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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. Assume all functions ff and gg are continuous over their domains.

439.

If f(x)>0,f′(x)>0f(x)>0,f′(x)>0 for all x,x, then the right-hand rule underestimates the integral ∫abf(x).∫abf(x). Use a graph to justify your answer.

440.

∫ a b f ( x ) 2 d x = ∫ a b f ( x ) d x ∫ a b f ( x ) d x ∫ a b f ( x ) 2 d x = ∫ a b f ( x ) d x ∫ a b f ( x ) d x

441.

If f(x)≤g(x)f(x)≤g(x) for all x∈[a,b],x∈[a,b], then ∫abf(x)≤∫abg(x).∫abf(x)≤∫abg(x).

442.

All continuous functions have an antiderivative.

Evaluate the Riemann sums L4andR4L4andR4 for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

443.

y=3x2−2x+1y=3x2−2x+1 over [−1,1][−1,1]

444.

y=ln(x2+1)y=ln(x2+1) over [0,e][0,e]

445.

y=x2sinxy=x2sinx over [0,π][0,π]

446.

y=x+1xy=x+1x over [1,4][1,4]

Evaluate the following integrals.

447.

∫ −1 1 ( x 3 − 2 x 2 + 4 x ) d x ∫ −1 1 ( x 3 − 2 x 2 + 4 x ) d x

448.

∫ 0 4 3 t 1 + 6 t 2 d t ∫ 0 4 3 t 1 + 6 t 2 d t

449.

∫ π / 3 π / 2 2 sec ( 2 θ ) tan ( 2 θ ) d θ ∫ π / 3 π / 2 2 sec ( 2 θ ) tan ( 2 θ ) d θ

450.

∫ 0 π / 4 e cos 2 x sin x cos x d x ∫ 0 π / 4 e cos 2 x sin x cos x d x

Find the antiderivative.

451.

∫ d x ( x + 4 ) 3 ∫ d x ( x + 4 ) 3

452.

∫ x ln ( x 2 ) d x ∫ x ln ( x 2 ) d x

453.

∫ 4 x 2 1 − x 6 d x ∫ 4 x 2 1 − x 6 d x

454.

∫ e 2 x 1 + e 4 x d x ∫ e 2 x 1 + e 4 x d x

Find the derivative.

455.

d d t ∫ 0 t sin x 1 + x 2 d x d d t ∫ 0 t sin x 1 + x 2 d x

456.

d d x ∫ 1 x 3 4 − t 2 d t d d x ∫ 1 x 3 4 − t 2 d t

457.

d d x ∫ 1 ln ( x ) ( 4 t + e t ) d t d d x ∫ 1 ln ( x ) ( 4 t + e t ) d t

458.

d d x ∫ 0 cos x e t 2 d t d d x ∫ 0 cos x e t 2 d t

The following problems consider the historic average cost per gigabyte of RAM on a computer.

Year 5-Year Change ($)
1980 0
1985 −5,468,750
1990 −755,495
1995 −73,005
2000 −29,768
2005 −918
2010 −177
459.

If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.

460.

The average cost per gigabyte of RAM can be approximated by the function C(t)=8,500,000(0.65)t,C(t)=8,500,000(0.65)t, where tt is measured in years since 1980, and CC is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

461.

Find the average cost of 1GB RAM for 2005 to 2010.

462.

The velocity of a bullet from a rifle can be approximated by v(t)=6400t2−6505t+2686,v(t)=6400t2−6505t+2686, where tt is seconds after the shot and vv is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: 0≤t≤0.5.0≤t≤0.5. What is the total distance the bullet travels in 0.5 sec?

463.

What is the average velocity of the bullet for the first half-second?

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