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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? In the following exercises, justify your answer with a proof or a counterexample.

253.

If the radius of convergence for a power series ∑n=0∞anxn∑n=0∞anxn is 5,5, then the radius of convergence for the series ∑n=1∞nanxn−1∑n=1∞nanxn−1 is also 5.5.

254.

Power series can be used to show that the derivative of exisex.exisex. (Hint: Recall that ex=∑n=0∞1n!xn.)ex=∑n=0∞1n!xn.)

255.

For small values of x,sinx≈x.x,sinx≈x.

256.

The radius of convergence for the Maclaurin series of f(x)=3xf(x)=3x is 3.3.

In the following exercises, find the radius of convergence and the interval of convergence for the given series.

257.

∑ n = 0 ∞ n 2 ( x − 1 ) n ∑ n = 0 ∞ n 2 ( x − 1 ) n

258.

∑ n = 0 ∞ x n n n ∑ n = 0 ∞ x n n n

259.

∑ n = 0 ∞ 3 n x n 12 n ∑ n = 0 ∞ 3 n x n 12 n

260.

∑ n = 0 ∞ 2 n e n ( x − e ) n ∑ n = 0 ∞ 2 n e n ( x − e ) n

In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.

261.

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

262.

f ( x ) = 8 x + 2 2 x 2 − 3 x + 1 f ( x ) = 8 x + 2 2 x 2 − 3 x + 1

In the following exercises, find the power series for the given function using term-by-term differentiation or integration.

263.

f ( x ) = tan −1 ( 2 x ) f ( x ) = tan −1 ( 2 x )

264.

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

In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?

265.

f ( x ) = x 3 − 2 x 2 + 4 , a = −3 f ( x ) = x 3 − 2 x 2 + 4 , a = −3

266.

f ( x ) = e 1 / ( 4 x ) , a = 4 f ( x ) = e 1 / ( 4 x ) , a = 4

In the following exercises, find the Maclaurin series for the given function.

267.

f ( x ) = cos ( 3 x ) f ( x ) = cos ( 3 x )

268.

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

In the following exercises, find the Taylor series at the given value.

269.

f ( x ) = sin x , a = π 2 f ( x ) = sin x , a = π 2

270.

f ( x ) = 3 x , a = 1 f ( x ) = 3 x , a = 1

In the following exercises, find the Maclaurin series for the given function.

271.

f ( x ) = e − x 2 − 1 f ( x ) = e − x 2 − 1

272.

f ( x ) = cos x − x sin x f ( x ) = cos x − x sin x

In the following exercises, find the Maclaurin series for F(x)=∫0xf(t)dtF(x)=∫0xf(t)dt by integrating the Maclaurin series of f(x)f(x) term by term.

273.

f ( x ) = sin x x f ( x ) = sin x x

274.

f ( x ) = 1 − e x f ( x ) = 1 − e x

275.

Use power series to prove Euler’s formula: eix=cosx+isinxeix=cosx+isinx

The following exercises consider problems of annuity payments.

276.

For annuities with a present value of $1$1 million, calculate the annual payouts given over 2525 years assuming interest rates of 1%,5%,and10%.1%,5%,and10%.

277.

A lottery winner has an annuity that has a present value of $10$10 million. What interest rate would they need to live on perpetual annual payments of $250,000?$250,000?

278.

Calculate the necessary present value of an annuity in order to support annual payouts of $15,000$15,000 given over 2525 years assuming interest rates of 1%,5%,and10%.1%,5%,and10%.

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