EVEN12 - CHAPTER 12 Functions of Several Variables Section...

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Unformatted text preview: CHAPTER 12 Functions of Several Variables Section 12.1 Introduction to Functions of Several Variables . . . . . . . 308 Section 12.2 Limits and Continuity Section 12.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . 312 . . . . . . . . . . . . . . . . . . . . . 315 Section 12.4 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . 321 Section 12.5 Chain Rules for Functions of Several Variables . . . . . .325 Section 12.6 Directional Derivatives and Gradients . . . . . . . . . . . 330 Section 12.7 Tangent Planes and Normal Lines . . . . . . . . . . . . . 334 Section 12.8 Extrema of Functions of Two Variables . . . . . . . . . . 340 Section 12.9 Applications of Extrema of Functions of Two Variables . 345 Section 12.10 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . 350 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 CHAPTER 12 Functions of Several Variables Section 12.1 Introduction to Functions of Several Variables Solutions to Even-Numbered Exercises 2. xz 2 2xy y2 4 4. z z x ln y 8 8 0 No, z is not a function of x and y. For example, x, y 1, 0 corresponds to both z ± 2 x ln y Yes, z is a function of x and y. 8. g x, y (a) g 2, 3 ln x y ln 2 ln 5 ln e ln 0 3 ln 2 ln e ln 2 e ln e 3 6 0 1 3 ln 5 ln 11 1 0 ln 1 ln 2e ln 2 1 0 6. f x, y (a) f 0, 0 (b) f 0, 1 (c) f 2, 3 (d) f 1, y (e) f x, 0 (f) f t, 1 4 x2 4 4 4 4 4 4 0 4 1 4y 2 4 36 4y 2 0 4 0 36 3 4 t2 4y 2 x2 (b) g 5, 6 (c) g e, 0 (d) g 0, 1 (e) g 2, (f) g e, e x2 t2 10. f x, y, z (a) f 0, 5, 4 (b) f 6, 8, y x y 0 z 5 6 8 4 3 3 11 12. V r, h (a) V 3, 10 (b) V 5, 2 r 2h 3 5 2 2 10 90 50 3 1 dt t 1 2 14. g x, y x (a) g 4, 1 4 1 dt t 1 3 ln t 4 ln 4 (b) g 6, 3 6 1 dt t 3 ln t 6 ln 3 ln 6 ln 1 2 16. f x, y (a) fx 3xy x, y x y2 f x, y 3x 3xy 3 xy xy y 3x y 2y y y2 x y2 x y y 3xy y 3x y y 2 3xy 3xy y 2 y2 y2 3xy y 3 xy x y2 2 3y, x 0 (b) f x, y y y f x, y 3x y 2y y y 3x 2y 3xy y2 y, y 0 308 S ection 12.1 18. f x, y Domain: 4 4 x2 x2 x2 x2 4 x, y : x2 4 4y 2 4y 2 ≥ 0 4y 2 ≤ 4 y2 1 ≤1 Domain: 20. f x, y arccos y x Introduction to Functions of Several Variables 22. f x, y 1≤ y ≤1 x Domain: ln xy xy 6 6>0 xy > 6 x, y : xy > 6 Range: all real numbers 309 x, y : Range: 0 ≤ z ≤ y2 ≤1 1 Range: 0 ≤ z ≤ 2 xy x y x, y : x y 24. z 26. f x, y Domain: x2 y2 28. g x, y Domain: xy x, y : y ≥ 0 Domain: x, y : x is any real number, y is any real number Range: all real numbers Range: all real numbers Range: z ≥ 0 30. (a) Domain: x, y : x is any real number, y is any real number 2≤z≤2 0 which represents points on the y-axis. Range: (b) z 0 when x (c) No. When x is positive, z is negative. When x is negative, z is positive. The surface does not pass through the first octant, the octant where y is negative and x and z are positive, the octant where y is positive and x and z are negative, and the octant where x, y and z are all negative. 1 2 Cone 32. f x, y Plane Range: z 6 6 2x 3y 34. g x, y Plane: z 1 2x 1 2x z 4 3 2 36. z x2 y2 Domain: entire xy-plane <z< 4 x −2 −3 −4 Domain of f : entire xy-plane Range: z ≥ 0 z −4 2 4 y 3 x 4 x 3 2 3 4 y 2 1 1 2 3 y 38. f x, y xy, x ≥ 0, y ≥ 0 0, elsewhere Domain of f : entire xy-plane Range: z ≥ 0 z 25 20 15 10 5 y 40. f x, y 1 12 144 16x 2 9y 2 42. f x, y x sin y z Semi-ellipsoid Domain: set of all points lying on or inside the ellipse x2 9 y 2 16 1 Range: 0 ≤ z ≤ 1 z 4 x −4 4 −4 −4 4 y 5 x 4 −2 −4 4 x y 310 Chapter 12 Functions of Several Variables 44. f x, y (a) 25 20 15 10 5 z xy, x ≥ 0, y ≥ 0 (b) g is a vertical translation of f 3 units downward (c) g is a reflection of f in the xy-plane y 5 x (d) The graph of g is lower than the graph of f. If z f x, y is on the graph of f, then 1 z is on the graph 2 of g. (e) 25 20 15 10 5 z y 5 x 46. z e1 x2 y2 48. z e1 1 1 x2 y2 cos x 4 2y 2 50. f x, y 6 2x 3y Level curves: c ln c x2 y2 Level curves: x2 ln c y2 cos x2 Ellipses Matches (a) 1 c c cos x2 x2 4 2y 2 4 2y 2 The level curves are of the form 6 2x 3y c or 2x 3y 6 c. Thus, the level curves are straight lines with a slope of 2 . 3 y 3 Hyperbolas centered at 0, 0 Matches (d) 2y 2 4 cos 1 c −2 x c = 10 c=8 c=0 c=2 c=4 c=6 52. f x, y x2 2y2 54. f x, y e xy 2 The level curves are ellipses of the form x2 2y2 y 3 The level curves are of the form e xy 2 c (except x2 c=8 c=6 c=4 c=2 c=0 x 3 2y2 0 is the point 0, 0 ). c, or ln c xy . 2 Thus, the level curves are hyperbolas. y 2 1 −3 c=4 c=3 c=2 −3 −2 −1 −1 −2 x c=1 2 1 2 c=1 4 c=1 3 S ection 12.1 56. f x, y ln x y Introduction to Functions of Several Variables 58. f x, y xy −6 311 y c = −2 1 c = −2 6 x 4 The level curves are of the form c ec y ln x x x y ec y −4 −6 6 c=0 c=1 c = −1 c=2 3 c = ±2 1 c= 2 −4 Thus, the level curves are parallel lines of slope 1 passing through the fourth quadrant. 60. h x, y 3 sin x 1 y −1 1 62. The graph of a function of two variables is the set of all points x, y, z for which z f x, y and x, y is in the domain of f. The graph can be interpreted as a surface in space. Level curves are the scalar fields f x, y c, for c, a constant. −1 64. f x, y x y x or y y 1 x c 66. The surface could be an ellipsoid centered at 0, 1, 0 . One possible function is f x, y x2 y 4 1 2 The level curves are the lines c 1. These lines all pass through the origin. 68. A r, t 1000e rt Number of years Rate 0.08 0.10 0.12 0.14 5 $1491.82 $1648.72 $1822.12 $2013.75 10 $2225.54 $2718.28 $3320.12 $4055.20 15 $3320.12 $4481.69 $6049.65 $8166.17 20 $4953.03 $7389.06 $11,023.18 $16,444.65 70. f x, y, z c 4 4 4x y 4x 2z y 2z 72. f x, y, z c 1 1 x2 x2 12 4y z 74. f x, y, z c 0 sin x 0 sin x z or z z 2 z sin x Plane z 3 2 12 4y z Elliptic paraboloid Vertex: 0, 0, z 5 3 x 2 1 4 y 1 4 x 8 y 3 x 5 y 312 Chapter 12 1 x y Functions of Several Variables 76. W x, y ,y<x 1 15 1 10 6 1 hr 5 1 hr 6 12 min 10 min (b) W 12, 9 (d) W 4, 2 1 12 1 4 2 9 1 hr 3 1 hr 2 20 min 30 min (a) W 15, 10 (c) W 12, 6 12 78. f x, y f 2x, 2y 100x 0.6 y 0.4 100 2x 100 2 43 r 3 0.6 2y 0.4 0.4y 0.4 0.6x 0.6 2 100 2 0.6 2 0.4x 0.6y 0.4 2 100x 0.6y 0.4 2f x, y 80. V r 2l r2 3l 3 4r r l 82. (a) Year z Model 1995 12.7 13.09 1996 14.8 14.79 1997 17.1 16.45 1998 18.5 18.47 1999 21.1 21.38 2000 25.8 25.78 (b) x has the greater influence because its coefficient 0.143 is larger than that of y 0.024 . (c) f x, 25 0.143x 0.143x 0.024 25 1.102 25. 0.502 This function gives the shareholder’s equity z in terms of net sales x and assumes constant assets of y 84. Southwest 86. Latitude and land versus ocean location have the greatest effect on temperature. 90. True 88. True Section 12.2 2. Let Limits and Continuity > 0 such that f x, y 2 > 0 be given. We need to find L 1 2 x 4< . whenever 0 < Then if 0 < x x 4 2 x x < a 4 2 2 y y 1 b 2 x 4 2 y < . Take < , we have 4<. 4. x, y → a, b lim 4f x, y g x, y 4 x, y → a, b x, y → a, b lim f x, y g x, y lim 45 3 20 3 S ection 12.2 lim f x, y x, y → a, b Limits and Continuity 313 6. x, y → a, b lim f x, y g x, y f x, y x, y → a, b x, y → a, b lim g x, y 5 5 3 lim f x, y 2 5 8. x, y → 0, 0 lim 5x y 1 0 0 1 1 10. Continuous everywhere x 1 xy 11 Continuous for x y > 0 x, y → 1, 1 lim 2 2 12. x, y → lim 4, 2 y cos xy 2 cos 2 0 14. x, y → 1, 1 lim xy x2 y2 1 2 Continuous everywhere Continuous except at 0, 0 x2 1 y2 x 1 y2 2 16. x, y, z → 2, 0, 1 lim xe yz 2e0 2 18. f x, y lim x2 x2 1 1 0 0 10 1 0 Continuous everywhere x, y → 0, 0 Continuous everywhere cos x 2 y 2 x2 y2 20. x, y → 0, 0 lim 1 The limit does not exist. Continuous except at 0, 0 y x2 y2 22. f x, y Continuous except at 0, 0 Path: y 0 x, y f x, y Path: y x x, y f x, y 1, 1 1 2 0.5, 0.5 1 0.5, 0 0 0.1, 0.1 5 0.1, 0 0 0.01, 0.01 50 0.01, 0 0 0.001, 0.001 500 1, 0 0 0.001, 0 0 x The limit does not exist because along the path y the function tends to infinity. y2 y 0 the function equals 0, whereas along the path y 24. f x, y 2x 2x 2 Continuous except at 0, 0 Path: y 0 x, y f x, y Path: y x x, y f x, y 1, 0 1 1, 1 1 3 0.25, 0 4 0.25, 0.25 1.17 0.01, 0 100 0.001, 0 1000 0.000001, 0 1,000,000 0.01, 0.01 1.95 0.001, 0.001 1.995 0.0001, 0.0001 2.0 x The limit does not exist because along the line y the function tends to 2. 