Page 154 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 6] PARTIAL DERIVATIVES 145

6.56. Evaluate each of the following limits where they exist.
3 x þ y y 2 2 x þ y 1
2
(a) lim ðcÞ lim x sin ðeÞ lim e 1=x ðy 1Þ
x!1 4 þ x 2y x!4 x x!0 ðgÞ lim p ffiffiffi p 1 y
ffiffiffiffiffiffiffiffiffiffiffi
x!0þ
y!2 y! y!1 y!1 x
3x 2y 2 2 2x y 1
lim ðdÞ lim x sinðx þ y Þ ð f Þ lim ðhÞ lim sin ðxy 2Þ
ðbÞ 2 2 2 2 1
x!0 2x 3y x!0 x þ y x!0 x þ y
y!0 y!0 y!0 y!1
x!2 tan ð3xy 6Þ
ffiffiffi
p
Ans. (a)4, (b) does not exist, (c)8 2; ðdÞ 0; ðeÞ 0; ð f Þ does not exist, (g)0, (h)1/3
6.57. Formulate a definition of limit for functions of (a)3, (b) n variables.
4x þ y 3z
6.58. Does lim as ðx; y; zÞ! ð0; 0; 0Þ exist? Justify your answer.
2x 5y þ 2z
6.59. Investigate the continuity of each of the following functions at the indicated points:
x 1
2
2
2
2
x þ y ; ðx 0 ; y 0 Þ: ; ð0; 0Þ: ðcÞðx þ y Þ sin if ðx; yÞ 6¼ð0; 0Þ,0 if ðx; yÞ¼ð0; 0Þ;
ðaÞ ðbÞ 2 2
3x þ 5y x þ y
ð0; 0Þ:
Ans. (a)continuous, (b)discontinuous, (c)continuous
6.60. Using the definition, prove that f ðx; yÞ¼ xy þ 6x is continuous at (a) ð1; 2Þ; ðbÞðx 0 ; y 0 Þ.
6.61. Prove that the function of Problem 6.60 is uniformly continuous in the square region defined by 0 @ x @ 1,
0 @ y @ 1.
PARTIAL DERIVATIVES
x y
6.62. If f ðx; yÞ¼ , find (a) @f =@x and (b) @f =@y at ð2; 1Þ from the definition and verify your answer by
x þ y
differentiation rules. Ans. (a) 2; ðbÞ 4

2
6.63. If f ðx; yÞ¼ ðx xyÞ=ðx þ yÞ for ðx; yÞ 6¼ð0; 0Þ , find (a) f x ð0; 0Þ; ðbÞ f y ð0; 0Þ.
0 for ðx; yÞ¼ð0; 0Þ
Ans. (a)1, (b)0
6.64. Investigate lim f x ðx; yÞ for the function in the preceding problem and explain why this limit (if it exists)
ðx;yÞ!ð0;0Þ
is or is not equal to f x ð0; 0Þ:
6.65. If f ðx; yÞ¼ðx yÞ sinð3x þ 2yÞ,compute (a) f x ; ðbÞ f y ; ðcÞ f xx ; ðdÞ f yy ; ð f Þ f yx at ð0; =3Þ.
ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi
p
Ans. (a) 1 3Þ; 1 ð2 3 3Þ; 3 ð 3 2Þ; 2 ði 3 þ 3Þ; 1 ð2 3 þ 1Þ,
2 ð þ ðbÞ 6 ðcÞ 2 ðdÞ 3 ðeÞ 2
( f ) 1 p ffiffiffi
2 ð2 3 þ 1Þ
6.66. (a)Prove by direct differentiation that z ¼ xy tanðy=xÞ satisfies the equation xð@z=@xÞþ yð@z=@yÞ¼ 2z if
ðx; yÞ 6¼ð0; 0Þ. (b)Discuss part (a)for all other points ðx; yÞ assuming z ¼ 0at ð0; 0Þ.
6.67. Verify that f xy ¼ f yx for the functions (a) ð2x yÞ=ðx þ yÞ,(b) x tan xy; and (c)coshðy þ cos xÞ,indicating
possible exceptional points and investigate these points.

2
2
2
2
2
2
6.68. Show that z ¼ lnfðx aÞ þðy bÞ g satisfies @ z=@x þ @ z=@y ¼ 0except at ða; bÞ.
2
2
6.69. Show that z ¼ x cosðy=xÞþ tanðy=xÞ satisfies x z xx þ 2xyz xy þ y z yy ¼ 0except at points for which x ¼ 0.
n
x y þ z
6.70. Show that if w ¼ ,then:
x þ y z