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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

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

408.

exsin(x)dxexsin(x)dx cannot be integrated by parts.

409.

1x4+1dx1x4+1dx cannot be integrated using partial fractions.

410.

In numerical integration, increasing the number of points decreases the error.

411.

Integration by parts can always yield the integral.

For the following exercises, evaluate the integral using the specified method.

412.

x2sin(4x)dxx2sin(4x)dx using integration by parts

413.

1x2x2+16dx1x2x2+16dx using trigonometric substitution

414.

xln(x)dxxln(x)dx using integration by parts

415.

3xx3+2x25x6dx3xx3+2x25x6dx using partial fractions

416.

x5(4x2+4)5/2dxx5(4x2+4)5/2dx using trigonometric substitution

417.

4sin2(x)sin2(x)cos(x)dx4sin2(x)sin2(x)cos(x)dx using a table of integrals or a CAS

For the following exercises, integrate using whatever method you choose.

418.

sin2(x)cos2(x)dxsin2(x)cos2(x)dx

419.

x3x2+2dxx3x2+2dx

420.

3x2+1x42x3x2+2xdx3x2+1x42x3x2+2xdx

421.

1x4+4dx1x4+4dx

422.

3+16x4x4dx3+16x4x4dx

For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals.

423.

[T] 12x5+2dx12x5+2dx

424.

[T] 0πesin(x2)dx0πesin(x2)dx

425.

[T] 14ln(1/x)xdx14ln(1/x)xdx

For the following exercises, evaluate the integrals, if possible.

426.

11xndx,11xndx, for what values of nn does this integral converge or diverge?

427.

1exxdx1exxdx

For the following exercises, consider the gamma function given by Γ(a)=0eyya1dy.Γ(a)=0eyya1dy.

428.

Show that Γ(a)=(a1)Γ(a1).Γ(a)=(a1)Γ(a1).

429.

Extend to show that Γ(a)=(a1)!,Γ(a)=(a1)!, assuming aa is a positive integer.

The fastest car in the world, the Bugati Veyron, can reach a top speed of 408 km/h. The graph represents its velocity.

This figure has a graph in the first quadrant. It increases to where x is approximately 03:00 mm:ss and then drops off steep. The maximum height of the graph, here the drop occurs is approximately 420 km/h.
430.

[T] Use the graph to estimate the velocity every 20 sec and fit to a graph of the form v(t)=aexpbxsin(cx)+d.v(t)=aexpbxsin(cx)+d. (Hint: Consider the time units.)

431.

[T] Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph.

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