0 the function tends to infinity, whereas along the line y 314 Chapter 12 4x2y2 x2 y2 lim Functions of Several Variables 1 x 1 x z 2 26. x, y → 0, 0 lim 0 x, y → 0, 0 28. lim g x, y 0. x, y → 0, 0 lim sin cos Hence, x, y → 0, 0 f x, y Does not exist 4 x 6 6 y f is continuous at 0, 0 , whereas g is not continuous at 0, 0 . 30. x, y → 0, 0 lim x2 x2y y2 32. f x, y x2 2xy y2 1 Does not exist z 18 The limit equals 0. z 5 x 4 x 4 y 5 5 y 34. x, y → 0, 0 x 2 lim xy 2 y2 r→0 lim r cos r 2 sin2 r2 r→0 lim r cos sin2 0 36. x, y → 0, 0 x 2 lim x 2y 2 y2 z y2 r 4 cos2 sin2 r→0 r2 lim r→0 lim r 2 cos2 sin2 0 38. f x, y, z x2 9 y2 9 40. f x, y, z xy sin z Continuous for x 2 1 t x2 f x2 1 x2 y2 y2 Continuous everywhere 42. ft g x, y f g x, y 44. f t g x, y 1 4 t x2 y2 f x2 y2 y2 4 4 1 x2 y2 y2 f g x, y Continuous for x 2 Continuous except at 0, 0 46. f x, y (a) lim x→0 x2 fx y2 x, y x f x, y lim x 2x x x x2 2y y y y y y 2 2 x 2 x→0 y2 x x 2 x2 y2 lim (b) lim f x, y y y f x, y lim x→0 lim 2x x→0 x y2 2x x2 y→0 y→0 y lim 2y y→0 lim y→0 y 2y S ection 12.3 48. f x, y (a) lim x→0 Partial Derivatives 315 yy fx f x, y 1 x, y x y y f x, y f x, y lim yy y y 1 y 2 y y 1 x 32 yy y y3 2 1 y y 12 x→0 0 y3 y 2 (b) lim y→0 lim y1 y 12 2 y→0 32 lim 31 y 2 y→0 y 12 lim y1 2 y→0 y 2 (L’Hôpital’s Rule) 3y 1 2y 50. See the definition on page 854. x2 xy ax: x, ax → 0, 0 52. x, y → 0, 0 lim y2 x2 ax x ax 2 (a) Along y lim (b) Along y 1 a a2 ,a 0 x2 : x, x 2 → 0, 0 lim x2 x2 x x2 2 x→0 lim 1 x x2 x→0 lim x2 1 a2 ax2 limit does not exist If a 0, then y 0 and the limit does not exist. (c) No, the limit does not exist. Different paths result in different limits. 54. Given that f x, y is continuous, then a > 0 such that f x, y 0< Let x f a, b f x, y a 2 x, y → a, b lim f x, y f a, b < 0, which means that for each > 0, there corresponds f a, b b 2 < . whenever y < 2, then f x, y < 0 for every point in the corresponding < f a, b ⇒ 2 ⇒ f a, b 2 < f x, y f a, b < neighborhood since f a, b 2 f a, b 3 1 f a, b < f x, y < f a, b < 0. 2 2 56. False. Let f x, y See Exercise 21. xy x2 y2 . 58. True Section 12.3 2. fy 1, 2 <0 Partial Derivatives 4. fx 1, 1 0 6. f x, y fx x, y fy x, y x2 2x 6y 3y 2 7 8. z z x z y 2y2 x y2 x 4y x 316 Chapter 12 Functions of Several Variables 1 ln xy 2 1 2x 1 2y 10. z z x z y y3 4xy2 4y2 1 12. z z x xe x xx e y xe x y 14. ex x y2 y z z x z y ln 1y 2 xy 1x 2 xy xy y ex y x y y 1 3y2 8xy z y y x2 x e y2 16. z z x z y ln x2 1 x2 x2 y2 2y y2 y2 2x 2x x2 y2 18. f x, y fx x, y fy x, y xy x2 x2 x2 y2 y2 y xy 2x x2 y2 2 y2 x xy 2y x2 y2 2 y3 x2 x3 x2 x 2y y2 2 xy 2 y2 2 20. g x, y gx x, y gy x, y ln x2 y2 1 ln x 2 2 x x 2 y2 22. f x, y f x f y 1 2x 2 1 2x 2 2x y3 y3 y3 12 1 2x 2 x2 y2 1 2y 2 x2 y2 2 3y2 y y 2 1 2x y3 12 x2 y2 3y2 2 2x y3 24. z z x z y sin 3x cos 3y 3 cos 3x cos 3y 3 sin 3x sin 3y 26. z z x z y cos x 2 y2 y2 y2 2x sin x 2 2y sin x 2 y x 28. f x, y x y 2t 2t x y 1 dt y y 2t 2t x y 1 dt 1 dt 2x 1 dt 2t x 2 dt x 2y fx x, y fy x, y 30. f x, y f x 2 2 x2 lim fx x x→0 2xy y2 x, y x x y 2 f x, y 2x xy x y2 x2 2xy y2 lim f y lim x 2 x→0 lim 2x x→0 x 2y 2x y f x, y x2 2x y y→0 y y y f x, y y y y 2 lim x2 2xy y2 y→0 lim y→0 2x 2y y 2y x S ection 12.3 1 x lim y fx x, y x y y f x, y lim x 1 x 1 y 1 y x y y x 1 x y lim y→0 Partial Derivatives 317 32. f x, y f x f y 34. h x, y hx x, y At y x→0 x→0 lim x→0 x x 1 yx 1 yx y y y x 1 y 1 y 2 lim f x, y f x, y y→0 lim x y→0 x y y x 2 x2 2x y2 36. z z x At 2 cos 2x 2 sin 2x z ,, 43 x z y sin 2x 2, 1 : hx hy x, y 2, 1 2y 2, 1 4 2 sin y 6 1 1 2 1 sin 2x y At 2, 1 : hy At z ,, 43 y sin 6 38. f x, y fx x, y arccos xy y 1 x2y2 40. f x, y fx x, y 6xy 4x2 5y2 30y3 2 4x 5y2 32 42. z x2 4y 2, y 1, 2, 1, 8 x2 4 Intersecting curve: z z x 2x z x 22 At 1, 1 , fx is undefined. fy x, y x 1 x2y2 At 1, 1 , fx 1, 1 fy x, y 30 27 10 9 32 At 2, 1, 8 : z 20 4 At 1, 1 , fy is undefined. 24x3 4x2 5y2 8 9 At 1, 1 , fy 1, 1 4 x 4 y 44. z 9x2 y 2, x 1, 1, 3, 0 9 y2 46. fx x, y fx fy 9x 2 0: 9x2 3y 2 12y, fy x, y 12y 12x 12x 3y 2 4y 4x y 2 4, you obtain Intersecting curve: z z y At 1, 3, 0 : z 40 0 ⇒ 3x 2 0 ⇒ y2 2y z y 23 6 Solving for x in the second equation, x 3 y 2 4 2 4y. 3y4 64y ⇒ y ⇒x Points: 0, 0 , 0 or y 0 or x 4 31 3 1 16 4 32 3 4 4 , 32 3 31 3 4 x 4 y 318 Chapter 12 2x y2 2y y2 Functions of Several Variables 48. fx x, y fy x, y x2 x2 1 1 0⇒x 0⇒y 0 0 50. (a) The graph is that of fx. (b) The graph is that of fy. Points: 0, 0 3xz x x y y 3z xy 3xz y2 y 3x 2 y 6xy 3x 2 5xy 5xyz 5yz 5xz 20yz 10z 2 10yz 2 58. z z x z x2 z yx z y z y2 z xy 2 2 2 2 52. w w x w y w z 54. G x, y, z 3xz 2 1 1 1 1 1 x x2 x2 x2 60. 2 y2 x y2 y y2 z y2 z z x 2z x2 2z yx z2 z2 z2 z2 32 x 3yz y 2 Gx x, y, z Gy x, y, z Gz x, y, z x x 32 3x 32 56. f x, y, z fx x, y, z fy x, y, z fz x, y, z x4 4x 3 12x 2 3x 2y 2 6xy 2 6y 2 y4 ln x 1 x x 1 x 1 x x 1 x y 2z yx y y 1 y y y 1 y 2 2z . xy 2 2 2 12xy 6x 2 y 6x 2 12xy 4y 3 12y 2 z y 2z y2 2z xy 1 y x Therefore, 62. z z x 2 z x2 2xey 2ey 3ye 2ey 2xey 2xey 2e y 3ye 3ye x x x 64. z z x 2z sin x cos x sin x 2 sin x 2y 2y 2y 2y 2y 2y 2y 66. z z x z x2 2z 2 9 9 2 x2 x x2 y x2 xy x2 y2 y2 9 y2 y2 32 x2 x 2z 9 9 9 z yx z y 2z 2 3ye 3e yx z y 2z yx z y 2z 32 x 2 cos x 4 sin x 2 sin x y x2 x2 x2 xy x2 2z y2 9 y2 y2 32 y 2 2 y2 2z y2 2z 9 9 z xy 3e x xy xy 32 Therefore, z x z y 2z yx 0 if x xy y . 0 S ection 12.3 xy x yx x x x xx x x x 2x 2 y y 3 2 Partial Derivatives 319 68. z z x 2z x2 2z y y y 3 2 2 70. xy y2 xy f x, y, z fx x, y, z x2 2x 3x 0 3 3 0 0 0 3xy 3y 4z 4yz z3 2 fy x, y, z fyy x, y, z fxy x, y, z 2y 2 y y 2y x y y 2 yx z y 2z y2 2z y2 2 x y4 x2 x y 2 y 1 2xy x y3 fyx x, y, z fyyx x, y, z fxyy x, y, z fyxy x, y, z xy Therefore, fxyy 2x x x2 y 4 fyxy fyyx 0. 2x y x zy 0. xy 2xy y3 There are no points for which zx 72. f x, y, z fx x, y, z fy x, y, z fyy x, y, z fxy x, y, z fyx x, y, z fyyx x, y, z fxyy x, y, z fyxy x, y, z y x 2z x x x x 4z x 4z x y 3 y 2z y 2z y 4z y y 3 2 74. z z x 2z sin x cos x ey 2 ey 2 ey e e y y 2 x2 z y 2z sin x sin x sin x ey e 2 e y y 2 ey 2 e y 3 y2 Therefore, 2z 2z 12z x y4 12z x y4 x 12z y4 78. z z t 2 x2 y2 0. sin x ey 2 e y sin x ey 2 e y 76. z arctan sin wct sin wx wc cos wct sin wx w 2c 2 sin wct sin wx w sin wct cos wx w 2 sin wct sin wx 2 From Exercise 53, we have z x2 2 z y2 2 x2 2xy y2 x2 2xy y2 2 0. 2 z t2 z x z x2 2 Therefore, z t2 c2 z . x2 2 320 Chapter 12 Functions of Several Variables 80. z z t z x 2z e t sin x c x c x c e t sin 1 e c t 82. If z f x, y , then to find fx you consider y constant and differentiate with respect to x. Similarly, to find fy, you consider x constant and differentiate with respect to y. 84. The plane z satisfies x y f x, y f f < 0 and > 0. x y z cos t 4 x2 1 e c2 x sin c c2 2z 4 x 2 2 z Therefore, t 4 y x . 2 86. In this case, the mixed partials are equal, fxy See Theorem 12.3. 88. f x, y (a) f x At x, y (b) f x At x, y 200x0.7y0.3 140x 0.3y0.3 fyx. 140 f x x y f x y x 0.3 1000, 500 , 60x0.7y 0.7 140 0.7 500 1000 0.3 140 1 2 0.3 113.72 60 1000, 500 , 60 1000 500 R R 10 0.7 60 2 0.7 97.47 90. V I, R VI I, R VI 0.03, 0.28 VR I, R VR 0.03, 0.28 1000 1 1 0.10 1 1I 0.10 1 1I 9 10,000 1 0.10 1 R 1 I2 10,000 1 0.10 1 1I R 11 10 14,478.99 10,000 1 0.10 1 1I R 9 0.10 1I 1000 1 0.10 1 1I R 10 9 1391.17 The rate of inflation has the greater negative influence on the growth of the investment. (See Exercise 61 in Section 12.1.) 