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Precalculus

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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

For the following exercises, use the graph of f fin Figure 1.

Graph of a piecewise function with two segments. The first segment goes from negative infinity to (-1, 0), an open point, and the second segment goes from (-1, 3), an open point, to positive infinity.
Figure 1
1.

f(1) f(1)

2.

lim x −1 + f(x) lim x −1 + f(x)

3.

lim x −1 f(x) lim x −1 f(x)

4.

lim x−1 f(x) lim x−1 f(x)

5.

lim x−2 f(x) lim x−2 f(x)

6.

At what values of x xis f fdiscontinuous? What property of continuity is violated?

For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as x xapproaches a. a. If the function has a limit as x xapproaches a, a, state it. If not, discuss why there is no limit

7.

f(x)={ 1 x 3, if x2 x 3 +1,if x>2   a=2 f(x)={ 1 x 3, if x2 x 3 +1,if x>2   a=2

8.

f(x)={ x 3 +1, if x<1 3 x 2 1, if x=1 x+3 +4, if x>1 a=1 f(x)={ x 3 +1, if x<1 3 x 2 1, if x=1 x+3 +4, if x>1 a=1

For the following exercises, evaluate each limit using algebraic techniques.

9.

lim x−5 ( 1 5 + 1 x 10+2x ) lim x−5 ( 1 5 + 1 x 10+2x )

10.

lim h0 ( h 2 +25 5 h 2 ) lim h0 ( h 2 +25 5 h 2 )

11.

lim h0 ( 1 h 1 h 2 +h ) lim h0 ( 1 h 1 h 2 +h )

For the following exercises, determine whether or not the given function f fis continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

12.

f(x)= x 2 4 f(x)= x 2 4

13.

f(x)= x 3 4 x 2 9x+36 x 3 3 x 2 +2x6 f(x)= x 3 4 x 2 9x+36 x 3 3 x 2 +2x6

For the following exercises, use the definition of a derivative to find the derivative of the given function at x=a. x=a.

14.

f(x)= 3 5+2x f(x)= 3 5+2x

15.

f(x)= 3 x f(x)= 3 x

16.

f(x)=2 x 2 +9x f(x)=2 x 2 +9x

17.

For the graph in Figure 2, determine where the function is continuous/discontinuous and differentiable/not differentiable.

Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-2, -1), an open point; the second segment goes from (-2, -4), an open point, to (0, 0), a closed point; the final segment goes from (0, 1), an open point, to positive infinity.
Figure 2

For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

18.

f(x)=| x2 || x+2 | f(x)=| x2 || x+2 |

19.

f(x)= 2 1+ e 2 x f(x)= 2 1+ e 2 x

For the following exercises, explain the notation in words when the height of a projectile in feet, s, s, is a function of time t tin seconds after launch and is given by the function s(t). s(t).

20.

s(0) s(0)

21.

s(2) s(2)

22.

s'(2) s'(2)

23.

s(2)s(1) 21 s(2)s(1) 21

24.

s(t)=0 s(t)=0

For the following exercises, use technology to evaluate the limit.

25.

lim x0 sin(x) 3x lim x0 sin(x) 3x

26.

lim x0 tan 2 (x) 2x lim x0 tan 2 (x) 2x

27.

lim x0 sin(x)(1cos(x)) 2 x 2 lim x0 sin(x)(1cos(x)) 2 x 2

28.

Evaluate the limit by hand.

lim x1 f(x), where  f(x)={ 4x7 x1 x 2 4 x=1 lim x1 f(x), where  f(x)={ 4x7 x1 x 2 4 x=1

At what value(s) of x xis the function below discontinuous?

f(x)={ 4x7x1 x 2 4x=1 f(x)={ 4x7x1 x 2 4x=1

For the following exercises, consider the function whose graph appears in Figure 3.

Graph of a positive parabola.
Figure 3
29.

Find the average rate of change of the function from x=1 to x=3. x=1 to x=3.

30.

Find all values of x xat which f'(x)=0. f'(x)=0.

31.

Find all values of x xat which f'(x) f'(x) does not exist.

32.

Find an equation of the tangent line to the graph of f fthe indicated point: f(x)=3 x 2 2x6,  x=2 f(x)=3 x 2 2x6,  x=2

For the following exercises, use the function f(x)=x ( 1x ) 2 5 f(x)=x ( 1x ) 2 5 .

33.

Graph the function f(x)=x ( 1x ) 2 5 f(x)=x ( 1x ) 2 5 by entering f(x)=x ( ( 1x ) 2 ) 1 5 f(x)=x ( ( 1x ) 2 ) 1 5 and then by entering f(x)=x ( ( 1x ) 1 5 ) 2 f(x)=x ( ( 1x ) 1 5 ) 2 .

34.

Explore the behavior of the graph of f(x) f(x) around x=1 x=1 by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at x=1. x=1.

For the following exercises, find the derivative of each of the functions using the definition: lim h0 f(x+h)f(x) h lim h0 f(x+h)f(x) h

35.

f(x)=2x8 f(x)=2x8

36.

f(x)=4 x 2 7 f(x)=4 x 2 7

37.

f(x)=x 1 2 x 2 f(x)=x 1 2 x 2

38.

f(x)= 1 x+2 f(x)= 1 x+2

39.

f(x)= 3 x1 f(x)= 3 x1

40.

f(x)= x 3 +1 f(x)= x 3 +1

41.

f(x)= x 2 + x 3 f(x)= x 2 + x 3

42.

f(x)= x1 f(x)= x1

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