Thoms calculus 11ed_ism_ch14

Thomas' Calculus, 11th Edition

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CHAPTER 14 PARTIAL DERIVATIVES 14.1 FUNCTIONS OF SEVERAL VARIABLES 1. (a) Domain: all points in the xy-plane (b) Range: all real numbers (c) level curves are straight lines y x c parallel to the line y x œ œ (d) no boundary points (e) both open and closed (f) unbounded 2. (a) Domain: set of all (x y) so that y x 0 y x ß   Ê   (b) Range: z 0   (c) level curves are straight lines of the form y x c where c 0 œ   (d) boundary is y x 0 y x, a straight line È œ Ê œ (e) closed (f) unbounded 3. (a) Domain: all points in the xy-plane (b) Range: z 0   (c) level curves: for f(x y) 0, the origin; for f(x y) c 0, ellipses with center ( 0) and major and minor ß œ ß œ axes along the x- and y-axes, respectively (d) no boundary points (e) both open and closed (f) unbounded 4. (a) Domain: all points in the xy-plane (b) Range: all real numbers (c) level curves: for f(x y) 0, the union of the lines y x; for f(x y) c 0, hyperbolas centered at ß œ œ „ ß œ Á (0 0) with foci on the x-axis if c 0 and on the y-axis if c 0 ß (d) no boundary points (e) both open and closed (f) unbounded 5. (a) Domain: all points in the xy-plane (b) Range: all real numbers (c) level curves are hyperbolas with the x- and y-axes as asymptotes when f(x y) 0, and the x- and y-axes ß Á when f(x y) 0 ß œ (d) no boundary points (e) both open and closed (f) unbounded 6. (a) Domain: all (x y) (0 y) ß Á ß (b) Range: all real numbers (c) level curves: for f(x y) 0, the x-axis minus the origin; for f(x y) c 0, the parabolas y cx minus the ß œ ß œ Á œ # origin (d) boundary is the line x 0 œ

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864 Chapter 14 Partial Derivatives (e) open (f) unbounded 7. (a) Domain: all (x y) satisfying x y 16 ß # # (b) Range: z   " 4 (c) level curves are circles centered at the origin with radii r 4 (d) boundary is the circle x y 16 # # œ (e) open (f) bounded 8. (a) Domain: all (x y) satisfying x y 9 ß Ÿ # # (b) Range: 0 z 3 Ÿ Ÿ (c) level curves are circles centered at the origin with radii r 3 Ÿ (d) boundary is the circle x y 9 # # œ (e) closed (f) bounded 9. (a) Domain: (x y) (0 0) ß Á ß (b) Range: all real numbers (c) level curves are circles with center ( 0) and radii r 0 (d) boundary is the single point (0 0) ß (e) open (f) unbounded 10. (a) Domain: all points in the xy-plane (b) Range: 0 z 1 Ÿ (c) level curves are the origin itself and the circles with center (0 0) and radii r 0 ß (d) no boundary points (e) both open and closed (f) unbounded 11. (a) Domain: all (x y) satisfying 1 y x 1 ß Ÿ Ÿ (b) Range: z Ÿ Ÿ 1 1 # # (c) level curves are straight lines of the form y x c where 1 c 1 œ Ÿ Ÿ (d) boundary is the two straight lines y 1 x and y 1 x œ œ (e) closed (f) unbounded 12. (a) Domain: all (x y), 0 ß B Á (b) Range: z 1 1 # # (c) level curves are the straight lines of the form y cx, c any real number and x 0 œ Á (d) boundary is the line x 0 œ (e) open (f) unbounded 13. f 14. e 15. a 16. c 17. d 18. b
Section 14.1 Functions of Several Variables 865 19. (a) (b) 20. (a) (b) 21. (a) (b) 22. (a) (b)

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866 Chapter 14 Partial Derivatives 23. (a) (b) 24. (a) (b) 25. (a) (b)
Section 14.1 Functions of Several Variables 867 26. (a) (b) 27. (a) (b) 28. (a) (b) 29. f(x y) 16 x y and 2 2 2 z 16 2 2 2 6 6 16 x y x y 10 ß œ ß Ê œ œ Ê œ Ê œ # # # # # # # # Š Š Š È È È È 30. f(x y) x 1 and (1 0) 1 1 0 x 1 0 x 1 or x 1 ß œ ß Ê D œ œ Ê œ

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