92. A (a) 0.885t A t 22.4h 1.20th 0.544 94. U (a) Ux (b) Uy 0.885 1.20t 22.4 1.20 30 13.6 1.20 0.80 1.845 x 5x 2 xy 10x 6y 3y 2 y 0.885 1.20h A 30 , 0.80 t A h 22.4 (c) Ux 2, 3 17 and Uy 2, 3 16. The person should consume one more unit of y because the rate of decrease of satisfaction is less for y. (d) 1 −2 x 2 1 2 y z A 30 , 0.80 h (b) The humidity has a greater effect on A since its coefficient 22.4 is larger than that of t. S ection 12.4 z x 2z x2 2z Differentials 2z 321 96. (a) 1.55x 1.55 0.014y 0.014 22.15 (b) Concave downward x2 <0 (c) Concave upward y2 >0 z y 2z 0.54 The rate of increase of Medicare expenses z is declining with respect to worker’s compensation expenses x . The rate of increase of Medicare expenses z is increasing with respect to public assistance expenses y . y2 98. False Let z x 100. True y y 1. 102. f x, y x 1 t 3 dt By the Second Fundamental Theorem of Calculus, f x f y d dx d dy y 1 x y t 3 dt t 3 dt d dx 1 x 1 y t 3 dt 1 x3 1 x y 3. Section 12.4 2. z dz x2 y 2x dx y 1 x2 e 2 2x ex 2 Differentials 4. w x z 1 z 2y y 2y dx z z 2x dy 2y 2 x z y dz 2y 2 x2 dy y2 dw 6. z dz y2 y2 e e 2 z2 x2 y2 x2 y2 2 dx 2y ex y2 e 2 x2 y2 dy ex 2 y2 e x2 y2 x dx y dy 8. w dw ey cos x ey 10. ey cos x dy 2z dz w dw x 2 yz 2 2xyz 2 dx sin yz x 2z 2 z cos yz dy sin x dx 2x 2yz 12. (a) f 1, 2 f 1.05, 2.1 z (b) dz 0.11180 x x2 y2 dx y x2 0.05 y2 dy 0.11180 5 2.2361 5.5125 2.3479 14. (a) f 1, 2 e2 y cos yz dz 7.3891 1.05e2.1 8.5745 f 1.05, 2.1 z (b) dz 1.1854 ey dx e2 0.05 xey dy e2 0.1 1.1084 x dx y dy x2 y2 2 0.1 5 322 Chapter 12 1 2 Functions of Several Variables 16. (a) f 1, 2 0.5 1.05 2.1 0.5 f 1.05, 2.1 z (b) dz 0 1 dx y 1 0.05 2 18. Let z x2 1 2 x dy y2 1 0.1 4 2, y 22 1 y 2 0 y 3, x 8.9 3 9, dx 9 3 0.03, dy 22 1 0.05, dy 2 1 cos 12 9 3 0.1. Then: dz 0.03 32 2 2x 1 9 2 y 3 dx 0.1 0 3x 2 1 y 2 dy 2.03 20. Let z 1 1 sin x 2 2 y2 , x 0.95 1, dx sin 2 0.05. Then: dz 12 0.05 2x cos x 2 12 y 2 dx 0.05 2y cos x 2 0 y 2 dy sin 1.05 2 1 cos 12 24. If z 22. In general, the accuracy worsens as x and y increase. f x, y , then z dz is the propagated error, z dz and is the relative error. z z 26. V dV r 2h 2 rh dr r 2 dh π r 2dh ∆h ∆V − dV 2πrhdr ∆r r 28. S r dS dr r r2 8, h r2 20 h2 20 h2 12 r2 r2 r2 h2 2r 2 r2 12 r 2 h2 r 2 h2 dS dh dS r r2 2r 2 r2 r S 8, 20 2 12 h2 h2 r 0.1 8 h2 12 h rh r 2 h2 rh r2 h2 dr h2 dh h 0.1 0.1 0.002 0.0002 dS 10.0341 5.3671 0.12368 0.00303 S 10.0768 5.3596 0.12368 0.00303 S 0.0427 0.0075 0.683 0.286 dS h2 dr h2 h 2 0.1 0.001 0.0001 2r 2 rh dh 10 10 5 7 541.3758 S ection 12.4 C v 1 0.0817 3.71 v 2 0.1516 v1 2 C T dC Differentials 323 30. 12 0.25 T 91.4 0.25v 91.4 0.0204 T 5.81 0.0817 3.71 v Cv dv 0.1516 231 2 CT dT 0.0204 8 91.4 ± 3 0.0817 3.71 23 5.81 0.25 23 ±1 ± 2.79 ± 1.46 ± 4.25 Maximum propagated error dC C 32. x, y r ± 4.25 30.24 ± 0.14 8.5, 3.2 , dx ≤ 0.05, dy ≤ 0.05 x2 y 2 ⇒ dr x x2 y2 dx y x2 y2 dy 0.9359 dx 0.3523 dy 8.5 dx 8.52 3.22 dr ≤ 1.288 0.05 y arctan x 0.064 y x2 y x y x2 y2 Using the worst case scenario, dx d v2 r 2v dv r 2 dv v v2 dr r2 dr r 2 0.03 0.02 0.08 ≤ 0.00194 0.00515 3.2 dy 8.52 3.22 ⇒d 1 2 dx 1 1 x y x x x2 y2 2 dy dx dy 8.52 3.2 dx 3.22 8.5 8.52 3.22 dy 0.05 and dy 0.0071. 0.05, you see that 34. a da da a 8% Note: The maximum error will occur when dv and dr differ in signs. 36. (a) Using the Law of Cosines: a2 b2 330 a (b) a da 2 c2 2bc cos A 4202 2 330 420 cos 9 330 ft 420 ft 9˚ 107.3 ft. b2 4202 2b 420 cos 12 12 b 2 1 3302 2 4202 4202 12 840b cos 840 330 cos ± 1774.79 2b 840 cos 12 db 840b sin d 840 cos 6 840 330 sin 20 2 330 20 20 180 1 11512.79 2 ± 8.27 ft 324 1 R R dR1 dR2 R Chapter 12 1 R1 1 R2 Functions of Several Variables 38. R1R 2 R1 R2 R1 R2 dR 0.5 2 R dR R1 10 and R2 R dR R2 2 R1 R22 R2 2 R1 R1 2 R12 R2 2 R2 2 0.14 ohm. When R1 15, we have R 152 10 15 0.5 102 10 15 2 40. T dg dL T 2 g L dT L g 32.24 2.48 T dg g 32.09 and L 32.09 2.5 T dL L 0.15 0.02 g L g g Lg 32.09 L 2.5 0.15 32.09 0.02 0.0111 sec. When g 2.5, we have T 2.5 32.09 42. z z f x, y fx x2 x2 x, y 2x x y2 y x 2 f x, y y2 xx y and y3 f x, y 10 y y y and y3 0x 1 1 2y y yy x 2 y 2 x2 y2 2x x fx x, y As 44. z z x 2y y fy x, y 1→0 y where 1 x and 2 y. x, y → 0, 0 , f x, y fx 5x 5x fx x, y x 5x x, y 5x 3y2 2 → 0. 10y y 10y 10 fy x, y 1→0 3y2 y 3y y 2 3y y y 1 2 2 y y 3 5x 10y y3 x y where 0 and 2 3y y y 2. As x, y → 0, 0 , 5x2y , y3 0, lim f x→0 2 → 0. 46. f x, y x3 x, y x, y x, 0 x f 0, y y 0, 0 0, 0 f 0, 0 f 0, 0 lim lim 0 x 0 y 0 0 x→0 (a) fx 0, 0 fy 0, 0 0 0 (b) Along the line y Along the line x x: 0, x, y → 0, 0 lim f x, y f x, y 5x3 x → 0 2x3 lim 0. 5 . 2 lim y→0 y→0 x, y → 0, 0 lim Thus, the partial derivatives exist at 0, 0 . Thus, f is not continuous at 0, 0 . Therefore f is not differentiable at 0, 0 . (See Theorem 12.5) Section 12.5 Chain Rules for Functions of Several Variables 325 Section 12.5 2. w x dw dt x2 cos t, y x x2 y2 y2 et Chain Rules for Functions of Several Variables 4. w x sin t y x2 y2 e t ln y x cos t sin t 1 x tan t sin t cot t 1 cos t y 1 sin t cos t y dw dt x sin t yet x2 y2 cos t sin t e2t cos2 t e2t 6. w (a) cos x dw dt y, x sin x 2t sin x t2, y y 2t y dw dt 1 sin x y0 1 1 2t sin t2 2t sin t2 (b) w cos t2 1, 8. w x y z (a) xy cos z t t2 arccos t dw dt y cos z 1 t2 t t t 2t 4t3 x cos z 2t t t2 1 t2 xy sin z 1 1 t2 1 1 t3 t2 2t3 t3 4t3 (b) w t 4, dw dt t2, y 10. w (a) xyz, x dw dt 2t, z xz 2 xy e t yz 2t 2t e 2t2e t t e t t 2t 1 t t2 e t 2t e e t 3 t 2 t t2 2t 3 t e t 2 2t2e (b) w dw dt t 2 2t e e 2t3 t 6t2 2t2e t t 3 12. Distance ft 48t x2 8 x1 2 y2 y1 2 48 3 2 2 48t 1 2 2 22 f1 26 ft 48 8 22 26 326 Chapter 12 x2 , y t2, t 1 w dx x dt 2x 2t y 2t 2 2t t1 t 3t4 t d 2w dt 2 At t t 1, Functions of Several Variables 14. w x y t dw dt 16. w x y w s y3 es et 3x2y 6xy es 6xy 0 3et e2t e2s 3y2 3y2 3x2 0 3x2 et 6e2s t w dy y dt x2 y2 t4 t 1 t4 2 w t 1 When s 0 and t 1, w s 6e and w t 3e e2 1. 1 4t3 t 12 4t3 12 1 d 2w dt 2 2 12t3 4 24 12t2 t 16 3t4 1 4 4t3 2 t 68 16 1 1: 74 4.25 18. w x y w s sin 2x s s t t 3y 2 cos 2x 5 cos 2x 3y 3y 3y 3y 2 , 3 cos 2x 5 cos 5s 3 cos 2x cos 5s w s 0 and 3y t 3y t w t 0. w t 2 cos 2x cos 2x When s 0 and t 20. w (a) w r 25 5x2 25 5y2, x 5x 5x2 5y2 r cos , y cos r sin 25 5r 25 5r2 25 0 5y 5x2 5y2 r cos 5y 5x2 5y2 sin 5r cos2 5r sin2 25 5x2 5y2 w 25 5x 5x2 5y 2 r sin 5r2 sin2 (b) w w r 25 25 5r2 5r cos 5r2 sin cos 2 25 5x 5y2 5r2 ; w 0 S ection 12.5 yz ,x x w r w Chain Rules for Functions of Several Variables 327 22. w (a) 2, y r z 1 x z 1 x r 4 ,z r y 1 x y x 1 r 2 24. w z x y 2r 2 x cos yz, x cos yz 2s cos st2 s2, y t2, z s 2t xy sin yz 1 2t3 xy sin yz 2s2t2 sin st2 2 2t3 yz 0 x2 yz 2 x2 r 2 yz x 2r 2 2 3 w s xz sin yz 0 s2t2 sin st2 2t3 2s 2 2 r 2 r w t cos yz 0 2s2t s 6s2t2 xz sin yz 2t 2t sin st2 2t3 2t3 r2 r 2r2 3 2s3t sin st2 (b) w w r w r2 2 1 2r2 3 26. w w s x2 2x y2 z2, x 2y t sin s, y t sin s t cos s, z 2z t2 2st4 st2 28. cos x dy dx tan xy Fx x, y Fy x, y 5 0 sin x y sec2 xy x sec2 xy cos s 2t2 sin s cos s w t 2x sin s 2t sin2 s x x2 y2 dy dx 2t2 sin s cos s 2z 2st 4s2t3 2t 2st4 2y cos s 2t cos2 s 4s2t3 30. y2 6 0 32. F x, y, z Fx Fy Fz z x z y 2y 5 36. x z z x xz y x y Fx Fz Fy Fz yz xy Fx x, y Fy x, y y2 2xy 2xy 2xy x2 x2 x2 y2 2 2 y2 2 2y y2 x2 2y x 2 y 2 y2 2yx 4 z z z z 1 x2 y x x x z y z y 4x2y 3 z 34. F x, y, z Fx Fy Fz z x z y ex ex ex ex sin y sin y cos y cos y Fx Fz Fy Fz sin y z 0 z (i) 1 cos y z x z x (ii) 1 1 cos y z cos y y z 0 implies sec y 0 implies z. z y 1. z ex sin y z 1 ex cos y x ex cos y z 1 ex cos y z 328 Chapter 12 y2z z2 Functions of Several Variables 8 0 y2 ln y 2z x y y2 2yz 2z x y3 2y2z 2yz 40. x2 Fx w x w y w z y2 z2 5yw 2y 5y 5w 20w 5y 10w2 5w, Fz 2x 20w 2y 5y 2 2z, Fw 5y 2x 20w F x, y, z, w 5y 20w 38. x ln y (i) z x z y Fx x, y, z Fz x, y, z Fy x, y, z Fz x, y, z 2x, Fy Fx Fw Fy Fw Fz Fw x3 tx (ii) 2z 20w y3 2 42. F x, y, z, w w x w y Fx Fw Fy Fw w 1x 2 x y 1 y 12 y z 1 2x 1 y 2 0 y z 12 44. f x, y f tx, ty 3xy2 3 3 tx ty 3xy2 ty 3 t3 x3 Degree: 3 x fx x, y y3 t3f x, y 1 x 2 1 y 2y y 12 2x w z Fz Fw x2 x2 1 2y 1 z y fy x, y x 3x2 3x3 3y2 9xy2 y 3y3 6xy 3y2 z 3f x, y 46. f x, y f tx, ty Degree: 1 x fx x, y y2 tx 2 2 tx ty 2 t x2 x2 y2 tf x, y y fy x, y x x4 x2 x3 x2 2xy2 y2 3 2 x2y2 y2 3 2 y x2 x2y y2 3 2 x2 x2 y2 x2 y2 3 2 f x, y x2 x2 w s w t w x w x x s x t wy ys wy (Page 878) yt y2 48. 