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Calculus Volume 2

Chapter Review Exercises

Calculus Volume 2Chapter Review Exercises
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  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

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

379.

If limnan=0,limnan=0, then n=1ann=1an converges.

380.

If limnan0,limnan0, then n=1ann=1an diverges.

381.

If n=1|an|n=1|an| converges, then n=1ann=1an converges.

382.

If n=12nann=12nan converges, then n=1(−2)nann=1(−2)nan converges.

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit.

383.

an=3+n21nan=3+n21n

384.

an=ln(1n)an=ln(1n)

385.

an=ln(n+1)n+1an=ln(n+1)n+1

386.

an=2n+15nan=2n+15n

387.

an=ln(cosn)nan=ln(cosn)n

Is the series convergent or divergent?

388.

n=11n2+5n+4n=11n2+5n+4

389.

n=1ln(n+1n)n=1ln(n+1n)

390.

n=12nn4n=12nn4

391.

n=1enn!n=1enn!

392.

n=1n(n+1/n)n=1n(n+1/n)

Is the series convergent or divergent? If convergent, is it absolutely convergent?

393.

n=1(−1)nnn=1(−1)nn

394.

n=1(−1)nn!3nn=1(−1)nn!3n

395.

n=1(−1)nn!nnn=1(−1)nn!nn

396.

n=1sin(nπ2)n=1sin(nπ2)

397.

n=1cos(πn)enn=1cos(πn)en

Evaluate

398.

n=12n+47nn=12n+47n

399.

n=11(n+1)(n+2)n=11(n+1)(n+2)

400.

A legend from India tells that a mathematician invented chess for a king. The king enjoyed the game so much he allowed the mathematician to demand any payment. The mathematician asked for one grain of rice for the first square on the chessboard, two grains of rice for the second square on the chessboard, and so on. Find an exact expression for the total payment (in grains of rice) requested by the mathematician. Assuming there are 30,00030,000 grains of rice in 11 pound, and 20002000 pounds in 11 ton, how many tons of rice did the mathematician attempt to receive?

The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula xn+1=bxn,xn+1=bxn, where xnxn is the population of houseflies at generation n,n, and bb is the average number of offspring per housefly who survive to the next generation. Assume a starting population x0.x0.

401.

Find limnxnlimnxn if b>1,b>1, b<1,b<1, and b=1.b=1.

402.

Find an expression for Sn=i=0nxiSn=i=0nxi in terms of bb and x0.x0. What does it physically represent?

403.

If b=34b=34 and x0=100,x0=100, find S10S10 and limnSnlimnSn

404.

For what values of bb will the series converge and diverge? What does the series converge to?

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