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Table of contents
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Functions
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Key Terms

average rate of change
the slope of the line connecting the two points (a,f(a)) (a,f(a)) and (a+h,f(a+h)) (a+h,f(a+h)) on the curve of f( x ); f( x ); it is given by AROC= f( a+h )−f( a ) h . AROC= f( a+h )−f( a ) h .
continuous function
a function that has no holes or breaks in its graph
derivative
the slope of a function at a given point; denoted f ′ (a), f ′ (a), at a point x=a x=a it is f ′ (a)= lim h→0 f( a+h )−f( a ) h , f ′ (a)= lim h→0 f( a+h )−f( a ) h , providing the limit exists.
differentiable
a function f( x ) f( x ) for which the derivative exists at x=a. x=a. In other words, if f ′ ( a ) f ′ ( a ) exists.
discontinuous function
a function that is not continuous at x=a x=a
instantaneous rate of change
the slope of a function at a given point; at x=a x=a it is given by f ′ (a)= lim h→0 f( a+h )−f( a ) h . f ′ (a)= lim h→0 f( a+h )−f( a ) h .
instantaneous velocity
the change in speed or direction at a given instant; a function s( t ) s( t ) represents the position of an object at time t ,t, and the instantaneous velocity or velocity of the object at time t=a t=a is given by s ′ (a)= lim h→0 s( a+h )−s( a ) h . s ′ (a)= lim h→0 s( a+h )−s( a ) h .
jump discontinuity
a point of discontinuity in a function f( x ) f( x ) at x=a x=a where both the left and right-hand limits exist, but lim x→ a − f(x)≠ lim x→ a + f(x) lim x→ a − f(x)≠ lim x→ a + f(x)
left-hand limit
the limit of values of f( x ) f( x ) as x x approaches from a a the left, denoted lim x→ a − f(x)=L. lim x→ a − f(x)=L. The values of f(x) f(x) can get as close to the limit L L as we like by taking values of x x sufficiently close to a a such that x<a x<a and x≠a. x≠a. Both a a and L L are real numbers.
limit
when it exists, the value, L, L, that the output of a function f( x ) f( x ) approaches as the input x x gets closer and closer to a a but does not equal a. a. The value of the output, f(x), f(x), can get as close to L L as we choose to make it by using input values of x x sufficiently near to x=a, x=a, but not necessarily at x=a. x=a. Both a a and L L are real numbers, and L L is denoted lim x→a f(x)=L. lim x→a f(x)=L.
properties of limits
a collection of theorems for finding limits of functions by performing mathematical operations on the limits
removable discontinuity
a point of discontinuity in a function f( x ) f( x ) where the function is discontinuous, but can be redefined to make it continuous
right-hand limit
the limit of values of f( x ) f( x ) as x x approaches a a from the right, denoted lim x→ a + f(x)=L. lim x→ a + f(x)=L. The values of f(x) f(x) can get as close to the limit L L as we like by taking values of x x sufficiently close to a a where x>a, x>a, and x≠a. x≠a. Both a a and L L are real numbers.
secant line
a line that intersects two points on a curve
tangent line
a line that intersects a curve at a single point
two-sided limit
the limit of a function f(x), f(x), as x x approaches a, a, is equal to L, L, that is, lim x→a f(x)=L lim x→a f(x)=L if and only if lim x→ a − f(x)= lim x→ a + f(x). lim x→ a − f(x)= lim x→ a + f(x).
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