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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
average velocity
the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval [t,a][t,a] (if t<at<a or [a,t][a,t] if t>a)t>a), with a position given by s(t),s(t), that is vave=s(t)s(a)tavave=s(t)s(a)ta
constant multiple law for limits
the limit law limxacf(x)=c·limxaf(x)=cLlimxacf(x)=c·limxaf(x)=cL
continuity at a point
A function f(x)f(x) is continuous at a point a if and only if the following three conditions are satisfied: (1) f(a)f(a) is defined, (2) limxaf(x)limxaf(x) exists, and (3) limxaf(x)=f(a)limxaf(x)=f(a)
continuity from the left
A function is continuous from the left at b if limxbf(x)=f(b)limxbf(x)=f(b)
continuity from the right
A function is continuous from the right at a if limxa+f(x)=f(a)limxa+f(x)=f(a)
continuity over an interval
a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function f(x)f(x) is continuous over a closed interval of the form [a,b][a,b] if it is continuous at every point in (a,b),(a,b), and it is continuous from the right at a and from the left at b
difference law for limits
the limit law limxa(f(x)g(x))=limxaf(x)limxag(x)=LMlimxa(f(x)g(x))=limxaf(x)limxag(x)=LM
differential calculus
the field of calculus concerned with the study of derivatives and their applications
discontinuity at a point
A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point
epsilon-delta definition of the limit
limxaf(x)=Llimxaf(x)=L if for every ε>0,ε>0, there exists a δ>0δ>0 such that if 0<|xa|<δ,0<|xa|<δ, then |f(x)L|<ε|f(x)L|<ε
infinite discontinuity
An infinite discontinuity occurs at a point a if limxaf(x)=±limxaf(x)=± or limxa+f(x)=±limxa+f(x)=±
infinite limit
A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a
instantaneous velocity
The instantaneous velocity of an object with a position function that is given by s(t)s(t) is the value that the average velocities on intervals of the form [t,a][t,a] and [a,t][a,t] approach as the values of t move closer to a,a, provided such a value exists
integral calculus
the study of integrals and their applications
Intermediate Value Theorem
Let f be continuous over a closed bounded interval [a,b];[a,b]; if z is any real number between f(a)f(a) and f(b),f(b), then there is a number c in [a,b][a,b] satisfying f(c)=zf(c)=z
intuitive definition of the limit
If all values of the function f(x)f(x) approach the real number L as the values of x(a)x(a) approach a, f(x)f(x) approaches L
jump discontinuity
A jump discontinuity occurs at a point a if limxaf(x)limxaf(x) and limxa+f(x)limxa+f(x) both exist, but limxaf(x)limxa+f(x)limxaf(x)limxa+f(x)
limit
the process of letting x or t approach a in an expression; the limit of a function f(x)f(x) as x approaches a is the value that f(x)f(x) approaches as x approaches a
limit laws
the individual properties of limits; for each of the individual laws, let f(x)f(x) and g(x)g(x) be defined for all xaxa over some open interval containing a; assume that L and M are real numbers so that limxaf(x)=Llimxaf(x)=L and limxag(x)=M;limxag(x)=M; let c be a constant
multivariable calculus
the study of the calculus of functions of two or more variables
one-sided limit
A one-sided limit of a function is a limit taken from either the left or the right
power law for limits
the limit law limxa(f(x))n=(limxaf(x))n=Lnlimxa(f(x))n=(limxaf(x))n=Ln for every positive integer n
product law for limits
the limit law limxa(f(x)·g(x))=limxaf(x)·limxag(x)=L·Mlimxa(f(x)·g(x))=limxaf(x)·limxag(x)=L·M
quotient law for limits
the limit law limxaf(x)g(x)=limxaf(x)limxag(x)=LMlimxaf(x)g(x)=limxaf(x)limxag(x)=LM for M0M0
removable discontinuity
A removable discontinuity occurs at a point a if f(x)f(x) is discontinuous at a, but limxaf(x)limxaf(x) exists
root law for limits
the limit law limxaf(x)n=limxaf(x)n=Lnlimxaf(x)n=limxaf(x)n=Ln for all L if n is odd and for L0L0 if n is even
secant
A secant line to a function f(x)f(x) at a is a line through the point (a,f(a))(a,f(a)) and another point on the function; the slope of the secant line is given by msec=f(x)f(a)xamsec=f(x)f(a)xa
squeeze theorem
states that if f(x)g(x)h(x)f(x)g(x)h(x) for all xaxa over an open interval containing a and limxaf(x)=L=limxah(x)limxaf(x)=L=limxah(x) where L is a real number, then limxag(x)=Llimxag(x)=L
sum law for limits
The limit law limxa(f(x)+g(x))=limxaf(x)+limxag(x)=L+Mlimxa(f(x)+g(x))=limxaf(x)+limxag(x)=L+M
tangent
A tangent line to the graph of a function at a point (a,f(a))(a,f(a)) is the line that secant lines through (a,f(a))(a,f(a)) approach as they are taken through points on the function with x-values that approach a; the slope of the tangent line to a graph at a measures the rate of change of the function at a
triangle inequality
If a and b are any real numbers, then |a+b||a|+|b||a+b||a|+|b|
vertical asymptote
A function has a vertical asymptote at x=ax=a if the limit as x approaches a from the right or left is infinite
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