50. dy dx z x z y fx x, y fy x, y fx x, y, z fz x, y, z fy x, y, z fz x, y, z 52. (a) V dV dt r 2h 2rh 2 rr 2 2r dr dt h h dr dt r dh dt 2 24 36 6 12 4 624 in.2 min r2 dh dt r 2h dr dt r dh dt 12 2 36 6 12 4 4608 in.3 min (b) S dS dt S ection 12.5 Chain Rules for Functions of Several Variables 329 54. (a) V dV dt 3 3 3 3 r2 2r rR Rh R2 h dr dt r 2R h 15 dR dt r2 rR R2 dh dt 15 2 2 15 19,500 R r R R r 25 25 25 10 4 2 25 10 4 15 25 25 2 12 6,500 cm3 min R r 2 (b) S dS dt r 2 h2 R r R R r r 2 h2 h r 2 dr h2 dt R r 2 h2 R r R R r r 2 dR h2 dt R 15 15 2 dh h2 dt 25 25 15 15 25 25 25 25 15 15 15 2 102 102 102 102 4 4 25 15 10 15 12 2 15 2 25 2 102 320 2 cm2 min 56. pV T dT dt mRT 1 pV mR dp 1 V mR dt p dV dt 58. g t Let u gt and g t Now, let t f x x 60. w w x w y w x w y x 0 x x y sin y y cos y y cos y x x x sin y sin y x x f xt, yt xt, v f u ntn t n f x, y yt, then du dt 1f f v x, y . dv dt f x u f y v 1 and we have u f y y nf x, y . x, v y. Thus, 62. w y arctan , x x arctan r sin r cos , 2 r cos , y r sin for 0, w < < arctan tan x x2 w y 2 2 1 2 w x y x2 y2 w x w r 2 w y y2 , w r y2 y x2 22 1 y2 2 x 2 2 x 2 x2 y2 1 r2 1 r2 w x 2 w 0 w y 2 1 1 r2 w r 2 1 r2 1 r2 w 2 Therefore, . 330 Chapter 12 Functions of Several Variables 64. Note first that u x u y u r v x2 Thus, v r u x2 Thus, v r u r x2 x y2 1u . r x x2 y y2 1v . r y y2 cos r sin x x2 y2 x2 sin y y2 r cos r sin cos r2 r 2 sin cos r2 r sin cos 0 0 y2 v y v x cos r sin x x2 x2 y2 y y2 x2 . y y2 x2 sin x y2 r cos2 r2 r cos r 2 sin2 r2 r sin2 1 r r 2 cos2 1 r 2 sin cos Section 12.6 2. f x, y f x, y f 4, 3 u Du f 4, 3 x3 3x2i 48i v v f 4, 3 Directional Derivatives and Gradients y3, v 3y2j 27j 2 i 2 u 2 j 2 24 2 27 2 2 21 2 2 2 i 2 j 4. f x, y v f x, y f 1, 1 u Du f 1, 1 x y j 1 i y i v v f 1, 1 e i x2 y2 x j y2 j j u 1 6. g x, y g x, y g 1, 0 u Dug 1, 0 arccos xy, v y 1 j v v g 1, 0 1 i 26 u xy 2 i i 5j x 1 xy 2 8. j h x, y v h h 0, 0 j 2xe x2 y2 i 2ye x2 y2 j 0 h 0, 0 u 0 5 j 26 5 26 z2 3k 5 26 26 12. Du h 0, 0 10. f x, y, z v f f 1, 2, 1 u Du f 1, 2, 1 x2 i 2x i 2i v v y2 2j h x, y, z v xyz 2, 1, 2 yz i i v v 2j 2 i 3 xz j 2k 1 j 3 u 2 k 3 8 3 xy k 2yj 4j 2z k 2k h h 2, 1, 1 2 j 14 3 k 14 6 7 14 u Du h 2, 1, 1 1 i 14 1 u f 1, 2, h 2, 1, 1 Section 12.6 y x y 3 i 2 y x f u 1 2x 18. f x, y v f u Du f v v cos x 2 i sin x y y j yi 1 i 5 1 sin x 5 1 sin x 5 At 0, , Du f 2xey x 2 Directional Derivatives and Gradients 331 14. f x, y u f Du f 16. g x, y 1 j 2 y 2 xey 1 i 2 ey i 1y e 2 3 j 2 xey j 3y xe 2 ey 2 3x 1 u x x 3y 2x y 3y y 2 2 g j Dug x 2x y 2 i x 20. g x, y, z v g sin x 2 j 5 y y 2 sin x 5 5 sin x 5 y y yj xyez 2i yez i g v v g u 4j xez j 4i xyez k 2j 1 i 5 4 5 8k. 2 j 5 4 5 8 5 At 2, 4, 0 , u Du g 0. ln x2 2x x2 4i y j i 22. g x, y g x, y g 2, 0 24. x z z x, y z 2, 3 y 1 x2 y j 2y y e x 2i w 2j 2ey x i 2ey xj 26. x tan y tan y tan 2i z zi 4 x sec2 y 4 zj sec2 2k 7 j 53 2j 50 53 50 53 53 x sec2 y zk w x, y, z w 4, 3, \ 1 sec2 2j 2 i 53 18i 14 53 28. PQ f x, y Du f 2i 6x i f u 7j, u 2yj, f 3, 1 36 53 1 i 5 \ 30. PQ f x, y f 0, 0 Du f 2 i j, u 2 j 5 2 cos 2x cos y i 2i f u 2 5 sin 2x sin yj 25 5 332 Chapter 12 Functions of Several Variables 32. h x, y h x, y h 0, h 0, 3 y cos x y sin x 3 i 6 32 36 y yi 3 6 9 63 36 cos x 3 j 3 2 y y sin x yj 32 2 3 23 6 3 34. g x, y g x, y g 0, 5 g 0, 5 ye x2 x2i 36. e x2 j w w 1 1 1 x2 1 x2 0 0 y2 z2 3 y2 z2 xi yj zk 2xye j 1 w 0, 0, 0 w 0, 0, 0 38. w w xy2z2 y2z2i 2xyz2j i 4j 33 2xy2zk 4k w 2, 1, 1 w 2, 1, 1 For Exercises 40 – 46, f x, y 3 x 3 1 2 y and D f x, y 2 2 2 1 2 3 2 52 12 2 33 12 1 cos 3 1 sin . 2 40. (a) D (b) D2 4 f 3, 2 f 3, 2 1 3 1 3 2 2 1 2 3 42. (a) u Du f f 1 i 2 u 1 3 1 2 4j 16 4 j 5 u 1 5 j 44. f 1 i 3 1 j 2 1 2 1 2 52 12 (b) v v u Du f 3i 9 3 i 5 f 5 2 5 3 5 46. f f f 1 i 3 1 13 1 j 2 2i 3j Therefore, u f u 0. f is the direction of greatest rate of change of f. 1 13 3i 2j and Du f 3, 2 Hence, in a direction orthogonal to f, the rate of change of f is 0. Section 12.6 For Exercises 48 and 50, f x, y 2 2 1 2 3 9 x2 y2 and D f x, y 2x cos Directional Derivatives and Gradients 2y sin 2 x cos y sin . 333 48. (a) D (b) D 4 f 1, 2 2 2 2 1 2 23 50. f 1, 2 f 1, 2 f 1, 2 Therefore, u 1 2i 1 5 5 4j i 2j 3 f 1, 2 2i f 1, 2 j and u 0. Du f 1, 2 52. (a) In the direction of the vector i (b) f 1 1 y i 22x 1 i 2 1 2 j. y i 1 2 xj 1 2 1 j 2 xj 4x f 1, 2 (Same direction as in part (a).) (c) 1 f 2i the gradient. j, the direction opposite that of 54. (a) f x, y 1 ⇒ 4y 4 8y x2 1 y2 3 y2 x2 4y 2 y2 4 x2 1 (b) f f 1 3, 2 16xy i x2 y2 2 3 2 i 8 1 8x2 x2 8y2 j y2 2 y 4 3 2 y 2 2 x2 Circle: center: 0, 2 , radius: 3 1 x 1 2 −2 −1 (c) The directional derivative of f is 0 in the directions ± j. (d) 6 z −6 x −6 6 y 56. f x, y c 6, P f x, y 6 0 2x 2x f 0, 0 6 2x 0, 0 2i 3y 58. f x, y c f x, y xy f 3 1, 3 xy 3, P yi 3i 1, 3 xj j 3j 6 3y 3y 2i 3j 334 60. 3x2 Chapter 12 2y2 3x2 6xi 6i 1 2y2 Functions of Several Variables y 62. xey f x, y y 5 xey ey i i 4j 1 i 17 17 i 17 4j y xey 1j 6 y f x, y f x, y f 1, 1 f 1, 1 f 1, 1 1 4y j 4j 1 3i 13 13 3i 13 2j 2j −1 x f x, y 1 4 f 5, 0 f 5, 0 f 5, 0 2 −1 x 2 4 6 4j 64. h x, y h 5000 0.002x i 0.001x2 0.008y j i 12j 0.004y2 66. The directional derivative gives the slope of a surface at a point in an arbitrary direction u cos i sin j. h 500, 300 5h 5i 2.4j or 68. See the definition, page 887. 70. The gradient vector is normal to the level curves. See Theorem 12.12. 72. The wind speed is greatest at A. 74. T x, y dx dt xt 4 xt 3x 2 16 100 2x C1e 2t x2 2y2, P dy dt yt 3 yt 4, 3 4y C2e y0 3e 4t 4t 76. (a) 500 z x0 4e e 2t C1 C2 x 6 6 y 4t y⇒u 32 x 16 (b) T x, y T 3, 5 400e 400e x2 7 y2 xi 1 2 1 2 j 3i j There will be no change in directions perpendicular to the gradient: ± i 6j (c) The greatest increase is in the direction of the gradi1 ent: 3i 2 j 78. False Du f x, y u cos 2 > 1 when 4 i sin 4 j. 80. True Section 12.7 2. F x, y, z x2 y2 x2 Tangent Planes and Normal Lines z2 y2 25 z2 0 25 4. F x, y, z 16x2 16x2 9y2 9y2 144z 144z 0 0 Hyperbolic paraboloid Sphere, radius 5, centered at origin. S ection 12.7 6. F x, y, z F x, y, z F 3, 1, 1 n F F x2 2xi 6i y2 2yj 2j 1 6i 44 1 3i 11 10. F x, y, z F x, y, z F 2, n 1, 2 F F F x, y, z F x, y, z F ,, 36 n 3 2 F F x2 2xi 4i 3y 3j 3j z2 11 2zk 2k 2j j k 2k 11 3i 11 j k 8. F x, y, z F x, y, z F 2, 1, 8 n F F Tangent Planes and Normal Lines x3 3x2i 12i z k k k k 335 1 12i 145 145 12i 145 zex 2 z3 3z2k 12k 3j 9 12. F x, y, z F x, y, z F 2, 2, 3 y2 2 3 y 2 2xzex 12i i 2yzex k 12j 2 y2j ex 2 y2k 12j 1 4i 13 sin x cos x 3 i 2 2 10 1 10 10 10 y , 1, 2, 2 x 12k n F F 1 12i 17 k 14. y yi 3 j 2 3 i 2 3i 3i z 2 cos x k 3 j 2 3j 3j k 2k 2k yj k 16. f x, y F x, y, z Fx x, y, z Fx 1, 2, 2 2x 1 y x z y x2 2 y 2x 2 y 2x Fy x, y, z Fy 1, 2, 2 z z y 2 2 z 0 0 2 