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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

Finding Limits: A Numerical and Graphical Approach

For the following exercises, use Figure 1.

Graph of a piecewise function with two segments. The first segment goes from (-1, 2), a closed point, to (3, -6), a closed point, and the second segment goes from (3, 5), an open point, to (7, 9), a closed point.
Figure 1
1.

lim x −1 + f(x) lim x −1 + f(x)

2.

lim x −1 f(x) lim x −1 f(x)

3.

lim x1 f(x) lim x1 f(x)

4.

lim x3 f(x) lim x3 f(x)

5.

At what values of x x is the function discontinuous? What condition of continuity is violated?

6.

Using Table 1, estimate lim x0 f(x). lim x0 f(x).

xxF(x) F(x)
−0.12.875
−0.012.92
−0.0012.998
0Undefined
0.0012.9987
0.012.865
0.12.78145
0.152.678
Table 1

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 limit as x xapproaches a, a, state it. If not, discuss why there is no limit.

7.

f(x)={ | x |1, if x1 x 3 , if x=1   a=1 f(x)={ | x |1, if x1 x 3 , if x=1   a=1

8.

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

9.

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

Finding Limits: Properties of Limits

For the following exercises, find the limits if lim xc f( x )=−3 lim xc f( x )=−3 and lim xc g( x )=5. lim xc g( x )=5.

10.

lim xc ( f(x)+g(x) ) lim xc ( f(x)+g(x) )

11.

lim xc f(x) g(x) lim xc f(x) g(x)

12.

lim xc ( f(x)g(x) ) lim xc ( f(x)g(x) )

13.

lim x 0 + f(x),f(x)={ 3 x 2 +2x+1 5x+3    x>0 x<0 lim x 0 + f(x),f(x)={ 3 x 2 +2x+1 5x+3    x>0 x<0

14.

lim x 0 f(x),f(x)={ 3 x 2 +2x+1 5x+3    x>0 x<0 lim x 0 f(x),f(x)={ 3 x 2 +2x+1 5x+3    x>0 x<0

15.

lim x 3 + ( 3x〚x〛 ) lim x 3 + ( 3x〚x〛 )

For the following exercises, evaluate the limits using algebraic techniques.

16.

lim h0 ( ( h+6 ) 2 36 h ) lim h0 ( ( h+6 ) 2 36 h )

17.

lim x25 ( x 2 625 x 5 ) lim x25 ( x 2 625 x 5 )

18.

lim x1 ( x 2 9x x ) lim x1 ( x 2 9x x )

19.

lim x4 7 12x+1 x4 lim x4 7 12x+1 x4

20.

lim x3 ( 1 3 + 1 x 3+x ) lim x3 ( 1 3 + 1 x 3+x )

Continuity

For the following exercises, use numerical evidence to determine whether the limit exists at x=a. x=a. If not, describe the behavior of the graph of the function at x=a. x=a.

21.

f(x)= 2 x4 ; a=4 f(x)= 2 x4 ; a=4

22.

f(x)= 2 ( x4 ) 2 ; a=4 f(x)= 2 ( x4 ) 2 ; a=4

23.

f(x)= x x 2 x6 ; a=3 f(x)= x x 2 x6 ; a=3

24.

f(x)= 6 x 2 +23x+20 4 x 2 25 ; a= 5 2 f(x)= 6 x 2 +23x+20 4 x 2 25 ; a= 5 2

25.

f(x)= x 3 9x ; a=9 f(x)= x 3 9x ; a=9

For the following exercises, determine where the given function f(x) f(x) is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

26.

f(x)= x 2 2x15 f(x)= x 2 2x15

27.

f(x)= x 2 2x15 x5 f(x)= x 2 2x15 x5

28.

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

29.

f(x)= x 3 125 2 x 2 12x+10 f(x)= x 3 125 2 x 2 12x+10

30.

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

31.

f(x)= x+2 x 2 3x10 f(x)= x+2 x 2 3x10

32.

f(x)= x+2 x 3 +8 f(x)= x+2 x 3 +8

Derivatives

For the following exercises, find the average rate of change f(x+h)f(x) h . f(x+h)f(x) h .

33.

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

34.

f(x)=5 f(x)=5

35.

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

36.

f(x)=ln(x) f(x)=ln(x)

37.

f(x)= e 2x f(x)= e 2x

For the following exercises, find the derivative of the function.

38.

f(x)=4x6 f(x)=4x6

39.

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

40.

Find the equation of the tangent line to the graph of f( x ) f( x ) at the indicated x xvalue.

f(x)= x 3 +4x f(x)= x 3 +4x ; x=2. x=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.

41.

f(x)= x | x | f(x)= x | x |

42.

Given that the volume of a right circular cone is V= 1 3 π r 2 h V= 1 3 π r 2 h and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of π π

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