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Precalculus

Review Exercises

PrecalculusReview Exercises
  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

Review Exercises

Sequences and Their Notation
1.

Write the first four terms of the sequence defined by the recursive formula a 1 =2, a n = a n1 +n. a 1 =2, a n = a n1 +n.

2.

Evaluate 6! (53)!3! . 6! (53)!3! .

3.

Write the first four terms of the sequence defined by the explicit formula a n = 10 n +3. a n = 10 n +3.

4.

Write the first four terms of the sequence defined by the explicit formula a n = n! n(n+1) . a n = n! n(n+1) .

Arithmetic Sequences
5.

Is the sequence 4 7 , 47 21 , 82 21 , 39 7 ,... 4 7 , 47 21 , 82 21 , 39 7 ,... arithmetic? If so, find the common difference.

6.

Is the sequence 2,4,8,16,... 2,4,8,16,... arithmetic? If so, find the common difference.

7.

An arithmetic sequence has the first term a 1 =18 a 1 =18 and common difference d=8. d=8. What are the first five terms?

8.

An arithmetic sequence has terms a 3 =11.7 a 3 =11.7 and a 8 =14.6. a 8 =14.6. What is the first term?

9.

Write a recursive formula for the arithmetic sequence 20,10,0,10,… 20,10,0,10,…

10.

Write a recursive formula for the arithmetic sequence 0, 1 2 ,1, 3 2 ,, 0, 1 2 ,1, 3 2 ,, and then find the 31st term.

11.

Write an explicit formula for the arithmetic sequence 7 8 , 29 24 , 37 24 , 15 8 , 7 8 , 29 24 , 37 24 , 15 8 ,

12.

How many terms are in the finite arithmetic sequence 12,20,28,,172? 12,20,28,,172?

Geometric Sequences
13.

Find the common ratio for the geometric sequence 2.5,5,10,20, 2.5,5,10,20,

14.

Is the sequence 4, 16, 28, 40 … geometric? If so find the common ratio. If not, explain why.

15.

A geometric sequence has terms a 7 =16,384 a 7 =16,384 and a 9 =262,144 a 9 =262,144 . What are the first five terms?

16.

A geometric sequence has the first term a 1 =3 a 1 =3 and common ratio r= 1 2 . r= 1 2 . What is the 8th term?

17.

What are the first five terms of the geometric sequence a 1 =3, a n =4 a n1 ? a 1 =3, a n =4 a n1 ?

18.

Write a recursive formula for the geometric sequence 1, 1 3 , 1 9 , 1 27 , 1, 1 3 , 1 9 , 1 27 ,

19.

Write an explicit formula for the geometric sequence 1 5 , 1 15 , 1 45 , 1 135 , 1 5 , 1 15 , 1 45 , 1 135 ,

20.

How many terms are in the finite geometric sequence 5,  5 3 ,  5 9 ,,  5 59,049 ? 5,  5 3 ,  5 9 ,,  5 59,049 ?

Series and Their Notation
21.

Use summation notation to write the sum of terms 1 2 m+5 1 2 m+5 from m=0 m=0 to m=5. m=5.

22.

Use summation notation to write the sum that results from adding the number 13 13 twenty times.

23.

Use the formula for the sum of the first n n terms of an arithmetic series to find the sum of the first eleven terms of the arithmetic series 2.5, 4, 5.5, … .

24.

A ladder has 15 15 tapered rungs, the lengths of which increase by a common difference. The first rung is 5 inches long, and the last rung is 20 inches long. What is the sum of the lengths of the rungs?

25.

Use the formula for the sum of the first n terms of a geometric series to find S 9 S 9 for the series 12,6,3, 3 2 , 12,6,3, 3 2 ,

26.

The fees for the first three years of a hunting club membership are given in Table 1. If fees continue to rise at the same rate, how much will the total cost be for the first ten years of membership?

Year Membership Fees
1 $1500
2 $1950
3 $2535
Table 1
27.