1 x 1 Fz x, y, z Fz 1, 2, 2 1 1 18. g x, y G x, y, z Gx x, y, z Gx 1, 0, 0 y z 0 y arctan , 1, 0, 0 x arctan y x z y x2 y2 Gy x, y, z Gy 1, 0, 0 1x y2 x2 x x2 y2 Gz x, y, z Gz 1, 0, 0 1 1 1 0 y x2 y2 x2 1 1 336 20. Chapter 12 f x, y F x, y, z Fx x, y, z 2 3 Functions of Several Variables 2 3x 2 3x 2 2 2 3, y, 3, y z 1, 1 Fy x, y, z 1 z y z 3y 1 2 3z 0 0 6 1, Fz x, y, z 1 x 3 y 2 3x 2x 22. z x2 2xy x2 2x 2 2y 2x 2 y2, 1, 2, 1 2xy 2y y2 z Fy x, y, z Fy 1, 2, 1 z 2y 2x z 2y 1 1 z 0 0 1 2 2x 2y Fz x, y, z Fz 1, 2, 1 1 1 F x, y, z Fx x, y, z Fx 1, 2, 1 2x 1 24. h x, y H x, y, z cos y, cos y 0 0 2 5, , 42 z Hy x, y, z 2 Hy 5, , 42 z 2 8 2 2 2 2 8z 0 0 2 4 sin y 2 2 Hz x, y, z 2 Hz 5, , 42 1 1 Hx x, y, z 2 Hx 5, , 42 2 y 2 2 y 2 4 z 4 2y 26. x2 2z2 y2, 1, 3, x2 y2 2x 2 6y 3y 3 3 x 28. x y 2z 3 , 4, 4, 2 x 1 1 1y 4 x x 2yz 3y 2 2z2 F x, y, z Fx x, y, z Fx 1, 3, 2x x 1 1 2 Fy x, y, z Fy 1, 3, 2 8z 4z 3y 2 2 4z 0 0 0 2y 6 Fz x, y, z Fz 1, 3, 2 4z 8 F x, y, z Fx x, y, z Fx 4, 4, 2 x 4 Fy x, y, z Fy 4, 4, 2 8z y y 2 8z 8z 0 2z 1 3 Fz x, y, z Fz 4, 4, 2 2y 8 16 16 S ection 12.7 30. x2 y2 z2 x2 2x 2 9, 1, 2, 2 y2 z2 9 2y 4 Fz x, y, z Fz 1, 2, 2 0, x 2y 2z 2z 4 Tangent Planes and Normal Lines 337 F x, y, z Fx x, y, z Fx 1, 2, 2 Plane: x Line: x 1 y2 Fy x, y, z Fy 1, 2, 2 2y y 2 2 2 z 2 12 2z 2 Direction numbers: 1, 2, 2 1 1 2 9 32. x2 z2 x2 2x 12 0, 5, 13, y2 z2 F x, y, z Fx x, y, z Fx 5, 13, Fy x, y, z 10 Fy 5, 13, 13, 12 12 2y 26 Fz x, y, z Fz x, y, z 2z 24 Direction numbers: 5, Plane 5x 5 13 y Line: 13 5x 12 z 13y 12 12z 0 0 x 5 5 y 13 13 z 12 12 34. xyz 10, 1, 2, 5 xyz yz 10 10 Fy x, y, z Fy 1, 2, 5 5y 2 5 z 2 38. For a sphere, the common object is the center of the sphere. For a right circular cylinder, the common object is the axis of the cylinder. G x, y, z k k k 1 1 4, i x 1 3 42 G x, y, z G 2, 4j 2 4k y 4 1 z 5 4 1, 5 4 j j y k k z 2 5 2z xz 5 Fz x, y, z Fz 1, 2, 5 5 0, 10x 5y xy 2 F x, y, z Fx x, y, z Fx 1, 2, 5 Plane: 10 x Line: x 10 Direction numbers: 10, 5, 2 1 y 2z 30 1 36. See the definition on page 897. 40. F x, y, z F x, y, z F 2, (a) F 1, 5 G x2 2xi 4i i 4 0 y2 2yj 2j j 2 1 z Direction numbers: 1, 4, (b) cos F F G G 3 21 2 42 ; not orthogonal 14 338 42. Chapter 12 F x, y, z F x, y, z F 3, 4, 5 3 i 5 x2 x x2 Functions of Several Variables y2 y2 4 j 5 i k k 1 3 17, z 13 2 i 5 34 j 5 26 k 5 z y x2 y2 j k G x, y, z G x, y, z G 3, 4, 5 5x 5i 5i 2y 2j 2j 3z 3k 3k 22 F G i j 3545 5 2 Direction numbers: 1, x 1 cos F F 3 y 4 17 G G x2 2xi 2i G i 2 1 y2 2y j 4j 5 Tangent line 13 85 2 38 z k k 8 Not orthogonal 5 76 G x, y, z G x, y, z G 1, 2, 5 k 1 6 13, 25i 2, x 25 13j 1 2k y 2 13 z 5 2 x i i y j j 6z 6k 6k 33 44. F x, y, z F x, y, z F 1, 2, 5 (a) F j 4 1 Direction numbers: 25, (b) cos F F 16 2 2 1 G G x2 0; orthogonal 46. (a) f x, y g x, y (b) 16 32 x2 2x2 y2 3x2 f x, y 2x y2 4y 5 z 6x g x, y 1 1 2 1 3y2 3y 3y 2 4y f g y2 2y2 x2 2x 4x 2x 1 1 2 4y 8y 31 42 42 3x2 3x2 12y 4y 2 2 y2 y2 6x 6x 4y 4y x 5 5 y x 2 2x x 4 To find points of intersection, let x 3y y 2 2 2 2 1. Then 42 14 2± 14 2 14 2 1 2 j. The normals to f and g at this point are 14 1 2 j and the normals are 2j 2j k and k and y f 1, 2 1 2j 14 2 j, g 1, k, which are orthogonal. Similarly, f 1, 2 14 2 j and g 1, 1 2 j k, which are also orthogonal. (c) No, showing that the surfaces are orthogonal at 2 points does not imply that they are orthogonal at every point of intersection. S ection 12.7 48. F x, y, z F 2yi 2xy 2xj 4i z3, 2, 2, 2 3z2 4j k 12k 12 176 3 11 11 50. F x, y, z F x, y, z F 2, 1, 3 cos Tangent Planes and Normal Lines x2 2xi 4i y2 5, 2, 1, 3 2yj 2j 0 339 F 2, 2, 2 cos arccos F 2, 2, 2 k F 2, 2, 2 3 11 11 3x2 6x 0, x 0, y 2 31 4 1 2 F 2, 1, 3 k F 2, 1, 3 arccos 0 90 25.24 52. F x, y, z F x, y, z 6x 4y z 1 2, 2y2 3i 3x 4y 4y 4j z k 5 30 25 z 3 4 3 12 2 1 1 2 −3 −2 −3 3 1 2 4 1 5 31 4 x 3 3 y 1, 54. T x, y, z dx dt xt x0 x 3 3t C1 3t 100 3x dy dt y z2, 2, 2, 5 1 t C2 t 2 C2 2 dz dt zt z0 z 5e 2z C3e C3 2t 2t 56. F x, y, z Fx x, y, z Fy x, y, z Fz x, y, z Plane: 2x0 x a2 x0x a2 x2 a2 2x a2 2y b2 2z c2 x0 y0 y b2 y2 b2 z2 c2 1 C1 2 2 yt y0 y 5 2y0 y b2 z0z c2 x02 a2 y0 y02 b2 2z0 z c2 z02 c2 z0 1 0 58. z F x, y, z Fx x, y, z Fy x, y, z Fx x, y, z xf xf f xf 1 y x y x y x z xf y x 1 x y x f y x y x2 f y x yy f xx Tangent plane at x0, y0, z0 : f f y0 x0 y0 y0 f x0 x0 x x0 f y0 x0 y0 x0 y0 y0 f x0 x0 y0 f y0 x0 x yf f Therefore, the plane passes through the origin x, y, z x0 y0 x0 y0 x0 f y0 y x0 y0 f y0 x0 x y0 z f z x0 f y0 y x0 z0 y0 x0 z 0 0 0 y0 y0 f x0 x0 0, 0, 0 . 340 Chapter 12 cos x sin x cos x f 0, 0 f 0, 0 1 (c) If x If y (d) x 0 0 0.2 0.2 1 y Functions of Several Variables 60. f x, y fx x, y fxx x, y y y, fy x, y fyy x, y fx 0, 0 x fx 0, 0 x xy 1 1 1 2 sin x cos x 1 1 2 fxx y y, fxy x, y cos x y (a) P1 x, y (b) P2 x, y fy 0, 0 y fy 0,0 y y2 0, 0 x2 fxy 0, 0 xy 1 2 fyy 0, 0 y2 12 2x 0, P2 0, y 0, P2 x, 0 y 0 0.1 0.1 0.5 0.5 f x, y 1 12 2 y . This is the second–degree Taylor polynomial for cos y. 12 2 x . This is the second–degree Taylor polynomial for cos x. P1 x, y 1 1 1 1 1 P2 x, y 1 0.9950 (e) 5 z 0.9950 0.9553 0.7648 0.0707 5 0.9950 0.7550 0.1250 x 5 y 62. Given z f x, y , then: f x, y fx x0, y0 i z 0 fy x0, y0 j k k 1 fx x0, y0 fx x0, y0 2 F x, y, z F x0, y0, z0 cos k F x0, y0, z0 F x0, y0, z0 fy x0, y0 1 fy x0, y0 2 1 1 2 2 2 Section 12.8 2. g x, y 9 x Extrema of Functions of Two Variables 3 2 y 2, 9 2 2 ≤9 (3, − 2, 9) 8 z Relative maximum: 3, gx gy 2x 2y 3 2 0⇒x 0⇒y 3 2 6 4 2 1 x 6 y 4. f x, y 25 x 2 2 y2 ≤ 5 (2, 0 , 5) 5 z Relative maximum: 2, 0, 5 Check: fx fy fxx 25 25 25 x x y x 2 2 2 2 2 y2 y2 y 232 0⇒x 0⇒y , fyy 2 x 5 5 y 0 25 25 x x 2 2 2 2 25 y2 x 22 y2 32 , fxy 25 yx 2 x 22 y2 32 At the critical point 2, 0 , fxx < 0 and fxx fyy fxy 2 > 0. Therefore, 2, 0, 5 is a relative maximum. S ection 12.8 6. f x, y x2 y2 4x 8y 11 x 2 2 Extrema of Functions of Two Variables z 8 6 341 y 4 2 9≤9 Relative maximum: 2, 4 , 9 Check: fx fy fxx 2x 2y 2, fyy 4 8 0⇒x 0⇒y 2, fxy 0 fxy 2 (2, 4 , 9) 2 4 6 x 4 2 8 y At the critical point 2, 4 , fxx < 0 and fxx fyy 8. f x, y fx fy fxx 2x 10y 2, fyy x2 10 30 5y2 10x 30y 5, y 0 3 , fxx < 0 and fxx fyy 62 3 > 0. Therefore, 2, 4, 9 is a relative maximum. 0 x 0 10, fxy At the critical point 5, Therefore, 5, 10. f x, y fx fy fxx 2x 6x x2 6y 20y 2 f xy > 0. 3, 8 is a relative maximum. 6xy 0 4 20, fxy 0 6 6, 2 , fxx > 0 and fxx fyy 3x 4y 1 2. 10y2 4y 4 Solving simultaneously yields x 6 and y 2. 2, fyy At the critical point 12. f x, y fx fy fxx 6x 4y 3x2 3 4 2y2 fxy 2 > 0. Therefore, 14. h x, y hx 6, 2, 0 is a relative minimum. x2 2x y2 2y y2 y2 23 13 5 2 0 x 0, y 0 0 0 when x 0 when y 4, fxy 0 1. hy 3 x2 3 x2 6, fyy 23 At the critical point 1 , 1 , fxx < 0 and 2 fxx fyy fxy 2 > 0. Therefore, 1 , 1, 341 is a relative 2 maximum. 16. f x, y x y 2 Since h x, y ≥ 2 for all x, y , 0, 0, 2 is a relative minimum. 18. f x, y y3 3yx2 3y2 3, ± 3x2 3, 1 1, 3 Since f x, y ≥ 2 for all x, y , the relative minima of f consist of all points x, y satisfying x y 0. Relative maximum: 0, 0, 1 Saddle points: 0, 2, z 40 20 3 x 3 y 20. z exy 100 z Saddle point: 0, 0, 1 x 3 3 y 342 Chapter 12 120x y x Functions of Several Variables 120y 2x 2y 0 0 2, gxy Solving simultaneously yields x 40 and y 40. 22. g x, y gx gy gxx 120 120 xy x2 y2 2, gyy 1 gxy 2 At the critical point 40, 40 , gxx < 0 and gxx gyy 24. g x, y gx gy gxx y x x 0 and y 0 > 0. Therefore, 40, 40, 4800 is a relative maximum. xy 0, gyy 0, gxy 1 gxy 2 At the critical point 0, 0 , gxx gyy 14 x 2 < 0. Therefore, 0, 0, 0 is a saddle point. 