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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.

427.

Vector field F(x,y)=x2yi+y2xjF(x,y)=x2yi+y2xj is conservative.

428.

For vector field F(x,y)=P(x,y)i+Q(x,y)j,F(x,y)=P(x,y)i+Q(x,y)j, if Py(x,y)=Qx(x,y)Py(x,y)=Qx(x,y) in open region D,D, then ∫∂DPdx+Qdy=0.∫∂DPdx+Qdy=0.

429.

The divergence of a vector field is a vector field.

430.

If curlF=0,curlF=0, then FF is a conservative vector field.

Draw the following vector fields.

431.

F ( x , y ) = 1 2 i + 2 x j F ( x , y ) = 1 2 i + 2 x j

432.

F ( x , y ) = y i + 3 x j x 2 + y 2 F ( x , y ) = y i + 3 x j x 2 + y 2

Are the following the vector fields conservative? If so, find the potential function ff such that F=∇f.F=∇f.

433.

F ( x , y ) = y i + ( x − 2 e y ) j F ( x , y ) = y i + ( x − 2 e y ) j

434.

F ( x , y ) = ( 6 x y ) i + ( 3 x 2 − y e y ) j F ( x , y ) = ( 6 x y ) i + ( 3 x 2 − y e y ) j

435.

F ( x , y , z ) = ( 2 x y + z 2 ) i + ( x 2 + 2 y z ) j + ( 2 x z + y 2 ) k F ( x , y , z ) = ( 2 x y + z 2 ) i + ( x 2 + 2 y z ) j + ( 2 x z + y 2 ) k

436.

F ( x , y , z ) = ( e x y ) i + ( e x + z ) j + ( e x + y 2 ) k F ( x , y , z ) = ( e x y ) i + ( e x + z ) j + ( e x + y 2 ) k

Evaluate the following integrals.

437.

∫Cx2dy+(2x−3xy)dx,∫Cx2dy+(2x−3xy)dx, along C:y=12xC:y=12x from (0, 0) to (4, 2)

438.

∫Cydx+xy2dy,∫Cydx+xy2dy, where C:x=t,y=t−1,0≤t≤1C:x=t,y=t−1,0≤t≤1

439.

∬Sxy2dS,∬Sxy2dS, where S is surface z=x2−y,0≤x≤1,0≤y≤4z=x2−y,0≤x≤1,0≤y≤4

Find the divergence and curl for the following vector fields.

440.

F ( x , y , z ) = 3 x y z i + x y e z j − 3 x y k F ( x , y , z ) = 3 x y z i + x y e z j − 3 x y k

441.

F ( x , y , z ) = e x i + e x y j + e x y z k F ( x , y , z ) = e x i + e x y j + e x y z k

Use Green’s theorem to evaluate the following integrals.

442.

∫C3xydx+2xy2dy,∫C3xydx+2xy2dy, where C is a square with vertices (0, 0), (0, 2), (2, 2) and (2, 0) oriented counterclockwise.

443.

∮C3ydx+(x+ey)dy,∮C3ydx+(x+ey)dy, where C is a circle centered at the origin with radius 3

Use Stokes’ theorem to evaluate ∫∫ScurlF·dS.∫∫ScurlF·dS.

444.

F(x,y,z)=yi−xj+zk,F(x,y,z)=yi−xj+zk, where SS is the upper half of the unit sphere

445.

F(x,y,z)=yi+xyzj−2zxk,F(x,y,z)=yi+xyzj−2zxk, where SS is the upward-facing paraboloid z=x2+y2z=x2+y2 lying in cylinder x2+y2=1x2+y2=1

Use the divergence theorem to evaluate ∫∫SF·dS.∫∫SF·dS.

446.

F(x,y,z)=(x3y)i+(3y−ex)j+(z+x)k,F(x,y,z)=(x3y)i+(3y−ex)j+(z+x)k, over cube SS defined by −1≤x≤1,−1≤x≤1, 0≤y≤2,0≤y≤2, 0≤z≤20≤z≤2

447.

F(x,y,z)=(2xy)i+(−y2)j+(2z3)k,F(x,y,z)=(2xy)i+(−y2)j+(2z3)k, where SS is bounded by paraboloid z=x2+y2z=x2+y2 and plane z=2z=2

448.

Find the amount of work performed by a 50-kg woman ascending a helical staircase with radius 2 m and height 100 m. The woman completes five revolutions during the climb.

449.

Find the total mass of a thin wire in the shape of an upper semicircle with radius 2,2, and a density function of ρ(x,y)=y+x2.ρ(x,y)=y+x2.

450.

Find the total mass of a thin sheet in the shape of a hemisphere with radius 2 for z≥0z≥0 with a density function ρ(x,y,z)=x+y+z.ρ(x,y,z)=x+y+z.

451.

Use the divergence theorem to compute the value of the flux integral over the unit sphere with F(x,y,z)=3zi+2yj+2xk.F(x,y,z)=3zi+2yj+2xk.

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