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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. 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

Non-right Triangles: Law of Sines

For the following exercises, assume α αis opposite side a,β a,βis opposite side b, b,and γ γis opposite side c. c.Solve each triangle, if possible. Round each answer to the nearest tenth.

1.

β=50°,a=105,b=45 β=50°,a=105,b=45

2.

α=43.1°,a=184.2,b=242.8 α=43.1°,a=184.2,b=242.8

3.

Solve the triangle.

Triangle with standard labels. Angle A is 36 degrees with opposite side a unknown. Angle B is 24 degrees with opposite side b = 16. Angle C and side c are unknown.
4.

Find the area of the triangle.

A triangle. One angle is 75 degrees with opposite side unknown. The adjacent sides to the 75 degree angle are 8 and 11.
5.

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25° and 49°, as shown in Figure 1. Find the distance of the plane from point A Aand the elevation of the plane.

Diagram of a plane flying over a highway. It is to the left and above points A and B on the ground in that order. There is a horizontal line going through the plan parallel to the ground. The angle formed by the horizontal line, the plane, and the line from the plane to point B is 25 degrees. The angle formed by the horizontal line, the plane, and point A is 49 degrees.
Figure 1

Non-right Triangles: Law of Cosines

6.

Solve the triangle, rounding to the nearest tenth, assuming α αis opposite side a,β a,βis opposite side b, b,and γ γs opposite side c:a=4, b=6,c=8. c:a=4, b=6,c=8.

7.

Solve the triangle in Figure 2, rounding to the nearest tenth.

A standardly labeled triangle. Angle A is 54 degrees with opposite side a unknown. Angle B is unknown with opposite side b=15. Angle C is unknown with opposite side C=13.
Figure 2
8.

Find the area of a triangle with sides of length 8.3, 6.6, and 9.1.

9.

To find the distance between two cities, a satellite calculates the distances and angle shown in Figure 3 (not to scale). Find the distance between the cities. Round answers to the nearest tenth.

Diagram of a satellite above and to the right of two cities. The distance from the satellite to the closer city is 210 km. The distance from the satellite to the further city is 250 km. The angle formed by the closer city, the satellite, and the other city is 1.8 degrees.
Figure 3

Polar Coordinates

10.

Plot the point with polar coordinates ( 3, π 6 ). ( 3, π 6 ).

11.

Plot the point with polar coordinates ( 5, 2π 3 ) ( 5, 2π 3 )

12.

Convert ( 6, 3π 4 ) ( 6, 3π 4 )to rectangular coordinates.

13.

Convert ( 2, 3π 2 ) ( 2, 3π 2 )to rectangular coordinates.

14.

Convert ( 7,2 ) ( 7,2 )to polar coordinates.

15.

Convert ( 9,4 ) ( 9,4 ) to polar coordinates.

For the following exercises, convert the given Cartesian equation to a polar equation.

16.

x=2 x=2

17.

x 2 + y 2 =64 x 2 + y 2 =64

18.

x 2 + y 2 =2y x 2 + y 2 =2y

For the following exercises, convert the given polar equation to a Cartesian equation.

19.

r=7cosθ r=7cosθ

20.

r= 2 4cosθ+sinθ r= 2 4cosθ+sinθ

For the following exercises, convert to rectangular form and graph.

21.

θ= 3π 4 θ= 3π 4

22.

r=5secθ r=5secθ

Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

23.

r=4+4sinθ r=4+4sinθ

24.

r=7 r=7

25.

Sketch a graph of the polar equation r=15sinθ. r=15sinθ.Label the axis intercepts.

26.

Sketch a graph of the polar equation r=5sin( 7θ ). r=5sin( 7θ ).

27.

Sketch a graph of the polar equation r=33cosθ r=33cosθ

Polar Form of Complex Numbers

For the following exercises, find the absolute value of each complex number.

28.

2+6i 2+6i

29.

43i 43i

Write the complex number in polar form.

30.

5+9i 5+9i

31.

1 2 3 2 i 1 2 3 2 i

For the following exercises, convert the complex number from polar to rectangular form.

32.

z=5cis( 5π 6 ) z=5cis( 5π 6 )

33.

z=3cis( 40° ) z=3cis( 40° )

For the following exercises, find the product z 1 z 2 z 1 z 2 in polar form.

34.

z 1 =2cis( 89° ) z 1 =2cis( 89° )

z 2 =5cis( 23° ) z 2 =5cis( 23° )

35.

z 1 =10cis( π 6 ) z 1 =10cis( π 6 )

z 2 =6cis( π 3 ) z 2 =6cis( π 3 )

For the following exercises, find the quotient z 1 z 2 z 1 z 2 in polar form.