26. f x, y fx fy fxx 2y 2x 2xy y2 1 2x3 Solving by substitution yields 3 critical points: 2y3 0, 0 , 1, 1 , 1, 1 6x2, fyy 6y2, fxy fxy fxy 2 2 2 At 0, 0 , fxx fyy At 1, 1 , fxx fyy At 1, 1 , fxx fyy 1 2 2x3 2x y 4x 4y4 4x3y 4 2 < 0 ⇒ 0, 0, 1 saddle point. > 0 and fxx < 0 ⇒ 1, 1, 2 relative maximum. fxy 2 > 0 and fxx < 0 ⇒ 1, 1, 2 relative maximum. 28. f x, y fx fy fxx fyy fxy x2 2xy2 2y 3 y2 e1 3x e1 ye 22 1 x2 y2 x2 x2 y2 y2 0 0 2y2 Solving yields the critical points 0, 0 , 0, ± 2 6 ,± ,0 . 2 2 4x y 4x2y2 12x 2x2 2 3 e1 x2 x2 y2 y2 8y2 x2 1 e1 y2 4xy3 2xy e1 fxy 2 < 0. Therefore, 0, 0, e 2 is a saddle point. At the critical points 0, ± 2 2 , At the critical point 0, 0 , fxx fyy fxx < 0 and fxx fyy fxy 2 > 0. Therefore, 0, ± 2 2, e are relative maxima. At the critical points ± 6 2, 0 , fxx > 0 2 and fxx fyy fxy > 0. Therefore, ± 6 2, 0, e e are relative minima. x2 x2 y2 2 ≥ 0. z y2 z 30. z 0 if x2 y2 0. x ,x 0. x 5 2 Relative minima at all points x, x and x, 5 y 32. fxx < 0 and fxx fyy fxy 2 3 8 22 > 0 34. fxx > 0 and fxx fyy fxy 2 25 8 102 > 0 f has a relative maximum at x0, y0 . 36. See Theorem 12.17. f has a relative minimum at x0, y0 . S ection 12.8 38. 4 3 z Extrema of Functions of Two Variables z 4 343 Extrema at all x, y 40. Relative maximum (2, 1, 4) 2 4 x 3 4 y 4 x 3 4 y 42. A and B are relative extrema. C and D are saddle points. 44. d fxx fyy fxy2 < 0 if fxx and fyy have opposite signs. Hence, a, b, f a, b is a saddle point. For example, consider f x, y x2 y2 and a, b 0, 0 . 46. f x, y fx fy fxx At 2, 48. f x, y fx fy fxx At 1, x x x 3x2 3y 2 6x x3 y3 18y 6x2 12 27 6y fxy 1 2 2 9y2 0 0 12x 27y 19 2 and y 3. 12x Solving yields x 12, fyy 3 , fxx fyy x 18, fxy 0 2, 0 is a saddle point. 0 and the test fails. 1, 2 2 2 y 2 2 2 ≥0 x1 12 y y2 12 y y 1 2 0 Solving yields x 1 and y 2. 2 0 fyy x x 1 2 2 y 2 2 3 2, 1 y 2 2 2 3 2, fxy x x 1 2 1y y 2 2 232 2 , fxx fyy fxy 2 is undefined and the test fails. 2, 0 Absolute minimum: 1, 50. f x, y fx fy fxx 3 x2 x2 4x y2 y2 13 23 ≥0 4y 3 x2 y2 4 x2 9 x2 fx and fy are undefined at x 13 0, y 0. The critical point is 0, 0 . 3y2 ,f y2 4 3 yy 4 3x2 y2 ,f 9 x2 y2 4 3 xy 9 x2 8xy y2 43 At 0, 0 , fxx fyy fxy 2 is undefined and the test fails. Absolute minimum: 0, 0, 0 52. f x, y, z fx fy fz 2x y 2x2 y 2x y 4 1 1 xy 2 1z 2 2 2 2 2 2 0 0 0 2 ≤4 Solving yields the critical points 0, a, b , c, 1, d , e, f, These points are all absolute maxima. 2. z z 1z 2 344 Chapter 12 2x y y x fx fx fx 2 2x f x, y f x, y f x, y y 2 Functions of Several Variables y 54. f x, y fx fy 4 2x 0 ⇒ 2x 0 ⇒ 2x 2x 2x 2x x y y 3 (1, 2) 2 On the line y 1, 0 ≤ x ≤ 1, 1 1 2 2 x 1 2 1 5 2 2 1 y = 2x (0, 1) 1 and the maximum is 1, the minimum is 0. On the line y x x 1 4 2 1 2 x 2x 1, 0 ≤ x ≤ 2, 4, 1 ≤ x ≤ 2, (2, 0) x 2 3 and the maximum is 16, the minimum is 0. On the line y 2x 4 2 4x 2 and the maximum is 16, the minimum is 0. Absolute maximum: 16 at 2, 0 Absolute minimum: 0 at 1, 2 and along the line y 56. f x, y fx fy 2 2y f x, y f x, y 2x 2y 2x 2xy y2 1 x⇒x 2x 2x x2 1 1. −1 x 1 2x. y 0 ⇒y 0⇒y 1, fx x2, 2x fx 2x 1 f 1, 1 1 (−1, 1) 2 On the line y On the curve y 1 ≤ x ≤ 1, 1≤x≤1 x2 2 (1, 1) x4 11 16 . 2x3 2x and the maximum is 1, the minimum is Absolute maximum: 1 at 1, 1 and on y Absolute minimum: x2 2y 2y x2 1, x2 x2 2, 2, 1, 2x 2x 11 16 1 11 2, 4 0.6875 at 58. f x, y fx fy f x, f Along y f Along x Along x 2x 2x x 2xy 0 0 y y2, R x x2 2x 2x 0 x, y : x ≤ 2, y ≤ 1 2 y 2x2 1, f 1, f Along y 2 ≤ x ≤ 2, 2 2 4 4 1 0⇒x 0⇒x 4y 4y y2, f 1, f 1, f 2y 2y 2, 4 4 2, 1 1 0. 0. x 0, 1 ≤ x ≤ 1. 1, f 9, f 1, 1, 1 1 0, f 2, 1 0, f 2, 1 9. 1. 2 ≤ x ≤ 2, 1 ≤ y ≤ 1, f 1 ≤ y ≤ 1, f 2, −1 x 1 −2 y2, f Thus, the maxima are f x2 4y 0 5 4x 9 and f 2, 1 9, and the minima are f x, 60. f x, y fx fy f 0, 0 Along y Along x Along y 2x 4xy 0 x 5, R y 0 x, y : 0 ≤ x ≤ 4, 0 ≤ y ≤ x 4 3 2 y 0, 0 ≤ x ≤ 4, f 4, 0 ≤ y ≤ 2, f x, 0 ≤ x ≤ 4, f x2 16 x2 5 and f 4, 0 16y 4x3 2 21. 16 2x 0 and f 4, 2 6x1 2 1 5, f 5, f 11. 1 2 3 4 x 0 on 0, 4 . 11. Thus, the maximum is f 4, 0 21 and the minimum is f 4, 2 S ection 12.9 4xy 1 y2 2 Applications of Extrema of Functions of Two Variables 345 62. f x, y fx fy For x For x For x2 x2 2 1 ,R 0⇒x 0⇒y x, y : x ≥ 0, y ≥ 0, x2 1 or y 1 or x 0. 0 0 y2 ≤ 1 4 1 x2 y y 1 x2 1 4 1 y2 x x 1 y2 1 2 2 0, y 0, also, and f 0, 0 1 and y y2 1, the point 1, 1 is outside R. f x, f 1 x2 4x 1 2 x2 x2 , and the maximum occurs at x x4 2 ,y 2 2 . 2 1, f x, y 8 9 Absolute maximum is 2 2 , . 2 2 f 0, 0 . In fact, f 0, y f x, 0 0 The absolute minimum is 0 y 1 R x 1 64. False Let f x, y x4 2x2 y2. 1 Relative minima: ± 1, 0, Saddle point: 0, 0, 0 Section 12.9 Applications of Extrema of Functions of Two Variables 4. A point on the paraboloid is given by x, y, x2 y2 . The square of the distance from 5, 0, 0 to a point on the paraboloid is given by S Sx Sy x 2x 2y 5 2 2. A point on the plane is given by x, y, 12 2x 3y . The square of the distance from 1, 2, 3 to a point on the plane is given by S Sx Sy x 2x 2y 1 2 y 29 29 2 2 9 3y 3y 2x 2 3. 3y 2 y2 4x x2 y2 x2 y2 0. y2 2 1 2 2x 2x 5 4y x2 0 From the equations Sx system 5x 6x 6y 10y 19 29. 0 and Sy 0, we obtain the From the equations Sx system 2x3 2xy2 2y 3 0 and Sy 5 y 0 0. 0, we obtain the x 2x y 2 Solving simultaneously, we have x and the distance is 16 14 2 16 14 , y 31 14 , z 43 14 Solving as in Exercise 3, we have x z 1.525 and the distance is 1.235 5 2 1.235, y 0, 1 31 14 2 2 43 14 2 3 1 . 14 1.525 2 4.06. 346 Chapter 12 y xy2z 32y2 64xy z Functions of Several Variables 32, z 32xy2 2xy2 2x2y 32 x2y2 y3 3xy2 x xy3 y2 32 y 64x 2x y 2x2 0 3xy 32 0. 2x y. Therefore, 8. Let x, y, and z be the numbers and let S Since x y z 1, we have S Sx Sy x2 2x 2y y2 21 21 1 x x x y y y 2 6. Since x P Px Py x2 y2 z2. 0 2x 0x 1 3, y 2y 1 3, 1 1. and z 1 3. Ignoring the solution y into Py 0, we have 64x 2x2 3x 32 4x x Therefore, x 8, y 0 and substituting y 2x 8 0 0. 8. Solving simultaneously yields x y 16, and z 10. Let x, y, and z be the length, width, and height, respectively. Then C0 The volume is given by V Vx Vy xyz C0 xy 1.5x2y2 2x y 6xy 2 1.5xy 2yz 2xz and z C0 1.5xy . 2x y y2 2C0 3x2 4x y x2 2C0 3y2 4x y 6xy 2 . 0, we note by the symmetry of the equations that y 1 3 z2 1 3 1 4 x. Substituting y x In solving the system Vx into Vx 0 yields x2 2C0 9x2 16x2 0 and Vy 0, 2C0 9x2, x 2C0 , y 2C0 , and z 2C0 . r2 x2 y2 . 12. Consider the sphere given by x2 Then the volume is given by V Vx Vy 2x 2y 2 r 2 8 xy 8 xy r2 r2 x x2 y x2 x2 y2 y2 y2 y2 r 2 and let a vertex of the rectangular box be x, y, x2 y2 y2 y2 8y r2 r2 x2 8x x2 y2 y2 r2 r2 2x2 x2 y2 2y2 0 0. 8xy r 2 x2 x2 y r2 x r2 Solving the system 2x2 x2 y2 2y2 r2 r2 y z r 3. yields the solution x 14. Let x, y, and z be the length, width, and height, respectively. Then the sum of the two perimeters of the two cross sections is given by 2x 2z 2y 2z 108 or x 54 y 2z. The volume is given by V Vy Vz xyz 54z 54y 54yz 2yz y2 y2z 2z2 4yz 2z 2yz2 z 54 y 54 2y y 2z 4z 4z 0. 54, we obtain the solution 0 Solving the system 2y x 18 inches, y 54 and y 18 inches, and z 9 inches. S ection 12.9 Applications of Extrema of Functions of Two Variables 347 16. A 1 30 2 30x sin 2x 30 2x2 sin 2x 2x cos x sin x2 sin cos 2x sin cos x2 2 cos2 2x x cos 0 1 0 2x x 15 . A x A 30 sin 30 cos A x A 2x x 4x sin 2x2 cos From From 0 we have 15 0 we obtain 15 30 2x 2x2 15 2x 0 ⇒ cos 30x 15 x x2 2 15 2x x 2 2x 15 15 3x2 2 1 2 0 0 0 10 2x 2x x2 30x x Then cos 1 ⇒ 2 60 . 18. P p, q, r p q r 2pq 2pr 2qr. 1 q 2pq 2p2 2p p 2q 1 2q 2q2 2 4q q. p 2pq q 2q2 20. R R p1 R p2 3p1 515p1 515 805 1.5p2 p2 805p2 1.5p2 1.5p1 515 805 1.5p1 p2 3p1 p2 0 0 1.5p12 p22 1 implies that r 2pq 2pq 2pq 2p 1 2p 2p 4p; P q 1 1 q 2 6 9 1 and 3 1 9 2 . 