Robert T Smith, Roland B Minton-Instructor's Solution Manuals to Calculus_ Early Transcendental Func - Contents 0 Preliminaries.1 0.1 Polynomials and

# Robert T Smith, Roland B Minton-Instructor's Solution Manuals to Calculus_ Early Transcendental Func

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Unformatted text preview: Contents 0 Preliminaries ................................................................................................................1 0.1 Polynomials and Rational Functions .................................................................1 0.2 Graphing Calculators and Computer Algebra Systems .....................................5 0.3 Inverse Functions .............................................................................................14 0.4 Trigonometric and Inverse Trigonometric Functions ......................................20 0.5 Exponential and Logarithmic Functions..........................................................25 0.6 Transformations of Functions ..........................................................................30 Chapter 0 Review Exercises ............................................................................34 1 Limits and Continuity................................................................................................40 1.1 A Brief Preview of Calculus ............................................................................40 1.2 The Concept of Limit.......................................................................................44 1.3 Computation of Limits.....................................................................................49 1.4 Continuity and its Consequences .....................................................................54 1.5 Limits Involving Infinity; Asymptotes ............................................................59 1.6 Formal Definition of the Limit ........................................................................63 1.7 Limits and Loss-of-Significance Errors...........................................................68 Chapter 1 Review Exercises ............................................................................72 2 Differentiation ............................................................................................................78 2.1 Tangent Line and Velocity...............................................................................78 2.2 The Derivative .................................................................................................85 2.3 Computation of Derivatives: The Power Rule.................................................94 2.4 The Product and Quotient Rules ....................................................................101 2.5 The Chain Rule ..............................................................................................108 2.6 Derivatives of Trigonometric Functions........................................................115 2.7 Derivatives of Exponential and Logarithmic Functions ................................119 2.8 Implicit Differentiation and Inverse Trigonometric Function .......................125 2.9 The Hyperbolic Functions..............................................................................135 2.10 The Mean Value Theorem .............................................................................138 Chapter 2 Review Exercises ..........................................................................143 3 Applications of Differentiation ...............................................................................150 3.1 Linear Approximations and Newton’s Method .............................................150 3.2 Indeterminate Forms and L’Hôpital’s Rule ...................................................158 3.3 Maximum and Minimum Values ...................................................................165 3.4 Increasing and Decreasing Functions ............................................................176 3.5 Concavity and the Second Derivative Test....................................................187 3.6 Overview of Curve Sketching........................................................................197 3.7 Optimization ..................................................................................................213 3.8 Related Rates .................................................................................................221 3.9 Rates of Change in Economics and the Sciences...........................................226 Chapter 3 Review Exercises ..........................................................................231 4 Integration ................................................................................................................240 4.1 Antiderivatives...............................................................................................240 4.2 Sums and Sigma Notation..............................................................................246 4.3 Area................................................................................................................251 4.4 The Definite Integral......................................................................................261 4.5 The Fundamental Theorem of Calculus.........................................................271 4.6 Integration by Substitution.............................................................................277 4.7 Numerical Integration ....................................................................................285 4.8 The Natural Logarithm as an Integral............................................................294 Chapter 4 Review Exercises ..........................................................................298 5 Applications of the Definite Integral ......................................................................304 5.1 Area Between Curves ....................................................................................304 5.2 Volume: Slicing, Disks, and Washers............................................................313 5.3 Volumes by Cylindrical Shells ......................................................................321 5.4 Arc Length and Surface Area ........................................................................328 5.5 Projectile Motion ...........................................................................................334 5.6 Applications of Integration to Physics and Engineering................................343 5.7 Probability......................................................................................................349 Chapter 5 Review Exercises ..........................................................................354 6 Integration