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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.

322.

The rectangular coordinates of the point (4,5π6)(4,5π6) are (23,−2).(23,−2).

323.

The equations x=cosh(3t),x=cosh(3t), y=2sinh(3t)y=2sinh(3t) represent a hyperbola.

324.

The arc length of the spiral given by r=θ2r=θ2 for 0≤θ≤3π0≤θ≤3π is 94π3.94π3.

325.

Given x=f(t)x=f(t) and y=g(t),y=g(t), if dxdy=dydx,dxdy=dydx, then f(t)=g(t)+C,f(t)=g(t)+C, where C is a constant.

For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.

326.

x=1+t,x=1+t, y=t2−1,y=t2−1, −1≤t≤1−1≤t≤1

327.

x=et,x=et, y=1−e3t,y=1−e3t, 0≤t≤10≤t≤1

328.

x=sinθ,x=sinθ, y=1−cscθ,y=1−cscθ, 0≤θ≤2π0≤θ≤2π

329.

x=4cosϕ,x=4cosϕ, y=1−sinϕ,y=1−sinϕ, 0≤ϕ≤2π0≤ϕ≤2π

For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any.

330.

r = 4 sin ( θ 3 ) r = 4 sin ( θ 3 )

331.

r = 5 cos ( 5 θ ) r = 5 cos ( 5 θ )

For the following exercises, find the polar equation for the curve given as a Cartesian equation.

332.

x + y = 5 x + y = 5

333.

y 2 = 4 + x 2 y 2 = 4 + x 2

For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line.

334.

x=ln(t),x=ln(t), y=t2−1,y=t2−1, t=1t=1

335.

r=3+cos(2θ),r=3+cos(2θ), θ=3π4θ=3π4

336.

Find dydx,dydx, dxdy,dxdy, and d2xdy2d2xdy2 of y=(2+e−t),y=(2+e−t), x=1−sin(t)x=1−sin(t)

For the following exercises, find the area of the region.

337.

x=t2,x=t2, y=ln(t),y=ln(t), 0≤t≤e0≤t≤e

338.

r=1−sinθr=1−sinθ in the first quadrant

For the following exercises, find the arc length of the curve over the given interval.

339.

x=3t+4,x=3t+4, y=9t−2,y=9t−2, 0≤t≤30≤t≤3

340.

r=6cosθ,r=6cosθ, 0≤θ≤2π.0≤θ≤2π. Check your answer by geometry.

For the following exercises, find the Cartesian equation describing the given shapes.

341.

A parabola with focus (2,−5)(2,−5) and directrix x=6x=6

342.

An ellipse with a major axis length of 10 and foci at (−7,2)(−7,2) and (1,2)(1,2)

343.

A hyperbola with vertices at (3,−2)(3,−2) and (−5,−2)(−5,−2) and foci at (−2,−6)(−2,−6) and (−2,4)(−2,4)

For the following exercises, determine the eccentricity and identify the conic. Sketch the conic.

344.

r = 6 1 + 3 cos ( θ ) r = 6 1 + 3 cos ( θ )

345.

r = 4 3 − 2 cos θ r = 4 3 − 2 cos θ

346.

r = 7 5 − 5 cos θ r = 7 5 − 5 cos θ

347.

Determine the Cartesian equation describing the orbit of Pluto, the most eccentric orbit around the Sun. The length of the major axis is 39.26 AU and minor axis is 38.07 AU. What is the eccentricity?

348.

The C/1980 E1 comet was observed in 1980. Given an eccentricity of 1.057 and a perihelion (point of closest approach to the Sun) of 3.364 AU, find the Cartesian equations describing the comet’s trajectory. Are we guaranteed to see this comet again? (Hint: Consider the Sun at point (0,0).)(0,0).)

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