Type:
Solution Manual
Resource:
Calculus Early Transcendentals
Edition:
11th Edition
Author(s):
Howard Anton Irl Bivens Stephen Davis
Instructor’s Manual to Accompany
Calculus Early Transcendentals 11th Edition
Howard Anton Drexel University
Irl C. Bivens Davidson College
Stephen L. Davis Davidson College
John Wiley & Sons, Inc.
i
Preface
Teaching calculus for the first time can be a daunting task. Decisions must be made about how much time to spend on each topic, what points need special emphasis, what problems to assign for homework, etc. The purpose of this Instructor’s Manual is to provide guidance in such choices for instructors using Calculus, Early Transcendentals, 11th ed., by Howard Anton, Irl Bivens, and Stephen Davis. While the manual should especially be of help to first-time instructors and to first-time users of Calculus, it is hoped that even experienced instructors will find something of value here. The layout of the Instructor’s Manual is straightforward. Each section of Calculus is given a suggested time allocation and a teaching plan. The teaching plans range from short, for relatively straightforward sections, to lengthy, for sections that involve more sophisticated ideas. Each teaching plan also contains a bulleted list of key points to emphasize. Responses to questions posed in Margin or Technology Mastery Notes are included at the end of a section’s entry, followed by a sample homework assignment. (Of course the suggested teaching plans are precisely that, suggestions meant to provide guidance for instructors. The authors would be delighted to hear from instructors about what works, what doesn’t, what advice should be kept, and what should be changed or dropped.) The Instructor Companion Site for Calculus at www.wiley.com/college/anton provides additional resources for the instructor. There are slides that reproduce important figures from the text and sample exams for each chapter. This Instructor’s Manual points out what slides are available for each section. It is the authors’ hope that Calculus will help both instructors and their students to have an enjoyable and worthwhile teaching and learning experience. Please let us know how it goes.
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Contents Limits and Continuity 1.1 Limits (An Intuitive Approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Computing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Limits at Infinity; End-Behavior of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Limits (Discussed More Rigorously) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Continuity of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 2 2 3 4 6
1.7 Inverse Trigonemtric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.8 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2
The Derivative 2.1 Tangent Lines and Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Derivative Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Introduction to Techniques of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 9 10 10 11 11 12
3
14 Topics in Differentiation 3.1 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Derivatives of Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Derivatives of Exponential and Inverse Trigonometric Functions . . . . . . . . . . . . . . . . 15 3.4 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.5 Local Linear Approximations; Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.6 L’Hˆopital’s Rule; Indeterminate Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4
18 The Derivative in Graphing and Applications 4.1 Analysis of Functions I: Increase, Decrease, and Concavity . . . . . . . . . . . . . . . . . . . 18 4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials . . . . . . . . . . . . . . . 18 4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents . . . . . . . . . 19 4.4 Absolute Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.5 Applied Maximum and Minimum Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.6 Rectilinear Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.7 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.8 Rolle’s Theorem; Mean-Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5
24 Integration 5.1 An Overview of the Area Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.3 Integration by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.4 The Definition of Area as a Limit; Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.5 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.6 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1
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5.7 Rectilinear Motion Revisited Using Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.8 Average Value of a Function and its Applications . . . . . . . . . . . . . . . . . . . . . . . . 29 5.9 Evaluating Definite Integrals by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.10 Logarithmic and Other Functions Defined by Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 29 6
Applications of the Definite Integral in Geometry, Science, and Engineering 6.1 Area Between Two Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Volumes by Slicing; Disks and Washers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Volumes by Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Length of a Plane Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Area of a Surface of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Moments, Centers of Gravity, and Centroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Fluid Pressure and Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Hyperbolic Functions and Hanging Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31 31 32 32 33 34 34 35 35 36
7
Principles of Integral Evaluation 7.1 An Overview of Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Integrating Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Integrating Rational Functions by Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . 7.6 Using Computer Algebra Systems and Tables of Integrals . . . . . . . . . . . . . . . . . . . . 7.7 Numerical Integration; Simpson’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37 37 37 38 39 40 41 41 42
8
43 Mathematical Modeling with Differential Equations 8.1 Modeling with Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 8.2 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 8.3 Slope Fields; Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 8.4 First Order Differential Equations and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9
Infinite Series 9.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Monotone Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 The Comparison, Ratio, and Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Alternating Series; Absolute and Conditional Convergence . . . . . . . . . . . . . . . . . . . 9.7 Maclaurin and Taylor Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Maclaurin and Taylor Series; Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Convergence of Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series . . . . . . . . . . . . .
