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Calculus Volume 1

Chapter Review Exercises

Calculus Volume 1Chapter Review Exercises
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  1. Preface
  2. 1 Functions and Graphs
    1. Introduction
    2. 1.1 Review of Functions
    3. 1.2 Basic Classes of Functions
    4. 1.3 Trigonometric Functions
    5. 1.4 Inverse Functions
    6. 1.5 Exponential and Logarithmic Functions
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Limits
    1. Introduction
    2. 2.1 A Preview of Calculus
    3. 2.2 The Limit of a Function
    4. 2.3 The Limit Laws
    5. 2.4 Continuity
    6. 2.5 The Precise Definition of a Limit
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  4. 3 Derivatives
    1. Introduction
    2. 3.1 Defining the Derivative
    3. 3.2 The Derivative as a Function
    4. 3.3 Differentiation Rules
    5. 3.4 Derivatives as Rates of Change
    6. 3.5 Derivatives of Trigonometric Functions
    7. 3.6 The Chain Rule
    8. 3.7 Derivatives of Inverse Functions
    9. 3.8 Implicit Differentiation
    10. 3.9 Derivatives of Exponential and Logarithmic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  5. 4 Applications of Derivatives
    1. Introduction
    2. 4.1 Related Rates
    3. 4.2 Linear Approximations and Differentials
    4. 4.3 Maxima and Minima
    5. 4.4 The Mean Value Theorem
    6. 4.5 Derivatives and the Shape of a Graph
    7. 4.6 Limits at Infinity and Asymptotes
    8. 4.7 Applied Optimization Problems
    9. 4.8 L’Hôpital’s Rule
    10. 4.9 Newton’s Method
    11. 4.10 Antiderivatives
    12. Key Terms
    13. Key Equations
    14. Key Concepts
    15. Chapter Review Exercises
  6. 5 Integration
    1. Introduction
    2. 5.1 Approximating Areas
    3. 5.2 The Definite Integral
    4. 5.3 The Fundamental Theorem of Calculus
    5. 5.4 Integration Formulas and the Net Change Theorem
    6. 5.5 Substitution
    7. 5.6 Integrals Involving Exponential and Logarithmic Functions
    8. 5.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Applications of Integration
    1. Introduction
    2. 6.1 Areas between Curves
    3. 6.2 Determining Volumes by Slicing
    4. 6.3 Volumes of Revolution: Cylindrical Shells
    5. 6.4 Arc Length of a Curve and Surface Area
    6. 6.5 Physical Applications
    7. 6.6 Moments and Centers of Mass
    8. 6.7 Integrals, Exponential Functions, and Logarithms
    9. 6.8 Exponential Growth and Decay
    10. 6.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  8. A | Table of Integrals
  9. B | Table of Derivatives
  10. C | Review of Pre-Calculus
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
  12. Index

True or False? Justify your answer with a proof or a counterexample.

310.

A function is always one-to-one.

311.

fg=gf,fg=gf, assuming f and g are functions.

312.

A relation that passes the horizontal and vertical line tests is a one-to-one function.

313.

A relation passing the horizontal line test is a function.

For the following problems, state the domain and range of the given functions:

f=x2+2x3,g=ln(x5),h=1x+4f=x2+2x3,g=ln(x5),h=1x+4

314.

h

315.

g

316.

hfhf

317.

gfgf

Find the degree, y-intercept, and zeros for the following polynomial functions.

318.

f(x)=2x2+9x5f(x)=2x2+9x5

319.

f(x)=x3+2x22xf(x)=x3+2x22x

Simplify the following trigonometric expressions.

320.

tan2xsec2x+cos2xtan2xsec2x+cos2x

321.

cos(2x)=sin2xcos(2x)=sin2x

Solve the following trigonometric equations on the interval θ=[−2π,2π]θ=[−2π,2π] exactly.

322.

6cos2x3=06cos2x3=0

323.

sec2x2secx+1=0sec2x2secx+1=0

Solve the following logarithmic equations.

324.

5x=165x=16

325.

log2(x+4)=3log2(x+4)=3

Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse f−1(x)f−1(x) of the function. Justify your answer.

326.

f(x)=x2+2x+1f(x)=x2+2x+1

327.

f(x)=1xf(x)=1x

For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.

328.

f(x)=9xf(x)=9x

329.

f(x)=x2+3x+4f(x)=x2+3x+4

330.

A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn 55° to keep up with the car. How fast is the car traveling?

For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts.

331.

a. Find the equation C=f(x)C=f(x) that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each.

332.

a. Find the inverse function x=f−1(C)x=f−1(C) and describe the meaning of this function. b. Determine how many shirts the owner can buy if he has $8000 to spend.

For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season.

333.

The population can be modeled by P(t)=82.567.5cos[(π/6)t],P(t)=82.567.5cos[(π/6)t], where tt is time in months (t=0(t=0 represents January 1) and PP is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?

334.

In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as P(t)=82.567.5cos[(π/6)t]+t,P(t)=82.567.5cos[(π/6)t]+t, where tt is time in months (t=0t=0 represents January 1) and PP is population (in thousands). When is the first time the population reaches 200,000?

For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation y=ert,y=ert, where yy is the percentage of radiocarbon still present in the material, tt is the number of years passed, and r=−0.0001210r=−0.0001210 is the decay rate of radiocarbon.

335.

If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?

336.

Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?

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