Student Solutions Manual Calculus of a Single Variable TENTH EDITION Ron Larson The Pennsylvania State University, The Behrend College Bruce Edwards University of Florida Prepared by Bruce Edwards University of Florida Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. 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C O N T E N T S Chapter P Preparation for Calculus......................................................................................................1 Chapter 1 Limits and Their Properties...............................................................................................27 Chapter 2 Differentiation....................................................................................................................57 Chapter 3 Applications of Differentiation........................................................................................110 Chapter 4 Integration........................................................................................................................187 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions....................................231 Chapter 6 Differential Equations......................................................................................................290 Chapter 7 Applications of Integration..............................................................................................321 Chapter 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals.................................371 Chapter 9 Infinite Series...................................................................................................................449 Chapter 10 Conics, Parametric Equations, and Polar Coordinates.....................................................514 iii Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. C H A P T E R P Preparation for Calculus Section P.1 Graphs and Models.................................................................................2 Section P.2 Linear Models and Rates of Change......................................................6 Section P.3 Functions and Their Graphs.................................................................12 Section P.4 Fitting Models to Data..........................................................................18 Review Exercises..........................................................................................................20 Problem Solving...........................................................................................................23 Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. C H A P T E R P Preparation for Calculus Section P.1 Graphs and Models 1. y = −3x + 3 9. y = x + 2 2 x-intercept: (2, 0) x −5 −4 −3 −2 −1 0 1 y-intercept: (0, 3) y 3 2 1 0 1 2 3 Matches graph (b). 2. y = 9 − x2 y 6 x-intercepts: (−3, 0), (3, 0) 4 (−5, 3) y-intercept: (0, 3) (1, 3) (−4, 2) 2 (0, 2) Matches graph (d). (−3, 1) (−1, 1) x −6 −4 (−2, 0) 2 3. y = 3 − x2 −2 ( ) ( ) x-intercepts: 3, 0, − 3, 0 11. y = x − 6 y-intercept: (0, 3) x 0 1 4 9 16 Matches graph (a). y −6 −5 −4 −3 −2 4. y = x3 − x x-intercepts: (0, 0), (−1, 0), (1, 0) y 2 y-intercept: (0, 0) x Matches graph (c). −4 4 8 12 16 −2 (9, −3) (16, −2) 5. y = 1x + 2 −4 (4, −4) 2 (1, −5) −6 (0, −6) x −4 −2 0 2 4 −8 y 0 1 2 3 4 3 13. y = x y 6 x −3 −2 −1 0 1 2 3 (4, 4) 4 (2, 3) y −1 −3 −3 Undef. 