Techniques............................................................................................360 6.1 Review of Formulas and Techniques.............................................................360 6.2 Integration by Parts ........................................................................................363 6.3 Trigonometric Techniques of Integration ......................................................372 6.4 Integration of Rational Functions Using Partial Fractions ............................380 6.5 Integration Table and Computer Algebra Systems........................................386 6.6 Improper Integrals..........................................................................................389 Chapter 6 Review Exercises ..........................................................................400 7 First-Order Differential Equations ........................................................................407 7.1 Modeling with Differential Equations ...........................................................407 7.2 Separable Differential Equations ...................................................................413 7.3 Direction Fields and Euler’s Method.............................................................423 7.4 Systems of First-Order Differential Equations ..............................................431 Chapter 7 Review Exercises ..........................................................................437 8 Infinite Series............................................................................................................443 8.1 Sequences of Real Numbers ..........................................................................443 8.2 Infinite Series .................................................................................................449 8.3 The Integral Test and Comparison Tests .......................................................457 8.4 Alternating Series...........................................................................................467 8.5 Absolute Convergence and the Ratio Test.....................................................471 8.6 Power Series...................................................................................................477 8.7 Taylor Series ..................................................................................................487 8.8 Applications of Taylor Series ........................................................................496 8.9 Fourier Series .................................................................................................500 Chapter 8 Review Exercises ..........................................................................509 9 Parametric Equations and Polar Coordinates ......................................................515 9.1 Plane Curves and Parametric Equations ........................................................515 9.2 Calculus and Parametric Equations ...............................................................524 9.3 Arc Length and Surface Area in Parametric Equations .................................532 9.4 Polar Coordinates...........................................................................................535 9.5 Calculus and Polar Coordinates .....................................................................550 9.6 Conic Sections ...............................................................................................563 Chapter 9 Review Exercises ..........................................................................571 10 Vectors and the Geometry of Space .......................................................................580 10.1 Vectors in the Plane .......................................................................................580 10.2 Vectors in Space ............................................................................................586 10.3 The Dot Product.............................................................................................591 10.4 The Cross Product..........................................................................................596 10.5 Lines and Planes in Space..............................................................................600 10.6 Surfaces in Space ...........................................................................................605 Chapter 10 Review Exercises ........................................................................613 11 Vector-Valued Functions.........................................................................................619 11.1 Vector-Valued Functions ...............................................................................619 11.2 The Calculus of Vector-Valued Functions ....................................................627 11.3 Motion in Space .............................................................................................632 11.4 Curvature........................................................................................................636 11.5 Tangent and Normal Vectors .........................................................................643 11.6 Parametric Surfaces .......................................................................................649 Chapter 11 Review Exercises ........................................................................655 12 Functions of Several Variables and Partial Differentiation.................................660 12.1 Functions of Several Variables ......................................................................660 12.2 Limits and Continuity ....................................................................................670 12.3 Partial Derivatives..........................................................................................676 12.4 Tangent Planes and Linear Approximations..................................................682 12.5 The Chain Rule ..............................................................................................689 12.6 The Gradient and Directional Derivatives .....................................................698 12.7 Extrema of Functions of Several Variables ...................................................705 12.8 Constrained Optimization and Lagrange Multipliers ....................................716 Chapter 12 Review Exercises ........................................................................729 13 Multiple Integrals.....................................................................................................738 13.1 Double Integrals.............................................................................................738 13.2 Area, Volume, and Center of Mass................................................................752 13.3 Double Integrals