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   (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 ßœ ßœ² ! ß 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 œ ßœÁ (0 0) with foci on the x-axis if c 0 and on the y-axis if c 0 ß² ³ (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 ßÁ when f(x y) 0 (d) no boundary points (e) both open and closed (f) unbounded 6. (a) Domain: all (x y) (0 y) ßÁß (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 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± ² ## (b) Range: z   " 4 (c) level curves are circles centered at the origin with radii r 4 ² (d) boundary is the circle x y 16 ±œ (e) open (f) bounded 8. (a) Domain: all (x y) satisfying x y 9 Ÿ (b) Range: 0 z 3 ŸŸ (c) level curves are circles centered at the origin with radii r 3 Ÿ (d) boundary is the circle x y 9 (e) closed (f) bounded 9. (a) Domain: (x y) (0 0) ßÁß (b) Range: all real numbers (c) level curves are circles with center ( 0) and radii r 0 ³ (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 ²Ÿ (c) level curves are the origin itself and the circles with center (0 0) and radii r 0 ß³ (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 ´ŸŸ 11 (c) level curves are straight lines of the form y x c where 1 c 1 ´œ (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 ßB Á (b) Range: z ´²² (c) level curves are the straight lines of the form y cx, c any real number and x 0 œÁ (d) boundary is the line x 0 œ (e) open (f) unbounded 13. f 14. e 15. a 16. c 17. d 18. b
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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)
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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 ßœ±± ß Êœ± ± œÊœ±± Ê ²œ ## Š‹ Š Š ÈÈ È È 30. f(x y) x 1 and (1 0) 1 1 0 x 1 0 x 1 or x 1 ßœ ± ß Ê D
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Thoms calculus 11ed_ism_ch14 - CHAPTER 14 PARTIAL...

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