3 0 y 0 24 2 2 P p, q p 2p2 2q P q 1.5p1 P p Solving q p 2q P p 2p 2q 2 Solving this system yields p1 $2296.67, p2 $4250. 0 gives and hence p P 11 , 33 2 1 3 2 1 3 2 1 9 2 1 9 22. S d1 d2 d3 y y 2 0 2 2 0 2 2 y 2 2 0 2 2 y 2 2 dS dy 1 2y 2 4 y2 0 when y 2 23 3 23 3 6 3 23 . 3 0.845. The sum of the distance is minimized when y 348 Chapter 12 x 4 2 Functions of Several Variables y2 x 1 2 24. (a) S y 6 2 x 12 2 y 2 2 30 z The surface appears to have a minimum near x, y (b) Sx Sy x x x 4 y 4 2 1, 5 . x 12 2 2 x 4 4 2 x y2 y2 x x 1 2 1 y 6 6 2 x x 12 y 12 y 2 y 2 2 −2 2 2 4 −4 y6 12 y 2 2 2 6 8 y (c) Let x1, y1 S 1, 5 Direction (d) t (e) t t 0.94 x2 3.56, 1.04, 1, 5 . Then 0.258i 0.03j 6.6 1.24 y2 x3 x4 1.24, 1.23, 5.03 y3 y4 5.06, 5.06 1.2335, 5.0694 Note: Minimum occurs at x, y (f ) S x, y points in the direction that S decreases most rapidly. 26. See the last paragraph on page 915 and Theorem 12.18. 1 10 1 5 2 28. (a) x 3 1 1 3 xi 46 4 20 3 x 10 x 3 1 2 3 4 4 5 6 xi a 28 0 04 02 1 y 0 1 1 2 yi 4 3 ,b 10 xy 0 1 1 6 xiyi 1 4 4 6 3 0 10 x2 9 1 1 9 xi2 20 (b) S 0 7 10 2 1 13 10 2 1 19 10 2 2 a y 1, 30. (a) y 0 0 0 1 1 2 2 2 yi 28 8 28 2 2 xy 0 0 0 3 4 8 10 12 8 72 144 xiyi 1 ,b 2 2 x2 9 1 4 9 16 16 25 36 37 1 8 8 2 xi2 1 28 2 3 4 116 3 ,y 4 2 8 37 8 116 3 4 0 1 x 2 2 3 4 5 4 2 (b) S 1 4 0 1 4 0 1 5 4 1 2 7 4 2 2 9 4 2 2 3 2 S ection 12.9 32. 1, 0 , 3, 3 , 5, 6 xi xiyi a b y 9, 39, 99 92 3 9 2 3 2 yi xi2 36 24 9 6 9, 35 3 2 3 2 Applications of Extrema of Functions of Two Variables 34. 6, 4 , 1, 2 , 3, 3 , 8, 6 , 11, 8 , 13, 8 ; n xi xi yi a b y 9 349 6 42 275 yi xi2 42 31 42 2 29 42 53 425 318 31 400 29 53 425 318 0.5472 1.3365 3 39 3 35 1 9 3 3 x 2 7 6 275 6 400 1 31 6 29 x 53 −1 −1 6 −1 −1 14 36. (a) 1.00, 450 , 1.25, 375 , 1.50, 330 xi xiyi a b y 3.75, yi 1,155, xi2 4.8125, 38. (a) y 1.8311x 47.1067 (b) For each 1 point increase in the percent x , y increases by about 1.83 (slope of line). 1,413.75 3.75 1,155 3.75 2 240 3.75 685 240 1.40 685 349. 685 240 3 1,413.75 3 4.8125 1 1,155 3 240x (b) When x n 1.40, y 40. S a, b i 1 axi n b xi2 yi n 2 n Sa a, b Sb a, b Saa a, b Sbb a, b Sab a, b 2a i n 1 2b i 1 xi n 2 i 1 xiyi 2a i n 1 xi xi2 1 2nb 2 i 1 yi 2 i 2n n 2 i xi 1 Saa a, b > 0 as long as xi d SaaSbb Sab2 4n i 0 for all i. (Note: If xi n n 2 0 for all i, then x n n 2 0 is the least squares regression line.) ≥ 0 since n i n xi2 1 4 i 1 xi 4n i xi2 1 i xi 1 xi2 ≥ 1 i n 1 2 xi . As long as d 0, the given values for a and b yield a minimum. 350 42. Chapter 12 4, 5 , xi yi xi2 xi3 xi4 xiyi xi2yi 544a a 0 19 40 0 544 12 160 40c 5 24 , Functions of Several Variables 44. 0, 10 , 1, 9 , 2, 6 , 3, 0 8 2, 6 , 2, 6 , 4, 2 (−2, 6) (− 4, 5) −9 xi (2, 6) (4, 2) 9 6 (0, 10) 11 yi xi2 xi3 xi4 xiyi xi2yi 25 14 −9 (1, 9) (2, 6) (3, 0) 9 −1 −4 36 98 21 33 36b 14b 6b 14c 6c 4c 9 20 , 160, 40b 3 10 , 12, 40a 41 6, 4c 3 10 19 x 41 6 98a 36a 14a a 33 21 25 c 199 20 , b c y 52 24 x 5 4, b y 52 4x 9 20 x 199 20 46. (a) y (b) y (c) 0.078x 2.96 0.07229x 2.9886 48. (a) 1 y y ax b 0.0029x 0.1640 0.0001429x2 7 1 0.0029x 50 0.1640 (b) −5 0 45 0 0 60 (d) For the linear model, x 50 gives y 6.86 billion. 6.96 billion. For the quadratic model, x 50 gives y (c) No. For x 60, y 100. Note that there is a vertical asymptote at x 56.6. As you extrapolate into the future, the quadratic model increases more rapidly. Section 12.10 2. Maximize f x, y Constraint: 2x f yi xj y x 2x y g 2i 2 j y Lagrange Multipliers xy. 4 4. Minimize f x, y Constraint: 2x f 2x i g 2y j 2x 2y 2i 2 4 4j ⇒x ⇒y 5 ⇒ 10 2 5 1 2, x2 4y 5 y 2. 4⇒4 4 1, x 1, y 2 f 1 2, 2x 4y x 1 2, y 1 f 1, 2 2 1 5 4 S ection 12.10 6. Maximize f x, y Constraint: 2y f 2x i 2x If x If 2y f 2, 1 g 2y j 2x 2x i ⇒x 2j 0 or 0 and f 0, 0 1 0. 1 2 2 1 2⇒y 1 ⇒ x2 2⇒x 2. x2y x2 ⇒ 6 ⇒ x2 3x 2 x3 x f 3 Lagrange Multipliers y 10. 351 x2 x2 0 y 2. 8. Minimize f x, y Constraint: x2y f 3i 3 j g 2 xy i 2 xy ⇒ 6 3x x2 j 3 2xy 1 x2 6 4 3 0, then y 1, 3x2 2 xy ⇒ y x 3x 2 0 1 Maximum. 4, y 33 4 2 4, 33 4 2 9 3 4 20 2 2x 32 2x 32 4, y 8 y. 10. Note: f x, y minimum. Minimize g x, y Constraint: 2x 2x 2y 2x 4y 2 4 y x2 x2 4y 2x y 2 is minimum when g x, y is y 2. 15 12. Minimize f x, y Constraint: xy 2 1 xy y x y 32 ⇒ 2x2 x 15 ⇒ 10 x x 15 3 ,y 2 35 2 e 1 ⇒ x2 2 2 e1 e 8 x y 4. f 4, 8 3 16 f 3 ,3 2 g 3 ,3 2 14. Maximize or minimize f x, y Constraint: x 2 y2 ≤ 1 y2 2x 2y ± Case 1: On the circle x 2 y 4e x 4e x2 y2 xy 4 xy 4 y2 1⇒x 2 , 2 Maxima: f ± Minima: f ± 2 2 1.1331 0.8825 2 2 ,± 2 2 xy 4 xy 4 18 Case 2: Inside the circle fx fy fxx y2 e 16 y 4e x 4e xy 4 0 0 ⇒x xy 4 y e xy 0 1 xy 16 1 4 , fyy x2 e 16 fxy 2 , fxy At 0, 0 , fxx fyy Saddle point: f 0, 0 < 0. 1 2 , 2 2 and a minimum of e 2 18 Combining the two cases, we have a maximum of e1 8 at ± at ± 2 2 ,± . 2 2 352 Chapter 12 Functions of Several Variables 16. Maximize f x, y, z Constraint: x yz xz xy x y z y z x yz. 6 18. Minimize x2 Constraint: x 2x 2y 10 14 xy 1 2 10x y x y 8 10 12 y2 10 14y 70 x y z y z 2 12 12 1 2 10 14 6⇒x 8 x y 14 4 f 2, 2, 2 8⇒ 5. Then x f 3, 5 3, y 9 30 25 70 70 4 20. Minimize f x, y, z Constraints: x x f 2x i 2x 2y 2z x x 2x x 2z y 2x 2 6 ⇒z 12 ⇒ y 2 12 6, z x 0 72 g 2y j h 2z k 2z y x2 6 12 y2 z 2. 22. Maximize f x, y, z Constraints: x2 x f g xz j yz h xyk 2x z 2 x yz. 5 0 2y i 2k i j yz i 2x i 2z k i 2j 2y 6 2 12 3 x 2 x z xz 3 x ⇒ 9 x 2 27 ⇒ x 6 x 2 x2 x xy z2 2y yz x5 2 x5 2x 5 x2 2z 5 0 2⇒ ⇒ ⇒z ⇒y xy 2z x3 25 x2 x2 x3 25 x3 3x 3 0 or x 10 , 3 5 5 3 10x xz 2 x 2 xy 2z xy 2 5 x2 2x f 6, 6, 0 x5 2 x2 x2 x2 0 x x 3x 2 10 ,y 3 10 1 2 10 ,z 3 5 3 f 10 1 , 32 5 15 9 Note: f 0, 0, 0 does not yield a maximum. S ection 12.10 24. Minimize the square of the distance f x, y 2x 2y x 10 4 2 Lagrange Multipliers 4 2 353 x2 y 5 2 x 10 2 subject to the constraint x 10 y2 4. 2x 2y y2 4 x x 4 y y 8x 10 ⇒y 25 2 x 4 29 2 x 4 4 ⇒ x2 16 50x 58x 100 112 4 0 Using a graphing utility, we obtain x x 58 ± 582 4 29 4 112 2 29 4 3.2572 and x 58 ± 2 29 29 2 41 29 29 4.7428 or, by the Quadratic Formula, 4± and 4 29 . 29 y 10 29 29 1.8570. Using the smaller value, we have x The point on the circle is 4 1 29 10 29 , 29 29 16 1 29 29 2 and the desired distance is d 10 29 29 2 10 8.77. The larger x-value does not yield a minimum. 26. Minimize the square of the distance f x, y, z x 4 2 y2 x2 z2 y2 x z 2x 4 2y 0, z 2 2x 2y z 0. z subject to the constraints 28. Maximize f x, y, z x2 y 2 z2 0 and x 2z 4. 0 0 2x 2y 2z 0 4⇒x 4 2z 3z2 3z 2 subject to the constraint 2x 2y 2z x2 y2 z 0, x 4 y x2 y2 x x2 y2 y z ⇒y 2 0 1 x2 y2 x z2 2z 4 02 16z 4z 2z z2 16 4 z 0 0 0 4 3 2, y The point on the plane is 2, 0, 2 and the desired distance is d 2 4 2 02 22 2 2. or z 4 The maximum value of f occurs when z 4, 0, 4 . of 30. See explanation at the bottom of page 922. 32. Maximize V x, y, z 1.5x y 2 xz 2yz yz xz xy 1.5xy 1.5y 2z 1.5x 2z 2x 2y 2xz 2yz x yz subject to the constraint C. x y and z 3 x 4 32 x 2 32 x 2 x Volume is maximum when x y 2C 3 and z 2C . 4 C Dimensions: r 2C 3 3 4 at the point 34. Minimize A , r constraint r 2h 2h 4r 2r 2 rh V0. 2 rh r2 V0 V0 2 h 2 r 2 subject to the 2r C ⇒ 1.5x 2 r 2h V0 ⇒ 2 r 3 and h 2 3 V0 2 354 Chapter 12 Functions of Several Variables x yz subject to the constraint (b) Maximize P n 36. (a) Maximize P x, y, z x yz xz xy x y z x y y z S. x1x2x3 . . . xn subject to the constraint S. xi i 1 z S 3 n S⇒x x2x3 . . . xn x1x3 . . . xn x1x2 . . . xn x1x2x3 . . . xn xi i 1 1 x1 x2 x3 ... xn y z Therefore, xyz ≤ xyz ≤ 3 S 3 S3 27 S 3 x S 3 S , x, y, z > 0 3 S ⇒ x1 x2 x3 ... xn S n Therefore, x1x2x3 . . . xn ≤ x1x2x3 . . . xn ≤ S n S n S n x1 x2 x3 n ... xn . n S n S ... S ,x ≥ 0 n ni xyz ≤ xyz ≤ 3 y 3 z . n x1x2x3 . . . xn ≤ x1x2x3 . . . xn ≤ n 38. Case 1: Minimize P l, h 1 h l 2h 2h l 4 ⇒ l l subject to the constraint lh 2 l2 8 A. 2 2 l 2 ,1 l 2 2h l 2 h Case 2: Minimize A l, h h l 4 l h 40. Maximize T x, y, z constraints x2 y 2 2x 2y 0 If y 2x 2y 2z 0, then z 1 and 0 and y 100 z2 50 2 2 lh l2 8 subject to the constraint 2h l l 2 P. l 2 ⇒ 2h 42. Maximize P x, y Constraint: 48x 40x 0.6y0.6 l ,h 2 l 4 l 2 l 4 l or l 2 100 x2 y 2 subject to the z 2 50 and x z 0. 