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  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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter 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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter 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. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter 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. Key Terms
    13. Key Equations
    14. Key Concepts
    15. Chapter 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter 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
absolute extremum
if ff has an absolute maximum or absolute minimum at c,c, we say ff has an absolute extremum at cc
absolute maximum
if f(c)f(x)f(c)f(x) for all xx in the domain of f,f, we say ff has an absolute maximum at cc
absolute minimum
if f(c)f(x)f(c)f(x) for all xx in the domain of f,f, we say ff has an absolute minimum at cc
antiderivative
a function FF such that F(x)=f(x)F(x)=f(x) for all xx in the domain of ff is an antiderivative of ff
concave down
if ff is differentiable over an interval II and ff is decreasing over I,I, then ff is concave down over II
concave up
if ff is differentiable over an interval II and ff is increasing over I,I, then ff is concave up over II
concavity
the upward or downward curve of the graph of a function
concavity test
suppose ff is twice differentiable over an interval I;I; if f>0f>0 over I,I, then ff is concave up over I;I; if f<0f<0 over I,I, then ff is concave down over II
critical point
if f(c)=0f(c)=0 or f(c)f(c) is undefined, we say that cc is a critical point of ff
differential
the differential dxdx is an independent variable that can be assigned any nonzero real number; the differential dydy is defined to be dy=f(x)dxdy=f(x)dx
differential form
given a differentiable function y=f(x),y=f(x), the equation dy=f(x)dxdy=f(x)dx is the differential form of the derivative of yy with respect to xx
end behavior
the behavior of a function as xx and xx
extreme value theorem
if ff is a continuous function over a finite, closed interval, then ff has an absolute maximum and an absolute minimum
Fermat’s theorem
if ff has a local extremum at c,c, then cc is a critical point of ff
first derivative test
let ff be a continuous function over an interval II containing a critical point cc such that ff is differentiable over II except possibly at c;c; if ff changes sign from positive to negative as xx increases through c,c, then ff has a local maximum at c;c; if ff changes sign from negative to positive as xx increases through c,c, then ff has a local minimum at c;c; if ff does not change sign as xx increases through c,c, then ff does not have a local extremum at cc
horizontal asymptote
if limxf(x)=Llimxf(x)=L or limxf(x)=L,limxf(x)=L, then y=Ly=L is a horizontal asymptote of ff
indefinite integral
the most general antiderivative of f(x)f(x) is the indefinite integral of f;f; we use the notation f(x)dxf(x)dx to denote the indefinite integral of ff
indeterminate forms
when evaluating a limit, the forms 00,00, /,/, 0·,0·, ,, 00,00, 0,0, and 11 are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is
infinite limit at infinity
a function that becomes arbitrarily large as x becomes large
inflection point
if ff is continuous at cc and ff changes concavity at c,c, the point (c,f(c))(c,f(c)) is an inflection point of ff
initial value problem
a problem that requires finding a function yy that satisfies the differential equation dydx=f(x)dydx=f(x) together with the initial condition y(x0)=y0y(x0)=y0
iterative process
process in which a list of numbers x0,x1,x2,x3x0,x1,x2,x3 is generated by starting with a number x0x0 and defining xn=F(xn1)xn=F(xn1) for n1n1
L’Hôpital’s rule
if ff and gg are differentiable functions over an interval a,a, except possibly at a,a, and limxaf(x)=0=limxag(x)limxaf(x)=0=limxag(x) or limxaf(x)limxaf(x) and limxag(x)limxag(x) are infinite, then limxaf(x)g(x)=limxaf(x)g(x),limxaf(x)g(x)=limxaf(x)g(x), assuming the limit on the right exists or is or
limit at infinity
the limiting value, if it exists, of a function as xx or xx
linear approximation
the linear function L(x)=f(a)+f(a)(xa)L(x)=f(a)+f(a)(xa) is the linear approximation of ff at x=ax=a
local extremum
if ff has a local maximum or local minimum at c,c, we say ff has a local extremum at cc
local maximum
if there exists an interval II such that f(c)f(x)f(c)f(x) for all xI,xI, we say ff has a local maximum at cc
local minimum
if there exists an interval II such that f(c)f(x)f(c)f(x) for all xI,xI, we say ff has a local minimum at cc
mean value theorem
if ff is continuous over [a,b][a,b] and differentiable over (a,b),(a,b), then there exists c(a,b)c(a,b) such that
f(c)=f(b)f(a)baf(c)=f(b)f(a)ba
Newton’s method
method for approximating roots of f(x)=0;f(x)=0; using an initial guess x0;x0; each subsequent approximation is defined by the equation xn=xn1f(xn1)f(xn1)xn=xn1f(xn1)f(xn1)
oblique asymptote
the line y=mx+by=mx+b if f(x)f(x) approaches it as xx or xx
optimization problems
problems that are solved by finding the maximum or minimum value of a function
percentage error
the relative error expressed as a percentage
propagated error
the error that results in a calculated quantity f(x)f(x) resulting from a measurement error dx
related rates
are rates of change associated with two or more related quantities that are changing over time
relative error
given an absolute error ΔqΔq for a particular quantity, ΔqqΔqq is the relative error.
rolle’s theorem
if ff is continuous over [a,b][a,b] and differentiable over (a,b),(a,b), and if f(a)=f(b),f(a)=f(b), then there exists c(a,b)c(a,b) such that f(c)=0f(c)=0
second derivative test
suppose f(c)=0f(c)=0 and ff is continuous over an interval containing c;c; if f(c)>0,f(c)>0, then ff has a local minimum at c;c; if f(c)<0,f(c)<0, then ff has a local maximum at c;c; if f(c)=0,f(c)=0, then the test is inconclusive
tangent line approximation (linearization)
since the linear approximation of ff at x=ax=a is defined using the equation of the tangent line, the linear approximation of ff at x=ax=a is also known as the tangent line approximation to ff at x=ax=a
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