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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

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

394.

The domain of f(x,y)=x3sin−1(y)f(x,y)=x3sin−1(y) is x=x= all real numbers, and −π≤y≤π.−π≤y≤π.

395.

If the function f(x,y)f(x,y) is continuous everywhere, then fxy=fyx.fxy=fyx.

396.

The linear approximation to the function of f(x,y)=5x2+xtan(y)f(x,y)=5x2+xtan(y) at (2,π)(2,π) is given by L(x,y)=22+21(x−2)+(y−π).L(x,y)=22+21(x−2)+(y−π).

397.

(34,916)(34,916) is a critical point of g(x,y)=4x3−2x2y+y2−2.g(x,y)=4x3−2x2y+y2−2.

For the following exercises, sketch the function in one graph and, in a second, sketch several level curves.

398.

f ( x , y ) = e − ( x 2 + 2 y 2 ) . f ( x , y ) = e − ( x 2 + 2 y 2 ) .

399.

f ( x , y ) = x + 4 y 2 . f ( x , y ) = x + 4 y 2 .

For the following exercises, evaluate the following limits, if they exist. If they do not exist, prove it.

400.

lim ( x , y ) → ( 1 , 1 ) 4 x y x − 2 y 2 lim ( x , y ) → ( 1 , 1 ) 4 x y x − 2 y 2

401.

lim ( x , y ) → ( 0 , 0 ) 4 x y x − 2 y 2 lim ( x , y ) → ( 0 , 0 ) 4 x y x − 2 y 2

For the following exercises, find the largest region of continuity for the function.

402.

f ( x , y ) = x 3 sin −1 ( y ) f ( x , y ) = x 3 sin −1 ( y )

403.

g ( x , y ) = ln ( 4 − x 2 − y 2 ) g ( x , y ) = ln ( 4 − x 2 − y 2 )

For the following exercises, find all first derivatives, full or partial, as appropriate.

404.

f ( x , y ) = x 2 − y 2 f ( x , y ) = x 2 − y 2

405.

u ( x , y ) = x 4 − 3 x y + 1 , x = 2 t , y = t 3 u ( x , y ) = x 4 − 3 x y + 1 , x = 2 t , y = t 3

For the following exercises, find all second partial derivatives.

406.

g ( t , x ) = 3 t 2 − sin ( x + t ) g ( t , x ) = 3 t 2 − sin ( x + t )

407.

h ( x , y , z ) = x 3 e 2 y z h ( x , y , z ) = x 3 e 2 y z

For the following exercises, find the equation of the tangent plane to the specified surface at the given point.

408.

z=x3−2y2+y−1z=x3−2y2+y−1 at point (1,1,−1)(1,1,−1)

409.

z=ex+2yz=ex+2y at point (0,1,3)(0,1,3)

410.

Approximate f(x,y)=ex2+yf(x,y)=ex2+y at (0.1,9.1).(0.1,9.1). Write down your linear approximation function L(x,y).L(x,y). How accurate is the approximation to the exact answer, rounded to four digits?

411.

Find the differential dzdz of h(x,y)=4x2+2xy−3yh(x,y)=4x2+2xy−3y and approximate ΔzΔz at the point (1,−2).(1,−2). Let Δx=0.1Δx=0.1 and Δy=0.01.Δy=0.01.

412.

Find the directional derivative of f(x,y)=x2+6xy−y2f(x,y)=x2+6xy−y2 in the direction v=i+4j.v=i+4j.

413.

Find the maximal directional derivative magnitude and direction for the function f(x,y)=x3+2xy−cos(πy)f(x,y)=x3+2xy−cos(πy) at point (3,0).(3,0).

For the following exercises, find the gradient.

414.

c ( x , t ) = e ( t − x ) 2 + 3 cos ( t ) c ( x , t ) = e ( t − x ) 2 + 3 cos ( t )

415.

f ( x , y ) = x + y 2 x y f ( x , y ) = x + y 2 x y

For the following exercises, find and classify the critical points.

416.

z = x 3 − x y + y 2 − 1 z = x 3 − x y + y 2 − 1

For the following exercises, use Lagrange multipliers to find the maximum and minimum values for the functions with the given constraints.

417.

f ( x , y ) = x 2 y , x 2 + y 2 = 4 f ( x , y ) = x 2 y , x 2 + y 2 = 4

418.

f ( x , y ) = x 2 − y 2 , x + 6 y = 4 f ( x , y ) = x 2 − y 2 , x + 6 y = 4

419.

A machinist is constructing a right circular cone out of a block of aluminum. The machine gives an error of 5%5% in height and 2%2% in radius. Find the maximum error in the volume of the cone if the machinist creates a cone of height 66 cm and radius 22 cm.

420.

A trash compactor is in the shape of a cuboid. Assume the trash compactor is filled with incompressible liquid. The length and width are decreasing at rates of 22 ft/sec and 33 ft/sec, respectively. Find the rate at which the liquid level is rising when the length is 1414 ft, the width is 1010 ft, and the height is 44 ft.

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