in Polar Coordinates...........................................................766 13.4 Surface Area...................................................................................................774 13.5 Triple Integrals...............................................................................................781 13.6 Cylindrical Coordinates .................................................................................792 13.7 Spherical Coordinates ....................................................................................799 13.8 Change of Variables in Multiple Integrals.....................................................808 Chapter 13 Review Exercises ........................................................................814 14 Vector Calculus ........................................................................................................822 14.1 Vector Fields..................................................................................................822 14.2 Line Integrals .................................................................................................829 14.3 Independence of Path and Conservative Vector Fields .................................837 14.4 Green’s Theorem ...........................................................................................844 14.5 Curl and Divergence ......................................................................................852 14.6 Surface Integrals ............................................................................................864 14.7 The Divergence Theorem ..............................................................................877 14.8 Stokes’ Theorem ............................................................................................884 14.9 Applications of Vector Calculus....................................................................892 Chapter 14 Review Exercises ........................................................................896 15 Second Order Differential Equations.....................................................................907 15.1 Second-Order Equations with Constant Coefficients ....................................907 15.2 Nonhomogeneous Equations: Undetermined Coefficients............................914 15.3 Applications of Second Order Equations.......................................................922 15.4 Power Series Solutions to Differential Equations..........................................931 Chapter 15 Review Exercises ........................................................................937 9. |x + 5 | < 2 −2 < x + 5 < 2 −2 − 5 < x < 2 − 5 −7 < x < −3. 10. |2x + 1 | < 4 −4 < 2x + 1 < 4 −4 − 1 < 2x < 4 − 1 −5 < 2x < 3 3 5 − <x< 2 2 11. Yes. The slope of the line joining the points 1 (2, 1) and (0, 2) is − , which is also the slope 2 of the line joining the Points (0, 2) and (4, 0). Chapter 0 Preliminaries 0.1 12. No. The slope of the line joining the points (3, 1) and (4, 4) is 3, while the slope of the line joining the points (4, 4) and (5, 8) is 4. Polynomials and Rational Fuctions 13. No. The slope of the line joining the points (4, 1) and (3, 2) is −1, while the slope of the 1 line joining the points (3, 2) and (1, 3) is − . 2 14. No. The slope of the line joining the points (1, 2) and (2, 5) is 3, but the slope of line join3 ing the points (2, 5) and (4, 8) is . 2 1. 3x + 2 < 8 3x < 8 − 2 3x < 6 x<2 2. 3 − 2x < 7 −2x < 4 x > −2 15. (a) d {(1, q 2) , (3, 6)} 2 2 = (3 − 1) + (6 − 2) √ √ = 4 + 16 = 20 y2 − y1 6−2 (b) m = =2 = x2 − x1 3−1 (c) The equation of line is y = m (x − x0 ) + y0 y = 2 (x − 1) + 2 y = 2x 3. 1 ≤ 2 − 3x < 6 1 − 2 ≤ −3x < 6 − 2 −1 ≤ −3x < 4 1 4 ≥x>− 3 3 4. −2 < 2x − 3 ≤ 5 1 < 2x ≤ 8 1 <x≤4 2 x+2 5. ≥0 x−4 x + 2 ≥ 0, x − 4 > 0 or x + 2 ≤ 0, x − 4 < 0 x ≥ −2, x > 4 or x ≤ −2, x < 4 x > 4 or x ≤ −2 6. 16. (a) d {(1, q −2) , (−1, −3)} 2 2 = (−1 − 1) + (−3 + 2) √ √ = 4+1= 5 −3 + 2 1 y2 − y1 = = (b) m = x2 − x1 −1 − 1 2 (c) The equation of line is y = m (x − x0 ) + y0 1 y = (x − 1) + (−1) 2 x−3 y= 2 2x + 1 <0 x+2 2x + 1 < 0, x + 2 > 0 or 2x + 1 > 0, x + 2 < 0 x < − 12 , x > −2 or x > − 12 , x < −2 1 −2 < x < − (Since x > − 12 , x < −2 is not 2 possible). 7. x2 + 2x − 3 ≥ 0 (x + 3) (x − 1) ≥ 0 x ≥ 1 or x ≤ −3 17. (a) d {(0.3, q −1.4) , (−1.1, −0.4)} 2 2 = (−1.1 − 0.3) + (−0.4 + 1.4) q √ 2 = (−1.4) + 1 = 2.96 y2 − y1 −0.4 + 1.4 1 (b) m = = =− x2 − x1 −1.1 − 0.3 1.4 8. x2 − 5x − 6 < 0 (x − 6) (x + 1) < 0 −1 < x < 6 1 2 CHAPTER 0. PRELIMINARIES (c) The equation of line is y = m (x − x0 ) + y0 1 y=− (x − 0.3) − 1.4 1.4 1.4y = −x − 1.66 x + 1.4y = −1.66 5 4 3 2 1 0 −5 −4 −3 −2 x −1 0 −1 1 2 3 4 5 4 5 −2 18. (a) d {(1.2, q 2.1) , (3.1, 2.4)} 2 2 = (3.1 − 1.2) + (2.4 − 2.1) q 2 2 = (1.9) + (0.3) √ √ = 3.61 + 0.09 = 3.7 (b) m = y −3 −4 −5 1 1 1 22. y = − (x + 2) + 1 = − x + 4 4 2 5 y2 − y1 2.4 − 2.1 0.3 = = ≈ 0.16 x2 − x1 3.1 − 1.2 1.9 4 3 2 (c) The equation of line is y = m (x − x0 ) + y0 y = (0.16) (x − 1.2) − 2.1 y = 0.16x − 2.292 1 0 −5 −4 −3 −2 −1 0 1 2 3 −1 x −2 y −3 19. y = 2 (x − 1) + 3 = 2x + 1 −4 −5 5 4 23. Parallel. Both have slope 3. 3 24. Neither. Slopes are 2 and 4. 2 1 25. Perpendicular. Slopes are −2 and 0 −5 −4 −3 −2 0 −1 1 2 3 4 5 −1 26. Neither. Slopes are 2 and −2. −2 1 27. Perpendicular. Slopes are 3and − . 3 1 28. Parallel. Both have slope − . 2 −3 −4 −5 29. (a) y = 2 (x − 2) + 1 1 (b) y = − (x − 2) + 1 2 20. y = 1 2.0 1.6 30. (a) y = 3x + 3 1 (b) y = − x + 3 3 1.2 0.8 0.4 0.0 −2 −1 0 1 −0.4 x 1 . 2 −0.8 y −1.2 −1.6 −2.0 2 31. (a) y = 2 (x − 3) + 1 1 (b) y = − (x − 3) + 1 2 32. (a) y = −1 (b) x = 0 3−1 2 = =2 2−1 1 Equation of line is y = 2 (x − 1) + 1 = 2x − 1. When x = 4, y = 7. 33. Slope m = 21. y = 1.2 (x − 2.3) + 1.1 = 1.2x − 1.66 0.1. POLYNOMIALS AND RATIONAL FUCTIONS 1 2 1 Equation of line is y = (x + 2) + 1. 2 When x = 4, y = 4. 34. Slope m = 35. Yes, passes vertical line test. 36. Yes, passes vertical line test. 37. No. The vertical line x = 0 meets the curve twice; nearby vertical lines meet it three times. 38. No, does not pass vertical line test. 39. Both: This is clearly a cubic polynomial, and also a rational function because it can be written as x3 − 4x + 1 . f (x) = 1 This shows that all polynomials are rational. 40. Rational. 41. Rational. 42. Neither: Contains square root. 43. We need the function under the square root to be non-negative. x + 2 ≥ 0 when x ≥ −2. The domain is {x ∈ R|x ≥ −2} = [−2, ∞) . 44. Negatives are permitted inside the cube root. There are no restrictions, so the domain is (−∞, ∞) or all real numbers. 45. The function is defined only if x2 − x − 6 ≥ 0 and x 6= 5 (x − 3) (x + 2) ≥ 0 and x 6= 5 x ≤ −2 or x ≥ 3 and x 6= 5 (−∞, −2] ∪ [3, 5) ∪ (5, ∞) 46. We need the numerator f...
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