100x0.4y0.6 36y 48 36 ⇒ ⇒ 100,000. y x x y y x 0.6 48 40 36 60 y x 0.4 0.4 60x0.4y 0, z 50. 50 2x 2 100 150 50 and x 50 4 z 50 2. 48x P 0. Thus, x T 0, If y T 0.4 0.6 48 40 60 36 50, 0 x2 0, then 50 , 0, 2 112.5 36y 2x 100,000 ⇒ x $126,309.71. y 2 ⇒ y 2x x 2500 5000 ,y 3 3 Therefore, the maximum temperature is 150. 2500 5000 , 3 3 R eview Exercises for Chapter 12 44. Minimize C x, y 48 36 60x 0.4y0.4 355 48x ⇒ ⇒ 36y subject to the constraint 100x0.6y0.4 y x x y y x 0.4 20,000. 48 60 36 40 y x 0.6 0.6 40x0.6y 0.6 0.4 48 60 40 36 8 x 9 y x 100x0.6y0.4 20,000 ⇒ x0.6 8 x 9 0.4 8 ⇒y 9 200 x y Therefore, C 209.65, 186.35 46. f x, y ax by, x, y > 0 y2 36 1 4x 200 8 9 0.4 8 200 9 8 9 0.4 209.65 186.35 $16,771.94. x2 Constraint: 64 (a) Level curves of f x, y y Using y x 7, y 8 3y are lines of form (b) Level curves of f x, y y 4 x 9 4 x 9 4, y C. 4x 9y are lines of form 4 x 3 4 x 3 C. 12.3, you obtain 3, and f 7, 3 28 9 37. Using y x 7, you obtain 5.2, and f 4, 5.2 62.8. − 10 10 −8 Constraint is an ellipse. Review Exercises for Chapter 12 2. Yes, it is the graph of a function. x x y c=−1 3 c = −1 2 c=−2 2 c = −2 c= 2 1 4. f x, y ln x y 2 6. f x, y c= 1 2 The level curves are of the form c ec ln x y x y. −3 c= 3 2 c=1 c=0 3 The level curves are of the form c y 1 c c c x x c y x. c=1 −3 3 The level curves are hyperbolas. −2 −2 c= 2 c=2 3 The level curves are passing through the origin with slope 1 . 356 Chapter 12 Functions of Several Variables xy x2 y2 ± x. 8. g x, y z 60 y 1 x 10. f x, y, z Elliptic cone z 2 9x2 y2 9z2 0 12. x, y → 1, 1 lim Does not exist Continuous except when y 2 x 5 y 5 x 5 y 14. x, y → 0, 0 lim y xe y 1 x2 2 0 1 0 0 0 16. f x, y fx fy xy x yx x x2 x y 2 y y y 2 18. xy x y2 y 2 z z x z y ln x 2 x2 x2 2x y2 2y y2 y2 1 1 1 Continuous everywhere 20. w w x w y w z x2 12 x 2 x2 x2 y2 y2 y y2 z y2 z2 z2 12 22. f x, y, z 2x x2 x y2 z2 fx 1 1 1 1 2 1 x2 x2 x2 1 1 . At 2, 0, 0 , z x 0. x2 x2 x y2 y y2 z y2 y2 y2 z2 z2 32 2x z2 z2 fy fz z2 z2 z2 32 1 1 32 32 24. u x, t u x u t c sin akx cos kt akc cos akx cos kt kc sin akx sin kt 26. z z x x2 ln y 2x ln y Slope in x-direction. z y x2 1 y . At 2, 0, 0 , z y 4. Slope in y-direction. x x y x x x x x x y x y 2y y 2x y y 3 2 2 28. h x, y hx hy hxx hyy hxy hyx y 30. g x, y gx gy gxx 2 cos x sin x 2 sin x cos x 2y 2y 2y 2y 2y 2y 2y gyy gxy gyx 4 cos x 2 cos x 2 cos x 3 x y2 x 2y x y4 2y x y4 y y x x x x y y3 y y3 R eview Exercises for Chapter 12 32. z z x 2z x2 357 x3 3x2 6x 3xy 2 3y 2 34. z z x 2z e x sin y e x sin y e x sin y e x cos y e x sin y 2z 2z x2 z y 2z y2 2z x2 2z y2 z y 2z y2 6xy 6x 0. Therefore, Therefore, x2 y2 0. 36. z dz xy x2 y 2 z dx x x2 z dy y y2 y xy x x2 y 2 x2 y2 dx x2 y 2x x2 xy y y2 x2 y2 dy x2 y3 y2 32 dx x2 x3 y2 32 dy 38. From the accompanying figure we observe tan dh Letting x h or h x h dx x 100, dx x tan h h d tan dx x sec2 d. θ 1 ±, 2 11 , and d 60 ± 180 . x (Note that we express the measurement of the angle in radians.) The maximum error is approximately dh tan 11 60 h2 h2 h2 dr h2 cos t, y ux xt 1 sin t sin t 1 Substitution: u du dt sin2 t r2 r2 r2 h2 rh h2 dr dh rh r2 8 h2 dh 1 10 ± 29 8 ± ± 1 2 100 sec2 11 60 ± 180 ± 0.3247 ± 2.4814 ± 2.81 feet. 40. A dA r r2 r2 2r 2 r2 25 1 ± 8 29 43 8 29 42. u y2 x, x du dt sin t uy yt sin t 2y cos t Chain Rule: 2 sin t cos t 2 cos t cos t sin t sin t 1 2 cos t 2 sin t cos t 358 Chapter 12 xy ,x z Functions of Several Variables 44. w 2r w r t, y rt, z 2r wy yr t wz zr 46. 2xz z x z2 xz2 y sin z y cos z z x z x 0 0 z2 y cos z 0 sin z y cos z Chain Rule: wx xr y 2 z 2rt 2r t 4r2t x t z 2r 2r xy 2 z2 tt t 2 2r 2r t rt t2 2xz z y y cos z z y 2xz sin z z y 4rt2 t 3 2r t 2 wy yt x r z rt 2 2r t 2 2xz w t wx xt y 1 z 4r2t xy z 4r2t 4r2t 2r wz zt 1 xy z2 4r 3 Substitution: w w r w t 12 y 4 2r t rt 2r t 4rt2 t 3 2r t 2 rt2 t 2 2r2t 2r rt 2 t 4r 3 48. f x, y f f 1, 4 u Du f 1, 4 x2 1 yj 2 2j 25 i 5 u 5 j 5 45 5 25 5 25 5 50. w w w 1, 0, 1 u Duw 1, 0, 1 6x2 12x 12 i 1 v 3 3xy 3y i 3j 3 i 3 4y 2z 3x 8yz j 4y 2 k 2x i 2i 1 v 5 f 1, 4 3 j 3 u 3 k 3 w 1, 0, 1 43 3 0 53 52. z z z 2, 1 z 2, 1 x2 x x2 x 4j 4 y2 3 4y sin x 4y cos xi 2j y2 4 sin x 2y j y 2xy i y2 x2 x y j 2 54. z z z 2, 1 z 2, 1 x2y 2xy i 4i 42 x2j 4j 56. 4y sin x 58. F x, y, z F F 2, 3, 4 y2 2 yj 6j z2 2z k 8k 25 0 f x, y f x, y f 2 ,1 2 3j 4k Therefore, the equation of the tangent plane is 3y 3 y 3 4z 3 z 4 4 4 0 or 3y 4z 25, Normal vector: j and the equation of the normal line is x 2, . R eview Exercises for Chapter 12 60. F x, y, z F F 1, 2, 2 x2 2x i 2i y2 2y j 4j z2 9 2z k 4k 2i 2j 2k 0 62. F x, y, z G x, y, z F G F 4, 4, 9 F G y2 x 2y i i 8i i 8 1 j k j 0 1 k 1 0 i j 8k z y k 25 0 0 359 Therefore, the equation of the tangent plane is x x 1 2y 2y 2z 9, 2 2z 2 0 or and the equation of the normal line is x 1 1 y 2 2 z 2 2 . Therefore, the equation of the tangent line is x 1 4 y 1 cos x, sin y, 0, 1 y 0 0.1 0.1 0.3 0.5 z 3 4 z 9 . 8 1 0 0 64. (a) f x, y fx fy P1 x, y cos x sin x, cos y, 1 y sin y, f 0, 0 fx 0, 0 fy 0, 0 1 0 1 (b) fxx fyy fxy fxx 0, 0 fyy 0, 0 fxy 0, 0 y 12 2x P2 x, y (d) x 0 0 0.2 0.5 1 (c) If y 0, you obtain the 2nd degree Taylor polynomial for cos x. f x, y 1.0 1.0998 1.0799 1.1731 1.0197 P1 x, y 1.0 1.1 1.1 1.3 1.5 P2 x, y 1.0 1.1 1.095 1.175 1.0 (e) 2 z z 2 2 −2 −1 1 y 1 x −1 1 y −2 2 x 1 −1 1 −1 2 y 1 x The accuracy lessens as the distance from 0, 0 increases. 66. f x, y fx fy 4 3y fxx fyy fxy fxx fyy 2x2 4x 6x 6y 4 18 6 f xy 2 6xy 6y 18y 8 9y 2 0 0, x 8x 14 3y 4 3, 8⇒y x 4 4 18 6 2 36 > 0 Therefore, 4, 4 , 3 2 is a relative minimum. 360 68. z zx zy Chapter 12 50 x 50 50 zxx y 0.3x2 0.15y 2 Functions of Several Variables 0.1x3 20 20.6 20x 0, x 0, y 14 , 0 4.2 6 4.2 6 2 150 ± 10 ± 14 0.05y3 20.6y 125 Critical Points: 10, 14 , 10, 0.6x, zyy zxy 2 10, 14 , 10, 14 0.3y, zxy 6 2 At 10, 14 , zxx zyy At 10, 10, At At 14 , zxx zyy 14, 02 > 0, zxx < 0. 02 < 0. 02 < 0. 02 > 0, zxx < 0. 10, 14, 199.4 is a relative maximum. zxy zxy 349.4 is a saddle point. 2 10, 14 , zxx zyy 10, 14, 10, 10, 14, 14 , zxx zyy 4.2 6 4.2 200.6 is a saddle point. xxy 749.4 is a relative minimum. 70. The level curves indicate that there is a relative extremum at A, the center of the ellipse in the second quadrant, and that there is a saddle point at B, the origin. 0.25x12 5x1 0.15x22 72. Minimize C x1, x2 0.50x1 0.30x2 x1 x2 10 12 10x1 3x2 3x2 3x2 x1 x2 20 12x2 subject to the constraint x1 x2 1000. 1000 ⇒ 3x1 5x1 8x1 3000 20 3020 377.5 622.5 C 377.5, 622.5 104,997.50 74. Minimize the square of the distance: f x, y, z fx fy 2x 2y 2 2 x 2 x2 2 x2 2 2 y y2 2x y2 2y 2 2 x2 2 2 y2 2x3 2y3 0 2. 2xy2 2x2y 0 0 0.6894. 0x 0y Clearly x y and hence: 4x3 x 0.6894 2 2 Thus, distance 2 distance 2.08 2 0. Using a computer algebra system, x 0.6894 2 2 2 0.6894 2 2 4.3389. 76. (a) 25, 28 , 50, 38 , 75, 54 , 100, 75 , 125, 102 xi xi4 375, 382,421,875, 3,515,625b 34,375b 375b 0.0717, c yi xi yi 297, 26,900, xi2 xi2 yi 34,375, 2,760,000, xi3 3,515,625 382,421,875a 3,515,625a 34,375a a 0.0045, b 34,375c 375c 5c 2,760,000 26,900 297 0.0045x2 0.0717x 23.2914 23.2914, y (b) When x 80 km/hr, y 57.8 km. ...
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This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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