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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.

435.

The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.

436.

If the force is constant, the amount of work to move an object from x=ax=a to x=bx=b is F(ba).F(ba).

437.

The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.

438.

If the half-life of seaborgium-266seaborgium-266 is 360360 ms, then k=(ln(2))/360.k=(ln(2))/360.

For the following exercises, use the requested method to determine the volume of the solid.

439.

The volume that has a base of the ellipse x2/4+y2/9=1x2/4+y2/9=1 and cross-sections of an equilateral triangle perpendicular to the y-axis.y-axis. Use the method of slicing.

440.

y=x2x,y=x2x, from x=1tox=4,x=1tox=4, rotated around they-axis using the washer method

441.

x=y2x=y2 and x=3yx=3y rotated around the y-axis using the washer method

442.

x=2y2y3,x=0,andy=0x=2y2y3,x=0,andy=0 rotated around the x-axis using cylindrical shells

For the following exercises, find

  1. the area of the region,
  2. the volume of the solid when rotated around the x-axis, and
  3. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.
443.

y=x3,x=0,y=0,andx=2y=x3,x=0,y=0,andx=2

444.

y=x2xandx=0y=x2xandx=0

445.

[T] y=ln(x)+2andy=xy=ln(x)+2andy=x

446.

y=x2y=x2 and y=xy=x

447.

y=5+x,y=5+x, y=x2,y=x2, x=0,x=0, and x=1x=1

448.

Below x2+y2=1x2+y2=1 and above y=1xy=1x

449.

Find the mass of ρ=exρ=ex on a disk centered at the origin with radius 4.4.

450.

Find the center of mass for ρ=tan2xρ=tan2x on x(π4,π4).x(π4,π4).

451.

Find the mass and the center of mass of ρ=1ρ=1 on the region bounded by y=x5y=x5 and y=x.y=x.

For the following exercises, find the requested arc lengths.

452.

The length of xx for y=cosh(x)y=cosh(x) from x=0tox=2.x=0tox=2.

453.

The length of yy for x=3yx=3y from y=0y=0 to y=4y=4

For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis.

454.

The shape created by revolving the region between y=4+x,y=4+x, y=3x,y=3x, x=0,x=0, and x=2x=2 rotated around the y-axis.

455.

The loudspeaker created by revolving y=1/xy=1/x from x=1x=1 to x=4x=4 around the x-axis.

For the following exercises, consider the Karun-3 dam in Iran. Its shape can be approximated as an isosceles triangle with height 205205 m and width 388388 m. Assume the current depth of the water is 180180 m. The density of water is 10001000 kg/m 3.3.

456.

Find the total force on the wall of the dam.

457.

You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and 45°F45°F outside and the temperature of the body is 78°F.78°F. You know the cooling constant is k=0.00824°F/min.k=0.00824°F/min. When did the victim die, assuming that a human’s temperature is 98°F98°F ?

For the following exercise, consider the stock market crash in 19291929 in the United States. The table lists the Dow Jones industrial average per year leading up to the crash.

Source: http://stockcharts.com/freecharts/historical/djia19201940.html
Years after 1920 Value ($)
11 63.9063.90
33 100100
55 110110
77 160160
99 381.17381.17
458.

[T] The best-fit exponential curve to these data is given by y=40.71+1.224x.y=40.71+1.224x. Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in 20142014 ?

For the following exercises, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings.

459.

Find the volume of the catenoid y=cosh(x)y=cosh(x) from x=−1tox=1x=−1tox=1 that is created by rotating this curve around the x-axis,x-axis, as shown here.

This figure is an image of a catenoid. It has been formed by rotating a catenary curve about a vertical axis.
460.

Find surface area of the catenoid y=cosh(x)y=cosh(x) from x=−1x=−1 to x=1x=1 that is created by rotating this curve around the x-axis.x-axis.

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