ISM_T11_C14_A

ISM_T11_C14_A - CHAPTER 14 PARTIAL DERIVATIVES 14.1...

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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)
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