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Precalculus

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PrecalculusKey Terms
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  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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. 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. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. 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. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. 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 Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. 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
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 h0 f( a+h )f( a ) h , f (a)= lim h0 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 h0 f( a+h )f( a ) h . f (a)= lim h0 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 h0 s( a+h )s( a ) h . s (a)= lim h0 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 xapproaches from a athe 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 Las we like by taking values of x xsufficiently close to a asuch that x<a x<a and xa. xa. Both a aand L Lare 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 xgets closer and closer to a abut does not equal a. a. The value of the output, f(x), f(x), can get as close to L Las we choose to make it by using input values of x xsufficiently near to x=a, x=a, but not necessarily at x=a. x=a. Both a aand L Lare real numbers, and L Lis denoted lim xa f(x)=L. lim xa 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 xapproaches a afrom 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 Las we like by taking values of x xsufficiently close to a awhere x>a, x>a, and xa. xa. Both a aand L Lare 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 xapproaches a, a, is equal to L, L, that is, lim xa f(x)=L lim xa 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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