical Applications
radians 1.3 Trigonometric Functions
range 1.1 Review of Functions
rate of change 2.1 A Preview of Calculus, 5.4 Integration Formulas and the Net Change Theorem
rational function 1.2 Basic Classes of Functions
Regiomontanus’ problem 4 Review Exercises
regular partition 5.1 Approximating Areas
related rates 4.1 Related Rates
relative error 4.2 Linear Approximations and Differentials
removable discontinuity 2.4 Continuity
restricted domain 1.4 Inverse Functions
Richter scale 1.5 Exponential and Logarithmic Functions
Riemann sum 5.1 Approximating Areas
right-endpoint approximation 5.1 Approximating Areas
Rolle’s theorem 4.4 The Mean Value Theorem
root function 1.2 Basic Classes of Functions
Root law for limits 2.3 The Limit Laws
S
secant 2.1 A Preview of Calculus
secant method 4.9 Newton’s Method
second derivative test 4.5 Derivatives and the Shape of a Graph
sigma notation 5.1 Approximating Areas
simple interest 6.8 Exponential Growth and Decay
skydiver 5.3 The Fundamental Theorem of Calculus
slicing method 6.2 Determining Volumes by Slicing
slope 1.2 Basic Classes of Functions
slope-intercept form 1.2 Basic Classes of Functions
smooth 6.4 Arc Length of a Curve and Surface Area
solid of revolution 6.2 Determining Volumes by Slicing
speed 3.4 Derivatives as Rates of Change
spring constant 6.5 Physical Applications
squeeze theorem 2.3 The Limit Laws
standard form of a line 1.2 Basic Classes of Functions
Sum law for limits 2.3 The Limit Laws
Sum Rule 3.3 Differentiation Rules
summation notation 5.1 Approximating Areas
sums and powers of integers 5.1 Approximating Areas
Surface area 6.4 Arc Length of a Curve and Surface Area
symmetry about the origin 1.1 Review of Functions
symmetry about the y-axis 1.1 Review of Functions
symmetry principle 6.6 Moments and Centers of Mass
T
table of values 1.1 Review of Functions
tangent 2.1 A Preview of Calculus
tangent line approximation 4.2 Linear Approximations and Differentials
Tangent Problem 2.1 A Preview of Calculus
theorem of Pappus for volume 6.6 Moments and Centers of Mass
total area 5.2 The Definite Integral
Tour de France 5.4 Integration Formulas and the Net Change Theorem
transcendental functions 1.2 Basic Classes of Functions
transformation of a function 1.2 Basic Classes of Functions
triangle inequality 2.5 The Precise Definition of a Limit
trigonometric functions 1.3 Trigonometric Functions
trigonometric identity 1.3 Trigonometric Functions
U
universal quantifier 2.5 The Precise Definition of a Limit
upper sum 5.1 Approximating Areas
V
variable of integration 5.2 The Definite Integral
velocity 5.4 Integration Formulas and the Net Change Theorem
vertical asymptote 2.2 The Limit of a Function
vertical line test 1.1 Review of Functions
W
washer method 6.2 Determining Volumes by Slicing
wingsuits 5.3 The Fundamental Theorem of Calculus
work 6.5 Physical Applications
Z
zeroes of functions 4.9 Newton’s Method
zeros of a function 1.1 Review of Functions
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