46 46 47 47 48 49 50 50 51 52 53
10
Parametric and Polar Curves; Conic Sections 10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves . . . . . . . . 10.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Tangent Lines, Arc Length, and Area for Polar Curves . . . . . . . . . . . . . . . . . . . . . 10.4 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Rotation of Axes; Second-Degree Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Conic Sections in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55 55 55 56 56 57 58
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11
59 Three-Dimensional Space; Vectors 11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces . . . . . . . . . . . . . . 59 11.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 11.3 Dot Product; Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 11.4 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 11.5 Parametric Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 11.6 Planes in 3-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 11.7 Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 11.8 Cylindrical and Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
12
Vector-Valued Functions 12.1 Introduction to Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Calculus of Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Change of Parameter; Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Unit Tangent, Normal, and Binormal Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Motion Along a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Kepler’s Laws of Planetary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64 64 64 66 66 67 67 68
13
Partial Derivatives 13.1 Functions of Two or More Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Differentiability, Differentials, and Local Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Directional Derivatives and Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Tangent Planes and Normal Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8 Maxima and Minima of Functions of Two Variables . . . . . . . . . . . . . . . . . . . . . . 13.9 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69 69 70 71 71 72 73 73 74 75
14
76 Multiple Integrals 14.1 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 14.2 Double Integrals Over Nonrectangular Regions . . . . . . . . . . . . . . . . . . . . . . . . . 76 14.3 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 14.4 Surface Area; Parametric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 14.5 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 14.6 Triple Integrals in Cylindrical and Spherical Coordinates . . . . . . . . . . . . . . . . . . . 80 14.7 Change of Variables in Multiple Integrals; Jacobians . . . . . . . . . . . . . . . . . . . . . . 80 14.8 Centers of Gravity Using Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
15
Topics in Vector Calculus 15.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Independence of Path; Conservative Vector Fields . . . . . . . . . . . . . . . . . . . . . . . 15.4 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Applications of Surface Integrals; Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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82 82 83 84 85 86 87 87 88
rch 26, arch 26, 2009 2009 06:39 06:39
”ISM ”ISM LT LT chapter chapter 1” 1”
Sheet Sheet number number 11 Page Page number number 26 26
black black
CHAPTER CHAPTER 11
Limits Limits and and Continuity Continuity Limits and Continuity EXERCISE EXERCISE SET SET 1.1 1.1 1. (b) Exercise 1. (a) (a) 33 Set 1.1 (b) 33
(c) (c) 33
(d) (d) 33
1. (a) 30 2. 2. (a) (a) 0
(b) 3 (b) (b) 00
(c) 3 (c) (c) 00
2. (a) 0 3. 3. (a) (a) −1 −1
(b) 0 (b) (b) 33
(c) 0 (d) 0 (c) (c) does does not not exist exist
3. (a) −1 4. 4. (a) (a) 22 4. (a) 2 5. 5. (a) (a) 00 5. (a) 0 6. 6. (a) (a) 11 6. (a) 1 7. 7. (a) (a) −∞ −∞ 7. (a) −∞
(c) does not exist (d) 1 (b) (c) (b) 00 (c) does does not not exist exist (b) 0 (c) does not exist (d) 2 (b) (c) (d) (b) 00 (c) 00 (d) 33 (b) 0 (c) 0 (d) 3 (b) 1 (c) 1 (d) (b) 1 (c) 1 (d) 00 (b) 1 (c) 1 (d) 0 (b) (c) (d) (b) −∞ −∞ (c) −∞ −∞ (d) 11 (b) −∞ (c) −∞ (d) 1 (b) +∞ (b) +∞ +∞ (b)
10. (a) +∞ 10. (a) (a) does +∞not exist
(ii) 12. (i) 12. (i) 12. (i)
(c) +∞ (c) +∞ +∞ (c)
(b) −∞ −∞ 2 (b) +∞ (b) (c)
9. (a) 11 9. (a) +∞ 9. (a)
(ii) (ii)
(b) 11 (b) (b)−∞
13. 13. (a) (a)
(d) (d) 22
(d) undef (d) can undef (d) not be found from graph
(c) does not −2 does−∞ not exist exist (f ) (d) −2x = 0, x = 2 (d) 2 (c) (e) x(d) = −2, (c) (d) (e)exist +∞ (c)−1does does not not exist
(c) 0
(d) (f ) 322 (d)
(g) x = −2, x = 2
−0.1 −0.01 −0.001 0.001 0.01 0.1 −0.1 −0.01 −0.001 0.001 0.01 0.1 −0.1 −0.01 −0.001 0.001 0.01 0.1 1.9866933 1.9998667 1.9999987 1.9999987 1.9998667 1.9866933 1.9866933 1.9998667 1.9999987 1.9999987 1.9998667 1.9866933 1.9866933 1.9998667 1.9999987 1.9999987 1.9998667 1.9866933 2. 2.
The The limit limit appears appears to to be be 2. 2.
1.986 -0.1 1.986 -0.1
0.1 0.1
The limit appears to be 2.
−0.5 −0.05 −0.005 0.005 0.05 0.5 −0.5 −0.05 −0.005 0.005 0.05 0.5 −0.489669752 −0.499895842 −0.499998958 −0.499998958 −0.499895842 −0.489669752 −0.489669752 −0.499895842 −0.499998958 −0.499998958 −0.499895842 −0.489669752
(ii) (ii) -0.4896698 -0.4896698
(ii)
(d) (d) 11
(b) 3
8. (a) +∞ 8. (a) +∞ +∞ 8. (a)
11. (i) 11. (i) 11. (i)
(d) 3 (d) (d) 00
The The limit limit appears appears to to be be −1/2. −1/2.
-0.5 -0.5 -0.5 -0.5
22 0.1429
1.5 1.5 0.2105
0.5 0.5
1.1 1.1 0.3021
1.01 1.01 0.3300
The limit appears to be −1/2. 1.001 1.001 0.3330
1 00 1.0000
0.5 0.5 0.5714
0.9 0.9 0.3690
0.99 0.99 0.3367
0.999 0.999 0.3337