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Calculus Volume 3

Chapter 7

Calculus Volume 3Chapter 7
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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

Checkpoint

7.1
  1. Nonlinear
  2. Linear, nonhomogeneous
7.4

Linearly independent

7.5

y ( x ) = c 1 e 3 x + c 2 x e 3 x y ( x ) = c 1 e 3 x + c 2 x e 3 x

7.6
  1. y(x)=ex(c1cos3x+c2sin3x)y(x)=ex(c1cos3x+c2sin3x)
  2. y(x)=c1e−7x+c2xe−7xy(x)=c1e−7x+c2xe−7x
7.7

y ( x ) = − e −2 x + e 5 x y ( x ) = − e −2 x + e 5 x

7.8

y(x)=ex(2cos3x−sin3x)y(x)=ex(2cos3x−sin3x)

This figure is the graph of y(x) = e^x(2 cos 3x − sin 3x) It has the positive x axis scaled in increments of even tenths. The y axis is scaled in increments of twenty. The graph itself starts at the origin. Its amplitude increases as x increases.
7.9

y(t)=te−7ty(t)=te−7t

This figure is the graph of y(t) = te^−7t. The horizontal axis is labeled with t and is scaled in increments of tenths. The y axis is scaled in increments of 0.5. The graph passes through the origin and has a horizontal asymptote of the positive t axis.


At time t=0.3,t=0.3, y(0.3)=0.3e(−7*0.3)=0.3e−2.1≈0.0367.y(0.3)=0.3e(−7*0.3)=0.3e−2.1≈0.0367. The mass is 0.0367 ft below equilibrium. At time t=0.1,t=0.1, y′(0.1)=0.3e−0.7≈0.1490.y′(0.1)=0.3e−0.7≈0.1490. The mass is moving downward at a speed of 0.1490 ft/sec.

7.10

y ( x ) = c 1 e − x + c 2 e 4 x − 2 y ( x ) = c 1 e − x + c 2 e 4 x − 2

7.11

y ( t ) = c 1 e 2 t + c 2 t e 2 t + sin t + cos t y ( t ) = c 1 e 2 t + c 2 t e 2 t + sin t + cos t

7.12
  1. y(x)=c1e4x+c2ex−xexy(x)=c1e4x+c2ex−xex
  2. y(t)=c1e−3t+c2e2t−5cos2t+sin2ty(t)=c1e−3t+c2e2t−5cos2t+sin2t
7.13

z1=3x+311x2,z1=3x+311x2, z2=2x+211xz2=2x+211x

7.14
  1. y(x)=c1cosx+c2sinx+cosxln|cosx|+xsinxy(x)=c1cosx+c2sinx+cosxln|cosx|+xsinx
  2. x(t)=c1et+c2tet+tetln|t|x(t)=c1et+c2tet+tetln|t|
7.15

x(t)=0.1cos(14t)x(t)=0.1cos(14t) (in meters); frequency is 142π142π Hz.

7.16

x(t)=17sin(4t+0.245),x(t)=17sin(4t+0.245), frequency=42π≈0.637,frequency=42π≈0.637, A=17A=17

7.17

x ( t ) = 0.6 e −2 t − 0.2 e −6 t x ( t ) = 0.6 e −2 t − 0.2 e −6 t

7.18

x ( t ) = 1 2 e −8 t + 4 t e −8 t x ( t ) = 1 2 e −8 t + 4 t e −8 t

7.19

x ( t ) = −0.24 e −2 t cos ( 4 t ) − 0.12 e −2 t sin ( 4 t ) x ( t ) = −0.24 e −2 t cos ( 4 t ) − 0.12 e −2 t sin ( 4 t )

7.20

x(t)=−12cos(4t)+94sin(4t)+12e−2tcos(4t)−2e−2tsin(4t)x(t)=−12cos(4t)+94sin(4t)+12e−2tcos(4t)−2e−2tsin(4t)
 Transient solution:12e−2tcos(4t)−2e−2tsin(4t)Transient solution:12e−2tcos(4t)−2e−2tsin(4t)
 Steady-state solution:−12cos(4t)+94sin(4t)Steady-state solution:−12cos(4t)+94sin(4t)

