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Calculus Volume 2

Review Exercises

Calculus Volume 2Review Exercises
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Table of contents
  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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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

Review Exercises

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

379.

If limn→∞an=0,limn→∞an=0, then ∑n=1∞an∑n=1∞an converges.

380.

If limn→∞an≠0,limn→∞an≠0, then ∑n=1∞an∑n=1∞an diverges.

381.

If ∑n=1∞|an|∑n=1∞|an| converges, then ∑n=1∞an∑n=1∞an converges.

382.

If ∑n=1∞2nan∑n=1∞2nan converges, then ∑n=1∞(−2)nan∑n=1∞(−2)nan converges.

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

383.

a n = 3 + n 2 1 − n a n = 3 + n 2 1 − n

384.

a n = ln ( 1 n ) a n = ln ( 1 n )

385.

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

386.

a n = 2 n + 1 5 n a n = 2 n + 1 5 n

387.

a n = ln ( cos n ) n a n = ln ( cos n ) n

Is the series convergent or divergent?

388.

∑ n = 1 ∞ 1 n 2 + 5 n + 4 ∑ n = 1 ∞ 1 n 2 + 5 n + 4

389.

∑ n = 1 ∞ ln ( n + 1 n ) ∑ n = 1 ∞ ln ( n + 1 n )

390.

∑ n = 1 ∞ 2 n n 4 ∑ n = 1 ∞ 2 n n 4

391.

∑ n = 1 ∞ e n n ! ∑ n = 1 ∞ e n n !

392.

∑ n = 1 ∞ n − ( n + 1 / n ) ∑ n = 1 ∞ n − ( n + 1 / n )

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

393.

∑ n = 1 ∞ ( −1 ) n n ∑ n = 1 ∞ ( −1 ) n n

394.

∑ n = 1 ∞ ( −1 ) n n ! 3 n ∑ n = 1 ∞ ( −1 ) n n ! 3 n

395.

∑ n = 1 ∞ ( −1 ) n n ! n n ∑ n = 1 ∞ ( −1 ) n n ! n n

396.

∑ n = 1 ∞ sin ( n π 2 ) ∑ n = 1 ∞ sin ( n π 2 )

397.

∑ n = 1 ∞ cos ( π n ) e − n ∑ n = 1 ∞ cos ( π n ) e − n

Evaluate

398.

∑ n = 1 ∞ 2 n + 4 7 n ∑ n = 1 ∞ 2 n + 4 7 n

399.

∑ n = 1 ∞ 1 ( n + 1 ) ( n + 2 ) ∑ n = 1 ∞ 1 ( 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, four grains of rice for the third 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 limn→∞xnlimn→∞xn 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 limn→∞Snlimn→∞Sn

404.

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

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