3 3 1 (0, 2) 2 2 (−2, 1) x −4 −2 2 4 y (−4, 0) −2 (1, 3) 3 2 (2, 32( 7. y = 4 − x2 (3, 1) (−3, −1) 1 x x −3 −2 0 2 3 −3 −2 −1 1 2 3 −1 y −5 0 4 0 −5 −2 (−2, − 32( (−1, −3) y 6 (0, 4) 2 (−2, 0) (2, 0) x −6 −4 4 6 −2 (−3, −5) −4 (3, −5) −6 2 Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Section P.1 Graphs and Models 3 15. y = 5 − x 27. Symmetric with respect to the y-axis because y = (−x)2 − 6 = x2 − 6. 5 (−4.00, 3) (2, 1.73) 29. Symmetric with respect to the x-axis because −6 6 (−y)2 = y2 = x3 − 8x. −3 31. Symmetric with respect to the origin because (a) (2, y) = (2,1.73) (y = 5 − 2 = 3 ≈ 1.73) (−x)(−y) = xy = 4. (b) (x, 3) = (−4, 3) (3 = 5 − (−4)) 33. y = 4 − x + 3 No symmetry with respect to either axis or the origin. 17. y = 2x − 5 35. Symmetric with respect to the origin because y-intercept: y = 2(0) − 5 = −5; (0, −5) −x −y = x-intercept: 0 = 2x − 5 (−x)2 +1 5 = 2x x y = . x = 5; (5, 0) x2 +1 2 2 37. y = x3 + x is symmetric with respect to the y-axis 19. y = x2 + x − 2 y-intercept: y = 02 + 0 − 2 because y = (−x)3 + (−x) = −(x3 + x) = x3 + x. y = −2; (0, −2) 39. y = 2 − 3x x-intercepts: 0 = x2 + x − 2 y = 2 − 3(0) = 2, y-intercept 0 = (x + 2)(x −1) 0 = 2 − 3(x) ⇒ 3x = 2 ⇒ x = 2, x-intercept x = −2,1;(−2, 0), (1, 0) 3 Intercepts: (0, 2), (2, 0) y 3 21. y = x 16 − x2 Symmetry: none y-intercept: y = 0 16 − 02 = 0; (0, 0) 2 (0, 2) 1 x-intercepts: 0 = x 16 − x2 (2, 0( 3 x 0 = x (4 − x)(4 + x) −1 2 3 −1 x = 0, 4, −4;(0, 0), (4, 0), (−4, 0) 41. y = 9 − x2 23. y = 2 − x y = 9 − (0)2 = 9, y-intercept 5x +1 0 = 9 − x2 ⇒ x2 = 9 ⇒ x = ±3, x-intercepts 2 − 0 y-intercept: y = 5(0) +1 = 2 ; (0,2) Intercepts: (0, 9), (3, 0), (−3, 0) y 10 (0, 9) 2 − x y = 9 − (−x)2 = 9 − x2 x-intercept: 0 = 5x +1 6 Symmetry: y-axis 4 0 = 2 − x 2 (−3, 0) (3, 0) x = 4 ; (4,0) x −6 −4 −2 2 4 6 −2 25. x2y − x2 + 4y = 0 y-intercept: 02(y) − 02 + 4y = 0 y = 0; (0, 0) x-intercept: x2(0) − x2 + 4(0) = 0 x = 0; (0, 0) Copyright 2013 Cengage Learning. 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May not be copied, scanned, or duplicated, in whole or in part. 4 Chapter P Preparation for Calculus 43. y = x3 + 2 51. y = 6 − x y = 03 + 2 = 2, y-intercept y = 6 − 0 = 6, y-intercept 0 = x3 + 2 ⇒ x3 = −2 ⇒ x = −3 2, x-intercept 6 − x = 0 Intercepts: (−3 2, 0), (0, 2) 6 = x y x = ±6, x-intercepts Symmetry: none 5 4 Intercepts: (0, 6), (−6, 0), (6, 0) 3 (0, 2) y = 6 − −x = 6 − x (− 3 2, 0) 1 x Symmetry: y-axis −3 −2 1 2 3 −1 y 8 6 (0, 6) 45. y = x x + 5 4 y = 0 0 + 5 = 0, y-intercept (−6, 0) 2 (6, 0) x −8 −4−2 2 4 6 8 −2 x x + 5 = 0 ⇒ x = 0,−5, x-intercepts −4 Intercepts: (0, 0), (−5, 0) y −6 −8 3 Symmetry: none 2 53. y2 − x = 9 (−5, 0) (0, 0) x y2 = x + 9 −4 −3 −2 −1 1 2 y = ± x + 9 −3 −4 y = ± 0 + 9 = ± 9 = ±3, y-intercepts ± x + 9 = 0 47. x = y3 x + 9 = 0 x = −9, x-intercept y3 = 0 ⇒ y = 0, y-intercept x = 0,x-intercept Intercepts: (0, 3), (0, −3), (−9, 0) y Intercept: (0, 0) 4 (−y)2 − x = 9 ⇒ y2 − x = 9 3 −x = (−y)3 ⇒ −x = −y3 2 Symmetry: x-axis (0, 0) Symmetry: origin −4−3−2−1 1 2 3 4 x y −2 6 −3 4 −4 (−9, 0) 2 (0, 3) x 8 −10 −6 −4 −2 2 49. y = −2 (0, −3) x −4 8 −6 y = ⇒ Undefined ⇒ no y-intercept 0 8 = 0 ⇒ Nosolution ⇒ no x-intercept x Intercepts: none y 8 8 8 −y = ⇒ y = 6 −x x 4 2 Symmetry: origin x −2 2 4 6 8 Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.