7.21

q ( t ) = −25 e − t cos ( 3 t ) − 7 e − t sin ( 3 t ) + 25 q ( t ) = −25 e − t cos ( 3 t ) − 7 e − t sin ( 3 t ) + 25

7.22
  1. y(x)=a0∑n=0∞(−1)nn!x2n=a0e−x2y(x)=a0∑n=0∞(−1)nn!x2n=a0e−x2
  2. y(x)=a0(x+1)3y(x)=a0(x+1)3

Section 7.1 Exercises

1.

linear, homogenous

3.

nonlinear

5.

linear, homogeneous

11.

y = c 1 e 5 x + c 2 e −2 x y = c 1 e 5 x + c 2 e −2 x

13.

y = c 1 e −2 x + c 2 x e −2 x y = c 1 e −2 x + c 2 x e −2 x

15.

y = c 1 e 5 x / 2 + c 2 e − x y = c 1 e 5 x / 2 + c 2 e − x

17.

y = e − x / 2 ( c 1 cos 3 x 2 + c 2 sin 3 x 2 ) y = e − x / 2 ( c 1 cos 3 x 2 + c 2 sin 3 x 2 )

19.

y = c 1 e −11 x + c 2 e 11 x y = c 1 e −11 x + c 2 e 11 x

21.

y = c 1 cos 9 x + c 2 sin 9 x y = c 1 cos 9 x + c 2 sin 9 x

23.

y = c 1 + c 2 x y = c 1 + c 2 x

25.

y = c 1 e ( ( 1 + 22 ) / 3 ) x + c 2 e ( ( 1 − 22 ) / 3 ) x y = c 1 e ( ( 1 + 22 ) / 3 ) x + c 2 e ( ( 1 − 22 ) / 3 ) x

27.

y = c 1 e − x / 6 + c 2 x e − x / 6 y = c 1 e − x / 6 + c 2 x e − x / 6

29.

y = c 1 + c 2 e 9 x y = c 1 + c 2 e 9 x

31.

y = −2 e −2 x + 2 e −3 x y = −2 e −2 x + 2 e −3 x

33.

y = 3 cos ( 2 x ) + 5 sin ( 2 x ) y = 3 cos ( 2 x ) + 5 sin ( 2 x )

35.

y = − e 6 x + 2 e −5 x y = − e 6 x + 2 e −5 x

37.

y = 2 e − x / 5 + 7 5 x e − x / 5 y = 2 e − x / 5 + 7 5 x e − x / 5

39.

y = ( 2 e 6 − e −7 ) e 6 x − ( 2 e 6 − e −7 ) e −7 x y = ( 2 e 6 − e −7 ) e 6 x − ( 2 e 6 − e −7 ) e −7 x

41.

No solutions exist.

43.

y = 2 e 2 x − 2 e 2 + 1 e 2 x e 2 x y = 2 e 2 x − 2 e 2 + 1 e 2 x e 2 x

45.

y = 4 cos 3 x + c 2 sin 3 x , infinitely many solutions y = 4 cos 3 x + c 2 sin 3 x , infinitely many solutions

47.

5 y ″ + 19 y ′ − 4 y = 0 5 y ″ + 19 y ′ − 4 y = 0

49.

a. y=3cos(8x)+2sin(8x)y=3cos(8x)+2sin(8x)
b.

This figure is a periodic graph. It has an amplitude of 3.5. Both the x and y axes are scaled in increments of 1.
51.

a. y=e(−5/2)x[−2cos(352x)+43535sin(352x)]y=e(−5/2)x[−2cos(352x)+43535sin(352x)]
b.

This figure is a graph of an oscillating function. The x and y axes are scaled in increments of even numbers. The amplitude of the graph is decreasing as x increases.

