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Calculus Volume 3

Chapter Review Exercises

Calculus Volume 3Chapter Review Exercises
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  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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter 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

True or False? Justify your answer with a proof or a counterexample.

117.

If yy and zz are both solutions to y+2y+y=0,y+2y+y=0, then y+zy+z is also a solution.

118.

The following system of algebraic equations has a unique solution:

6z1+3z2=84z1+2z2=4.6z1+3z2=84z1+2z2=4.

119.

y=excos(3x)+exsin(2x)y=excos(3x)+exsin(2x) is a solution to the second-order differential equation y+2y+10=0.y+2y+10=0.

120.

To find the particular solution to a second-order differential equation, you need one initial condition.

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation.

121.

y2y=0y2y=0

122.

y3y+2y=cos(t)y3y+2y=cos(t)

123.

(dydt)2+yy=1(dydt)2+yy=1

124.

d2ydt2+tdydt+sin2(t)y=etd2ydt2+tdydt+sin2(t)y=et

For the following problems, find the general solution.

125.

y+9y=0y+9y=0

126.

y+2y+y=0y+2y+y=0

127.

y2y+10y=4xy2y+10y=4x

128.

y=cos(x)+2y+yy=cos(x)+2y+y

129.

y+5y+y=x+e2xy+5y+y=x+e2x

130.

y=3y+xexy=3y+xex

131.

yx2=−3y94y+3xyx2=−3y94y+3x

132.

y=2cosx+yyy=2cosx+yy

For the following problems, find the solution to the initial-value problem, if possible.

133.

y+4y+6y=0,y+4y+6y=0, y(0)=0,y(0)=0, y(0)=2y(0)=2

134.

y=3ycos(x),y=3ycos(x), y(0)=94,y(0)=94, y(0)=0y(0)=0

For the following problems, find the solution to the boundary-value problem.

135.

4y=−6y+2y,4y=−6y+2y, y(0)=0,y(0)=0, y(1)=1y(1)=1

136.

y=3xyy,y=3xyy, y(0)=−3,y(0)=−3, y(1)=0y(1)=0

For the following problem, set up and solve the differential equation.

137.

The motion of a swinging pendulum for small angles θθ can be approximated by d2θdt2+gLθ=0,d2θdt2+gLθ=0, where θθ is the angle the pendulum makes with respect to a vertical line, g is the acceleration resulting from gravity, and L is the length of the pendulum. Find the equation describing the angle of the pendulum at time t,t, assuming an initial displacement of θ0θ0 and an initial velocity of zero.

The following problems consider the “beats” that occur when the forcing term of a differential equation causes “slow” and “fast” amplitudes. Consider the general differential equationay+by=cos(ωt)ay+by=cos(ωt) that governs undamped motion. Assume that baω.baω.

138.

Find the general solution to this equation (Hint: call ω0=b/aω0=b/a).

139.

Assuming the system starts from rest, show that the particular solution can be written as y=2a(ω02ω2)sin(ω0ωt2)sin(ω0+ωt2).y=2a(ω02ω2)sin(ω0ωt2)sin(ω0+ωt2).

140.

[T] Using your solutions derived earlier, plot the solution to the system 2y+9y=cos(2t)2y+9y=cos(2t) over the interval t=[−50,50].t=[−50,50]. Find, analytically, the period of the fast and slow amplitudes.

For the following problem, set up and solve the differential equations.

141.

An opera singer is attempting to shatter a glass by singing a particular note. The vibrations of the glass can be modeled by y+ay=cos(bt),y+ay=cos(bt), where y+ay=0y+ay=0 represents the natural frequency of the glass and the singer is forcing the vibrations at cos(bt).cos(bt). For what value bb would the singer be able to break that glass? (Note: in order for the glass to break, the oscillations would need to get higher and higher.)

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