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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.

202.

A parametric equation that passes through points P and Q can be given by r(t)=〈t2,3t+1,t−2〉,r(t)=〈t2,3t+1,t−2〉, where P(1,4,−1)P(1,4,−1) and Q(16,11,2).Q(16,11,2).

203.

d d t [ u ( t ) × u ( t ) ] = 2 u ′ ( t ) × u ( t ) d d t [ u ( t ) × u ( t ) ] = 2 u ′ ( t ) × u ( t )

204.

The curvature of a circle of radius rr is constant everywhere. Furthermore, the curvature is equal to 1/r.1/r.

205.

The speed of a particle with a position function r(t)r(t) is (r′(t))/(|r′(t)|).(r′(t))/(|r′(t)|).

Find the domains of the vector-valued functions.

206.

r ( t ) = 〈 sin ( t ) , ln ( t ) , t 〉 r ( t ) = 〈 sin ( t ) , ln ( t ) , t 〉

207.

r ( t ) = 〈 e t , 1 4 − t , sec ( t ) 〉 r ( t ) = 〈 e t , 1 4 − t , sec ( t ) 〉

Sketch the curves for the following vector equations. Use a calculator if needed.

208.

[T] r(t)=〈t2,t3〉r(t)=〈t2,t3〉

209.

[T] r(t)=〈sin(20t)e−t,cos(20t)e−t,e−t〉r(t)=〈sin(20t)e−t,cos(20t)e−t,e−t〉

Find a vector function that describes the following curves.

210.

Intersection of the cylinder x2+y2=4x2+y2=4 with the plane x+z=6x+z=6

211.

Intersection of the cone z=x2+y2z=x2+y2 and plane z=y−4z=y−4

Find the derivatives of u(t),u(t), u′(t),u′(t), u′(t)×u(t),u′(t)×u(t), u(t)×u′(t),u(t)×u′(t), and u(t)·u′(t).u(t)·u′(t). Find the unit tangent vector.

212.

u ( t ) = 〈 e t , e − t 〉 u ( t ) = 〈 e t , e − t 〉

213.

u ( t ) = 〈 t 2 , 2 t + 6 , 4 t 5 − 12 〉 u ( t ) = 〈 t 2 , 2 t + 6 , 4 t 5 − 12 〉

Evaluate the following integrals.

214.

∫ ( tan ( t ) sec ( t ) i − t e 3 t j ) d t ∫ ( tan ( t ) sec ( t ) i − t e 3 t j ) d t

215.

∫14u(t)dt,∫14u(t)dt, with u(t)=〈ln(t)t,1t,sin(tπ4)〉u(t)=〈ln(t)t,1t,sin(tπ4)〉

Find the length for the following curves.

216.

r(t)=〈3t,4cos(t),4sin(t)〉r(t)=〈3t,4cos(t),4sin(t)〉 for 1≤t≤41≤t≤4

217.

r(t)=2i+tj+3t2kr(t)=2i+tj+3t2k for 0≤t≤10≤t≤1

Reparameterize the following functions with respect to their arc length measured from t=0t=0 in direction of increasing t.t.

218.

r ( t ) = 2 t i + ( 4 t − 5 ) j + ( 1 − 3 t ) k r ( t ) = 2 t i + ( 4 t − 5 ) j + ( 1 − 3 t ) k

219.

r ( t ) = cos ( 2 t ) i + 8 t j − sin ( 2 t ) k r ( t ) = cos ( 2 t ) i + 8 t j − sin ( 2 t ) k

Find the curvature for the following vector functions.

220.

r ( t ) = ( 2 sin t ) i − 4 t j + ( 2 cos t ) k r ( t ) = ( 2 sin t ) i − 4 t j + ( 2 cos t ) k

221.

r ( t ) = 2 e t i + 2 e − t j + 2 t k r ( t ) = 2 e t i + 2 e − t j + 2 t k

222.

Find the unit tangent vector, the unit normal vector, and the binormal vector for r(t)=2costi+3tj+2sintk.r(t)=2costi+3tj+2sintk.

223.

Find the tangential and normal acceleration components with the position vector r(t)=〈cost,sint,et〉.r(t)=〈cost,sint,et〉.

224.

A Ferris wheel car is moving at a constant speed vv and has a constant radius r.r. Find the tangential and normal acceleration of the Ferris wheel car.

225.

The position of a particle is given by r(t)=〈t2,ln(t),sin(πt)〉,r(t)=〈t2,ln(t),sin(πt)〉, where tt is measured in seconds and rr is measured in meters. Find the velocity, acceleration, and speed functions. What are the position, velocity, speed, and acceleration of the particle at 1 sec?

The following problems consider launching a cannonball out of a cannon. The cannonball is shot out of the cannon with an angle θθ and initial velocity v0.v0. The only force acting on the cannonball is gravity, so we begin with a constant acceleration a(t)=−gj.a(t)=−gj.

226.

Find the velocity vector function v(t).v(t).

227.

Find the position vector r(t)r(t) and the parametric representation for the position.

228.

At what angle do you need to fire the cannonball for the horizontal distance to be greatest? What is the total distance it would travel?

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