Find the sum of the infinite geometric series k=1 45 ( 1 3 ) k1 . k=1 45 ( 1 3 ) k1 .

28.

A ball has a bounce-back ratio of 3 5 3 5 the height of the previous bounce. Write a series representing the total distance traveled by the ball, assuming it was initially dropped from a height of 5 feet. What is the total distance? (Hint: the total distance the ball travels on each bounce is the sum of the heights of the rise and the fall.)

29.

Alejandro deposits $80 of his monthly earnings into an annuity that earns 6.25% annual interest, compounded monthly. How much money will he have saved after 5 years?

30.

The twins Sarah and Scott both opened retirement accounts on their 21st birthday. Sarah deposits $4,800.00 each year, earning 5.5% annual interest, compounded monthly. Scott deposits $3,600.00 each year, earning 8.5% annual interest, compounded monthly. Which twin will earn the most interest by the time they are 55 55 years old? How much more?

Counting Principles
31.

How many ways are there to choose a number from the set {10,6, 4, 10, 12, 18, 24, 32} {10,6, 4, 10, 12, 18, 24, 32} that is divisible by either 4 4 or 6? 6?

32.

In a group of 20 20 musicians, 12 12 play piano, 7 7 play trumpet, and 2 2 play both piano and trumpet. How many musicians play either piano or trumpet?

33.

How many ways are there to construct a 4-digit code if numbers can be repeated?

34.

A palette of water color paints has 3 shades of green, 3 shades of blue, 2 shades of red, 2 shades of yellow, and 1 shade of black. How many ways are there to choose one shade of each color?

35.

Calculate P( 18,4 ). P( 18,4 ).

36.

In a group of 5 5 freshman, 10 10 sophomores, 3 3 juniors, and 2 2 seniors, how many ways can a president, vice president, and treasurer be elected?

37.

Calculate C( 15,6 ). C( 15,6 ).

38.

A coffee shop has 7 Guatemalan roasts, 4 Cuban roasts, and 10 Costa Rican roasts. How many ways can the shop choose 2 Guatemalan, 2 Cuban, and 3 Costa Rican roasts for a coffee tasting event?

39.

How many subsets does the set { 1,3,5,,99 } { 1,3,5,,99 } have?

40.

A day spa charges a basic day rate that includes use of a sauna, pool, and showers. For an extra charge, guests can choose from the following additional services: massage, body scrub, manicure, pedicure, facial, and straight-razor shave. How many ways are there to order additional services at the day spa?

41.

How many distinct ways can the word DEADWOOD be arranged?

42.

How many distinct rearrangements of the letters of the word DEADWOOD are there if the arrangement must begin and end with the letter D?

Binomial Theorem
43.

Evaluate the binomial coefficient ( 23 8 ). ( 23 8 ).

44.

Use the Binomial Theorem to expand ( 3x+ 1 2 y ) 6 . ( 3x+ 1 2 y ) 6 .

45.

Use the Binomial Theorem to write the first three terms of ( 2a+b ) 17 . ( 2a+b ) 17 .

46.

Find the fourth term of ( 3 a 2 2b ) 11 ( 3 a 2 2b ) 11 without fully expanding the binomial.

Probability

For the following exercises, assume two die are rolled.

47.

Construct a table showing the sample space.

48.

What is the probability that a roll includes a 2? 2?

49.

What is the probability of rolling a pair?

50.

What is the probability that a roll includes a 2 or results in a pair?

51.

What is the probability that a roll doesn’t include a 2 or result in a pair?

52.

What is the probability of rolling a 5 or a 6?

53.

What is the probability that a roll includes neither a 5 nor a 6?

For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

54.

What is the percent chance that all the children attending the party prefer soda?

55.

What is the percent chance that at least one of the children attending the party prefers milk?

56.

What is the percent chance that exactly 3 of the children attending the party prefer soda?

57.

What is the percent chance that exactly 3 of the children attending the party prefer milk?

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