Page 153 - Schaum's Outline of Theory and Problems of Advanced Calculus

144 PARTIAL DERIVATIVES [CHAP. 6

Supplementary Problems

FUNCTIONS AND GRAPHS
2x þ y
6.45. If f ðx; yÞ¼ , find (a) f ð1; 3Þ; ðbÞ f ð2 þ h; 3Þ f ð2; 3Þ ; ðcÞ f ðx þ y; xyÞ.
1 xy h
11 2x þ 2y þ xy
1
Ans. ðaÞ ; ;
2
4 ðbÞ ðcÞ 1 x y xy 2
5ð3h þ 5Þ
2
2
6.46. If gðx; y; zÞ¼ x yz þ 3xy, find (a) gð1; 2; 2Þ; ðbÞ gðx þ 1; y 1; z Þ; ðcÞ gðxy; xz; x þ yÞ.
2
2
2 2
2
2
2
Ans. ðaÞ 1; ðbÞ x x 2 yz þ z þ 3xy þ 3y; ðcÞ x y x z xyz þ 3x yz
6.47. Give the domain of definition for which each of the following functional rules are defined and real, and
indicate this domain graphically.
1 1 2x y

; ðbÞ f ðx; yÞ¼ lnðx þ yÞ; ðcÞ f ðx; yÞ¼ sin :
x þ y 1 x þ y
ðaÞ f ðx; yÞ¼ 2 2

2
2
Ans: ðaÞ x þ y 6¼ 1; ðbÞ x þ y > 0; ðcÞ 2x y @ 1
x þ y
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x þ y þ z 1
6.48. (a) What is the domain of definition for which f ðx; y; zÞ¼ 2 2 2 is defined and real? (b)Indi-
cate this domain graphically. x þ y þ z 1
2
2
2
2
2
2
Ans.(a) x þ y þ z @ 1; x þ y þ z < 1 and x þ y þ z A 1; x þ y þ z > 1
6.49. Sketch and name the surface in three-dimensional space represented by each of the following.
2
2
2
2
2
3x þ 2z ¼ 12; x þ z ¼ y ; x þ y ¼ 2y;
ðaÞ ðdÞ ðgÞ
2
2
2
2
2
4z ¼ x þ y ; x þ y þ z ¼ 16; z ¼ x þ y;
ðbÞ ðeÞ ðhÞ
2
2
2
2
2
2
z ¼ x 4y ; x 4y 4z ¼ 36; y ¼ 4z;
ðcÞ ð f Þ ðiÞ
2
2
2
x þ y þ z 4x þ 6y þ 2z 2 ¼ 0:
ð jÞ
Ans. ðaÞ plane, (b) paraboloid of revolution, (c) hyperbolic paraboloid, (d) right circular cone,
(e)sphere, ( f ) hyperboloid of two sheets, (g) right circular cylinder, (h)plane, (i) parabolic cylinder,
( j)sphere, center at ð2; 3; 1Þ and radius 4.
2
2
2
2
2
2
6.50. Construct a graph of the region bounded by x þ y ¼ a and x þ z ¼ a , where a is a constant.
6.51. Describe graphically the set of points ðx; y; zÞ such that:
2
2
2
2
2
2
2
2
(a) x þ y þ z ¼ 1; x þ y ¼ z ; ðbÞ x þ y < z < x þ y.
52. The level curves for a function z ¼ f ðx; yÞ are curves in the xy plane defined by f ðx; yÞ¼ c, where c is any
constant. They provide a way of representing the function graphically. Similarly, the level surfaces of
w ¼ f ðx; y; zÞ are the surfaces in a rectangular ðxyz)coordinate system defined by f ðx; y; zÞ¼ c, where c is
any constant. Describe and graph the level curves and surfaces for each of the following functions:
2
2
(a) f ðx; yÞ¼ lnðx þ y 1Þ; ðbÞ f ðx; yÞ¼ 4xy; ðcÞ f ðx; yÞ¼ tan 1 y=ðx þ 1Þ; ðdÞ f ðx; yÞ¼ x 2=3 þ
2
2
2
y 2=3 ; ðeÞ f ðx; y; zÞ¼ x þ 4y þ 16z ; ð f Þ sinðx þ zÞ=ð1 yÞ:
LIMITS AND CONTINUITY
6.53. Prove that (a) lim ð3x 2yÞ¼ 14 and (b) lim ðxy 3x þ 4Þ¼ 0byusing the definition.
x!4
y! 1 ðx;yÞ!ð2;1Þ
6.54. If lim f ðx; yÞ¼ A and lim gðx; yÞ¼ B, where lim denotes limit as ðx; yÞ! ðx 0 ; y 0 Þ,prove that:
(a) lim f f ðx; yÞþ gðx; yÞg ¼ A þ B; ðbÞ lim f f ðx; yÞ gðx; yÞg ¼ AB.
6.55. Under what conditions is the limit of the quotient of two functions equal to the quotient of their limits?
Prove your answer.