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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 value function
f(x)={x,x<0x,x0f(x)={x,x<0x,x0
algebraic function
a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable xx
base
the number bb in the exponential function f(x)=bxf(x)=bx and the logarithmic function f(x)=logbxf(x)=logbx
composite function
given two functions ff and g,g, a new function, denoted gf,gf, such that (gf)(x)=g(f(x))(gf)(x)=g(f(x))
cubic function
a polynomial of degree 3; that is, a function of the form f(x)=ax3+bx2+cx+d,f(x)=ax3+bx2+cx+d, where a0a0
decreasing on the interval II
a function decreasing on the interval II if, for all x1,x2I,f(x1)f(x2)x1,x2I,f(x1)f(x2) if x1<x2x1<x2
degree
for a polynomial function, the value of the largest exponent of any term
dependent variable
the output variable for a function
domain
the set of inputs for a function
even function
a function is even if f(x)=f(x)f(x)=f(x) for all xx in the domain of ff
exponent
the value xx in the expression bxbx
function
a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output
graph of a function
the set of points (x,y)(x,y) such that xx is in the domain of ff and y=f(x)y=f(x)
horizontal line test
a function ff is one-to-one if and only if every horizontal line intersects the graph of f,f, at most, once
hyperbolic functions
the functions denoted sinh,cosh,tanh,csch,sech,sinh,cosh,tanh,csch,sech, and coth,coth, which involve certain combinations of exex and exex
increasing on the interval II
a function increasing on the interval II if for all x1,x2I,f(x1)f(x2)x1,x2I,f(x1)f(x2) if x1<x2x1<x2
independent variable
the input variable for a function
inverse function
for a function f,f, the inverse function f−1f−1 satisfies f−1(y)=xf−1(y)=x if f(x)=yf(x)=y
inverse hyperbolic functions
the inverses of the hyperbolic functions where coshcosh and sechsech are restricted to the domain [0,);[0,); each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function
inverse trigonometric functions
the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions
linear function
a function that can be written in the form f(x)=mx+bf(x)=mx+b
logarithmic function
a function of the form f(x)=logb(x)f(x)=logb(x) for some base b>0,b1b>0,b1 such that y=logb(x)y=logb(x) if and only if by=xby=x
mathematical model
A method of simulating real-life situations with mathematical equations
natural exponential function
the function f(x)=exf(x)=ex
natural logarithm
the function lnx=logexlnx=logex
number e
as mm gets larger, the quantity (1+(1/m)m(1+(1/m)m gets closer to some real number; we define that real number to be e;e; the value of ee is approximately 2.7182822.718282
odd function
a function is odd if f(x)=f(x)f(x)=f(x) for all xx in the domain of ff
one-to-one function
a function ff is one-to-one if f(x1)f(x2)f(x1)f(x2) if x1x2x1x2
periodic function
a function is periodic if it has a repeating pattern as the values of xx move from left to right
piecewise-defined function
a function that is defined differently on different parts of its domain
point-slope equation
equation of a linear function indicating its slope and a point on the graph of the function
polynomial function
a function of the form f(x)=anxn+an1xn1++a1x+a0f(x)=anxn+an1xn1++a1x+a0
power function
a function of the form f(x)=xnf(x)=xn for any positive integer n1n1
quadratic function
a polynomial of degree 2; that is, a function of the form f(x)=ax2+bx+cf(x)=ax2+bx+c where a0a0
radians
for a circular arc of length ss on a circle of radius 1, the radian measure of the associated angle θθ is ss
range
the set of outputs for a function
rational function
a function of the form f(x)=p(x)/q(x),f(x)=p(x)/q(x), where p(x)p(x) and q(x)q(x) are polynomials
restricted domain
a subset of the domain of a function ff
root function
a function of the form f(x)=x1/nf(x)=x1/n for any integer n2n2
slope
the change in y for each unit change in x
slope-intercept form
equation of a linear function indicating its slope and y-intercept
symmetry about the origin
the graph of a function ff is symmetric about the origin if (x,y)(x,y) is on the graph of ff whenever (x,y)(x,y) is on the graph
symmetry about the y-axis
the graph of a function ff is symmetric about the yy-axis if (x,y)(x,y) is on the graph of ff whenever (x,y)(x,y) is on the graph
table of values
a table containing a list of inputs and their corresponding outputs
transcendental function
a function that cannot be expressed by a combination of basic arithmetic operations
transformation of a function
a shift, scaling, or reflection of a function
trigonometric functions
functions of an angle defined as ratios of the lengths of the sides of a right triangle
trigonometric identity
an equation involving trigonometric functions that is true for all angles θθ for which the functions in the equation are defined
vertical line test
given the graph of a function, every vertical line intersects the graph, at most, once
zeros of a function
when a real number xx is a zero of a function f,f(x)=0f,f(x)=0
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