E t 8 8 15 sin 2 16 impossible 523 chapter 14 17 r

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Unformatted text preview: is as axis hyperbolic paraboloid straddling x- and z -axes paraboloid opening along the negative y -axis 6. (a) same (b) same (c) same 2 (d) same 7. (a) x = 0 : (e) y = 2 z x −2 2 a c (f ) y = z2 x2 z2 y2 + = 1; y = 0 : + = 1; 25 4 9 4 x2 z2 +2 a2 c z x2 z2 + =1 9 4 x2 y2 z=0: + =1 9 25 y2 z2 + =1 4 25 y x2 y2 + =1 9 25 x (b) x = 0 : z = 4y 2 ; y = 0 : z = x2 ; z z=0:x=y=0 z = 4 y2 z = x2 x2 + 4 y2 = 0 (0, 0, 0) x y Exercise Set 13.7 (c) x = 0 : 476 z2 x2 z2 y2 − = 1; y = 0 : − = 1; 16 4 9 4 z y2 z2 – =1 16 4 x2 y2 z=0: + =1 9 16 y y2 x2 + =1 9 16 x x2 z2 – =1 9 4 8. (a) x = 0 : y = z = 0; y = 0 : x = 9z 2 ; z = 0 : x = y 2 z x = 9z2 y x (b) x = 0 : −y 2 + 4z 2 = 4; y = 0 : x2 + z 2 = 1; x = y2 z=0 z z = 0 : 4x2 − y 2 = 4 y=0 x=0 y x y (c) x = 0 : z = ± ; y = 0 : z = ±x; z = 0 : x = y = 0 2 x=0 z z=0 x y y=0 9. (a) 4x2 + z 2 = 3; ellipse (b) y 2 + z 2 = 3; circle (c) y 2 + z 2 = 20; circle (d) 9x2 − y 2 = 20; hyperbola (e) z = 9x2 + 16; parabola (f ) 9x2 + 4y 2 = 4; ellipse 10. (a) y 2 − 4z 2 = 27; hyperbola (b) 9x2 + 4z 2 = 25; ellipse (c) 9z 2 − x2 = 4; hyperbola (d) x2 + 4y 2 = 9; ellipse (e) z = 1 − 4y 2 ; parabola (f ) x2 − 4y 2 = 4; hyperbola 477 Chapter 13 z 11. z 12. z 13. (0, 0, 3) (0, 0, 2) (1, 0, 0) x x y (0, 3, 0) (2, 0, 0) (0, 3, 0) (0, 2, 0) y x y (6, 0, 0) Ellipsoid Ellipsoid Hyperboloid of one sheet z 14. z 15. z 16. (0, 3, 0) (3, 0, 0) x y x Hyperboloid of one sheet (0, 0, 2) Elliptic cone Elliptic cone z 17. z 18. y x y z 19. (0, 0, –2) x y x y Hyperboloid of two sheets x y Hyperboloid of two sheets Hyperbolic paraboloid z 20. z 21. z 22. y x Hyperbolic paraboloid y x y x Elliptic paraboloid Circular paraboloid Exercise Set 13.7 478 z 23. z 24. z 25. (0, 0, 2) y x x y x (0, 2, 0) Elliptic paraboloid Hyperboloid of one sheet Circular cone z 26. z 27. y z 28. (0, 0, 2) (- 3, 0, 0) x (3, 0, 0) y Hyperboloid of two sheets (2, 0, 0) Hyperboloid of one sheet Hyperbolic paraboloid z 29. x y x 30. z (0, 0, 1) (0, 1, 0) y x y x z 31. (1, 0, 0) z 32. (0, 0, 1) x x (1, 0, 0) (0, 1, 0) y y y 479 Chapter 13 z 33. z 34. (0 , 0, 2 ) x y x y Hyperboloid of one sheet (–2, 3, –9) Circular paraboloid z 35. z 36. (-1 , 1 , 2 ) (1, –1, –2) y y x x Ellipsoid Hyperboloid of one sheet y2 x2 + =1 (b) 6, 4 9 4 (d) The focal axis is parallel to the x-axis. √ √ (c) (± 5, 0, 2) √ z2 y2 + =1 (b) 4, 2 2 4 2 (d) The focal axis is parallel to the y -axis. √ (c) (3, ± 2, 0) x2 