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Calculus Volume 3

Review Exercises

Calculus Volume 3Review Exercises
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Table of contents
  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. 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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