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Precalculus

Key Equations

PrecalculusKey Equations
  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 Equations
    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 Equations

definition of the exponential function f(x)= b x ,  where  b>0, b1 f(x)= b x ,  where  b>0, b1
definition of exponential growth f(x)=a b x ,where a>0, b>0, b1 f(x)=a b x ,where a>0, b>0, b1
compound interest formula A(t)=P ( 1+ r n ) nt  ,where A(t)is the account value at time t tis the number of years Pis the initial investment, often called the principal ris the annual percentage rate (APR), or nominal rate nis the number of compounding periods in one year A(t)=P ( 1+ r n ) nt  ,where A(t)is the account value at time t tis the number of years Pis the initial investment, often called the principal ris the annual percentage rate (APR), or nominal rate nis the number of compounding periods in one year
continuous growth formula A(t)=a e rt ,where A(t)=a e rt ,where
t t is the number of unit time periods of growth
a a is the starting amount (in the continuous compounding formula a is replaced with P, the principal)
e e is the mathematical constant, e2.718282 e2.718282
General Form for the Translation of the Parent Function f(x)= b x f(x)= b x f(x)=a b x+c +d f(x)=a b x+c +d
Definition of the logarithmic function For  x>0,b>0,b1,  x>0,b>0,b1,
y= log b ( x ) y= log b ( x ) if and only if b y =x. b y =x.
Definition of the common logarithm For x>0, x>0, y=log( x ) y=log( x ) if and only if 10 y =x. 10 y =x.
Definition of the natural logarithm For x>0, x>0, y=ln( x ) y=ln( x ) if and only if e y =x. e y =x.
General Form for the Translation of the Parent Logarithmic Function f(x)= log b ( x ) f(x)= log b ( x )  f(x)=a log b ( x+c )+d  f(x)=a log b ( x+c )+d
The Product Rule for Logarithms log b (MN)= log b ( M )+ log b ( N ) log b (MN)= log b ( M )+ log b ( N )
The Quotient Rule for Logarithms log b ( M N )= log b M log b N log b ( M N )= log b M log b N
The Power Rule for Logarithms log b ( M n )=n log b M log b ( M n )=n log b M
The Change-of-Base Formula log b M= log n M log n b         n>0,n1,b1 log b M= log n M log n b         n>0,n1,b1
One-to-one property for exponential functions For any algebraic expressions S S and T T and any positive real number b, b, where
b S = b T b S = b T if and only if S=T. S=T.
Definition of a logarithm For any algebraic expression S and positive real numbers b  b  and c, c, where b1, b1,
log b (S)=c log b (S)=c if and only if b c =S. b c =S.
One-to-one property for logarithmic functions For any algebraic expressions S and T and any positive real number b, b, where b1, b1,
log b S= log b T log b S= log b T if and only if S=T. S=T.
Half-life formula If A= A 0 e kt , A= A 0 e kt , k<0, k<0, the half-life is t= ln(2) k . t= ln(2) k .
Carbon-14 dating t= ln( A A 0 ) 0.000121 . t= ln( A A 0 ) 0.000121 .
A 0 A 0 is the amount of carbon-14 when the plant or animal died
A A is the amount of carbon-14 remaining today
t t is the age of the fossil in years
Doubling time formula If A= A 0 e kt , A= A 0 e kt , k>0, k>0, the doubling time is t= ln2 k t= ln2 k
Newton’s Law of Cooling T(t)=A e kt + T s , T(t)=A e kt + T s , where T s T s is the ambient temperature, A=T(0) T s , A=T(0) T s , and k k is the continuous rate of cooling.
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