Section 7.2 Exercises

55.

y = c 1 e −4 x / 3 + c 2 e x − 2 y = c 1 e −4 x / 3 + c 2 e x − 2

57.

y = c 1 cos 4 x + c 2 sin 4 x + 1 20 e −2 x y = c 1 cos 4 x + c 2 sin 4 x + 1 20 e −2 x

59.

y = c 1 e 2 x + c 2 x e 2 x + 2 x 2 + 5 x + 4 y = c 1 e 2 x + c 2 x e 2 x + 2 x 2 + 5 x + 4

61.

y = c 1 e − x + c 2 x e − x + 1 2 sin x − 1 2 cos x y = c 1 e − x + c 2 x e − x + 1 2 sin x − 1 2 cos x

63.

y = c 1 cos x + c 2 sin x − 1 3 x cos 2 x − 5 9 sin 2 x y = c 1 cos x + c 2 sin x − 1 3 x cos 2 x − 5 9 sin 2 x

65.

y = c 1 e −5 x + c 2 x e −5 x + 1 6 x 3 e −5 x + 4 25 y = c 1 e −5 x + c 2 x e −5 x + 1 6 x 3 e −5 x + 4 25

67.

a. yp(x)=Ax2+Bx+Cyp(x)=Ax2+Bx+C
b. yp(x)=−13x2+43x−359yp(x)=−13x2+43x−359

69.

a. yp(x)=(Ax2+Bx+C)e−xyp(x)=(Ax2+Bx+C)e−x
b. yp(x)=(14x2−58x−3332)e−xyp(x)=(14x2−58x−3332)e−x

71.

a. yp(x)=(Ax2+Bx+C)excosxyp(x)=(Ax2+Bx+C)excosx +(Dx2+Ex+F)exsinx+(Dx2+Ex+F)exsinx
b. yp(x)=(−110x2−1125x−27250)excosxyp(x)=(−110x2−1125x−27250)excosx +(−310x2+225x+39250)exsinx+(−310x2+225x+39250)exsinx

73.

y = c 1 + c 2 e −2 x + 1 15 e 3 x y = c 1 + c 2 e −2 x + 1 15 e 3 x

75.

y = c 1 e 2 x + c 2 e −4 x + x e 2 x y = c 1 e 2 x + c 2 e −4 x + x e 2 x

77.

y = c 1 e 3 x + c 2 e −3 x − 8 x 9 y = c 1 e 3 x + c 2 e −3 x − 8 x 9

79.

y = c 1 cos 2 x + c 2 sin 2 x − 3 2 x cos 2 x + 3 4 sin 2 x ln ( sin 2 x ) y = c 1 cos 2 x + c 2 sin 2 x − 3 2 x cos 2 x + 3 4 sin 2 x ln ( sin 2 x )

81.

y = − 347 343 + 4 343 e 7 x + 2 7 x 2 e 7 x − 4 49 x e 7 x y = − 347 343 + 4 343 e 7 x + 2 7 x 2 e 7 x − 4 49 x e 7 x

83.

y = − 57 25 + 3 25 e 5 x + 1 5 x e 5 x + 4 25 e −5 x y = − 57 25 + 3 25 e 5 x + 1 5 x e 5 x + 4 25 e −5 x

85.

y p = 1 2 + 10 3 x 2 ln x y p = 1 2 + 10 3 x 2 ln x

Section 7.3 Exercises

87.

x″+16x=0,x″+16x=0, x(t)=16cos(4t)−2sin(4t),x(t)=16cos(4t)−2sin(4t), period =π2sec,=π2sec, frequency =2πHz=2πHz

89.

x″+196x=0,x″+196x=0, x(t)=0.15cos(14t),x(t)=0.15cos(14t), period =π7sec,=π7sec, frequency =7πHz=7πHz

91.

a. x(t)=5sin(2t)x(t)=5sin(2t)
b. period =πsec,=πsec, frequency =1πHz=1πHz
c.

This figure is the graph of a function. It is a periodic function with consistent amplitude. The horizontal axis is labeled in increments of 1. The vertical axis is labeled in increments of 1.5.


d. t=π2sect=π2sec

93.

a. x(t)=e−t/5(20cos(3t)+15sin(3t))x(t)=e−t/5(20cos(3t)+15sin(3t))
b. underdamped

95.

a. x(t)=5e−4t+10te−4tx(t)=5e−4t+10te−4t
b. critically damped

97.

x(π)=7e−π/46x(π)=7e−π/46 ft below

99.