y2 − =1 (b) (0, ±2, 4) 4 4 (d) The focal axis is parallel to the y -axis. √ (c) (0, ±2 2, 4) 37. (a) 38. (a) 39. (a) y2 x2 − =1 (b) (±2, 0, −4) 40. (a) 4 4 (e) The focal axis is parallel to the x-axis. 41. (a) z + 4 = y 2 (d) (c) (2, 0, −15/4) The focal axis is parallel to the z -axis. 42. (a) z − 4 = −x2 (d) (b) (2, 0, −4) √ (c) (±2 2, 0, −4) (b) (0, 2, 4) The focal axis is parallel to the z -axis. (c) (0, 2, 15/4) Exercise Set 13.7 480 43. x2 + y 2 = 4 − x2 − y 2 , x2 + y 2 = 2; circle of radius in the plane z = 2, centered at (0, 0, 1) √ 2 z 4 2 2 x +y = 2 (z = 2) x 44. y 2 + z = 4 − 2(y 2 + z ), y 2 + z = √ 3; 4/ parabolas in the planes x = ±2/ 3 which open in direction of the negative y -axis y z x y z = 4 – y2, x = 4 3 3 45. y = 4(x2 + z 2 ) 47. |z − (−1)| = paraboloid 46. y 2 = 4(x2 + z 2 ) x2 + y 2 + (z − 1)2 , z 2 + 2z + 1 = x2 + y 2 + z 2 − 2z + 1, z = (x2 + y 2 )/4; circular 48. |z + 1| = 2 x2 + y 2 + (z − 1)2 , z 2 + 2z + 1 = 4 x2 + y 2 + z 2 − 2z + 1 , 4x2 + 4y 2 + 3z 2 − 10z + 3 = 0, y2 (z − 5/3)2 x2 + + = 1; ellipsoid, center at (0, 0, 5/3). 4/3 4/3 16/9 x2 y2 x2 z2 + 2 = 1; if y = 0 then 2 + 2 = 1; since c < a the major axis has length 2a, the 2 a a a c minor axis length 2c. 49. If z = 0, 50. x2 y2 z2 + 2 + 2 = 1, where a = 6378.1370, b = 6356.5231. 2 a a b 51. Each slice perpendicular to the z -axis for |z | < c is an ellipse whose equation is y2 y2 c2 − z 2 x2 x2 +222 = 1, the area of which is + 2= , or 2 2 2 a2 b c2 (a /c )(c − z 2 ) (b /c )(c − z 2 ) π a c c2 − z 2 b c c2 − z 2 =π ab 2 c − z 2 so V = 2 c2 c π 0 ab 2 4 c − z 2 dz = πabc. 2 c 3 481 Chapter 13 EXERCISE SET 13.8 √ 5 2, 3π/4, 6 1. (a) (8, π/6, −4) (b) 2. (a) (2, 7π/4, 1) (b) (1, π/2, 1) (c) (2, π/2, 0) (d) (8, 5π/3, 6) √ (c) (4 2, 3π/4, −7) √ (d) (2 2, 7π/4, −2) (c) (5, 0, 4) (d) (−7, 0, −9) 3. (a) √ 2 3, 2, 3 (b) 4. (a) √ 3, −3 3, 7 (b) (0, 1, 0) (c) (0, 3, 5) (d) (0, 4, −1) 5. (a) √ 2 2, π/3, 3π/4 (b) (2, 7π/4, π/4) (c) (6, π/2, π/3) (d) (10, 5π/6, π/2) 6. (a) √ 8 2, π/4, π/6 (b) (c) (2, 0, π/2) (d) (4, π/6, π/6) √√ −4 2, 4 2, −2 √ 2 2, 5π/3, 3π/4 √ √ √ 7. (a) (5 6/4, 5 2/4, 5 2/2) (c) (0,0,1) (b) (7,0,0) (d) (0, −2, 0) √ √ √ − 2/4, 6/4, − 2/2 √√√ (c) (2 6, 2 2, 4 2) √ √ √ 3 2/4, −3 2/4, −3 3/2 √ (d) 0, 2 3, 2 8. (a) 9. (a) (b) √ 2 3, π/6, π/6 (c) (2, 3π/4, π/2) 10. (a) √ (b) 2, π/4, 3π/4 √ 4 3, 1, 2π/3 (d) √ 4 2, 5π/6, π/4 √ 2 2, 0, 3π/4 √...
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This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.

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