36.

z 1 =12cis( 55° ) z 1 =12cis( 55° )

z 2 =3cis( 18° ) z 2 =3cis( 18° )

37.

z 1 =27cis( 5π 3 ) z 1 =27cis( 5π 3 )

z 2 =9cis( π 3 ) z 2 =9cis( π 3 )

For the following exercises, find the powers of each complex number in polar form.

38.

Find z 4 z 4 when z=2cis( 70° ) z=2cis( 70° )

39.

Find z 2 z 2 when z=5cis( 3π 4 ) z=5cis( 3π 4 )

For the following exercises, evaluate each root.

40.

Evaluate the cube root of z zwhen z=64cis( 210° ). z=64cis( 210° ).

41.

Evaluate the square root of z zwhen z=25cis( 3π 2 ). z=25cis( 3π 2 ).

For the following exercises, plot the complex number in the complex plane.

42.

62i 62i

43.

1+3i 1+3i

Parametric Equations

For the following exercises, eliminate the parameter t tto rewrite the parametric equation as a Cartesian equation.

44.

{ x( t )=3t1 y( t )= t { x( t )=3t1 y( t )= t

45.

{ x(t)=cost y(t)=2 sin 2 t  { x(t)=cost y(t)=2 sin 2 t 

46.

Parameterize (write a parametric equation for) each Cartesian equation by using x( t )=acost x( t )=acostand y(t)=bsint y(t)=bsintfor x 2 25 + y 2 16 =1. x 2 25 + y 2 16 =1.

47.

Parameterize the line from (2,3) (2,3)to (4,7) (4,7)so that the line is at (2,3) (2,3)at t=0 t=0and (4,7) (4,7)at t=1. t=1.

Parametric Equations: Graphs

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

48.

{ x( t )=3 t 2 y( t )=2t1 { x( t )=3 t 2 y( t )=2t1

49.

{ x(t)= e t y(t)=2 e 5t { x(t)= e t y(t)=2 e 5t

50.

{ x(t)=3cost y(t)=2sint { x(t)=3cost y(t)=2sint

51.

A ball is launched with an initial velocity of 80 feet per second at an angle of 40° to the horizontal. The ball is released at a height of 4 feet above the ground.

  1. Find the parametric equations to model the path of the ball.
  2. Where is the ball after 3 seconds?
  3. How long is the ball in the air?

Vectors

For the following exercises, determine whether the two vectors, u uand v, v,are equal, where u uhas an initial point P 1 P 1 and a terminal point P 2 , P 2 ,and v vhas an initial point P 3 P 3 and a terminal point P 4 . P 4 .

52.

P 1 =( 1,4 ), P 2 =( 3,1 ), P 3 =( 5,5 ) P 1 =( 1,4 ), P 2 =( 3,1 ), P 3 =( 5,5 )and P 4 =( 9,2 ) P 4 =( 9,2 )

53.

P 1 =( 6,11 ), P 2 =( 2,8 ), P 3 =( 0,1 ) P 1 =( 6,11 ), P 2 =( 2,8 ), P 3 =( 0,1 )and P 4 =( 8,2 ) P 4 =( 8,2 )

For the following exercises, use the vectors u=2ij,v=4i3j, u=2ij,v=4i3j,and w=2i+5j w=2i+5jto evaluate the expression.

54.

uv

55.

2vu + w

For the following exercises, find a unit vector in the same direction as the given vector.

56.

a = 8i − 6j

57.

b = −3ij

For the following exercises, find the magnitude and direction of the vector.

58.

6,−2 6,−2

59.

−3,−3 −3,−3

For the following exercises, calculate uv. uv.

60.

u = −2i + j and v = 3i + 7j

61.

u = i + 4j and v = 4i + 3j

62.

Given v = −3,4 = −3,4 draw v, 2v, and 1 2 1 2 v.

63.

Given the vectors shown in Figure 4, sketch u + v, uv and 3v.

Diagram of vectors v, 2v, and 1/2 v. The 2v vector is in the same direction as v but has twice the magnitude. The 1/2 v vector is in the same direction as v but has half the magnitude.
Figure 4
64.

Given initial point P 1 =( 3,2 ) P 1 =( 3,2 )and terminal point P 2 =( 5,1 ), P 2 =( 5,1 ),write the vector v vin terms of i iand j. j.Draw the points and the vector on the graph.

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