x ( t ) = 32 9 sin ( 4 t ) + cos ( 128 t ) − 16 9 2 sin ( 128 t ) x ( t ) = 32 9 sin ( 4 t ) + cos ( 128 t ) − 16 9 2 sin ( 128 t )

101.

q ( t ) = e −6 t ( 0.051 cos ( 8 t ) + 0.03825 sin ( 8 t ) ) − 1 20 cos ( 10 t ) q ( t ) = e −6 t ( 0.051 cos ( 8 t ) + 0.03825 sin ( 8 t ) ) − 1 20 cos ( 10 t )

103.

q ( t ) = e −10 t ( −32 t − 5 ) + 5 , I ( t ) = 2 e −10 t ( 160 t + 9 ) q ( t ) = e −10 t ( −32 t − 5 ) + 5 , I ( t ) = 2 e −10 t ( 160 t + 9 )

Section 7.4 Exercises

105.

y = c 0 + 5 c 1 ∑ n = 1 ∞ ( − x / 5 ) n n ! = c 0 + 5 c 1 e − x / 5 y = c 0 + 5 c 1 ∑ n = 1 ∞ ( − x / 5 ) n n ! = c 0 + 5 c 1 e − x / 5

107.

y = c 0 ∑ n = 0 ∞ ( x ) 2 n ( 2 n ) ! + c 1 ∑ n = 0 ∞ ( x ) 2 n + 1 ( 2 n + 1 ) ! y = c 0 ∑ n = 0 ∞ ( x ) 2 n ( 2 n ) ! + c 1 ∑ n = 0 ∞ ( x ) 2 n + 1 ( 2 n + 1 ) !

109.

y = c 0 ∑ n = 0 ∞ x 2 n n ! = c 0 e x 2 y = c 0 ∑ n = 0 ∞ x 2 n n ! = c 0 e x 2

111.

y = c 0 ∑ n = 0 ∞ x 2 n 2 n n ! + c 1 ∑ n = 0 ∞ x 2 n + 1 1 ⋅ 3 ⋅ 5 ⋅ 7 ⋯ ( 2 n + 1 ) y = c 0 ∑ n = 0 ∞ x 2 n 2 n n ! + c 1 ∑ n = 0 ∞ x 2 n + 1 1 ⋅ 3 ⋅ 5 ⋅ 7 ⋯ ( 2 n + 1 )

113.

y = c 1 x 3 + c 2 x y = c 1 x 3 + c 2 x

115.

y = 1 − 3 x + 2 x 3 3 ! − 12 x 4 4 ! + 16 x 6 6 ! − 120 x 7 7 ! + ⋯ y = 1 − 3 x + 2 x 3 3 ! − 12 x 4 4 ! + 16 x 6 6 ! − 120 x 7 7 ! + ⋯

Review Exercises

117.

True

119.

False

121.

second order, linear, homogeneous, λ2−2=0λ2−2=0

123.

first order, nonlinear, nonhomogeneous

125.

y = c 1 sin ( 3 x ) + c 2 cos ( 3 x ) y = c 1 sin ( 3 x ) + c 2 cos ( 3 x )

127.

y = c 1 e x sin ( 3 x ) + c 2 e x cos ( 3 x ) + 2 5 x + 2 25 y = c 1 e x sin ( 3 x ) + c 2 e x cos ( 3 x ) + 2 5 x + 2 25

129.

y = c 1 e − x + c 2 e −4 x + x 4 + e 2 x 18 − 5 16 y = c 1 e − x + c 2 e −4 x + x 4 + e 2 x 18 − 5 16

131.

y = c 1 e ( −3 / 2 ) x + c 2 x e ( −3 / 2 ) x + 4 9 x 2 + 4 27 x − 16 27 y = c 1 e ( −3 / 2 ) x + c 2 x e ( −3 / 2 ) x + 4 9 x 2 + 4 27 x − 16 27

133.

y = e −2 x sin ( 2 x ) y = e −2 x sin ( 2 x )

135.

y = e 1 − x e 4 − 1 ( e 4 x − 1 ) y = e 1 − x e 4 − 1 ( e 4 x − 1 )

137.

θ ( t ) = θ 0 cos ( g l t ) θ ( t ) = θ 0 cos ( g l t )

141.

b = a b = a

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