Page 214 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 8] APPLICATIONS OF PARTIAL DERIVATIVES 205

8.72. (a)Prove that the shortest distance from the point ða; b; cÞ to the plane Ax þ By þ Cz þ D ¼ 0is

Aa þ Bb þ Cc þ D
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
A þ B þ C 2
(b)Find the shortest distance from ð1; 2; 3Þ to the plane 2x 3y þ 6z ¼ 20. Ans. (b)6
8.73. The potential V due to a charge distribution is given in spherical coordinates ðr; ; Þ by
p cos
V ¼ 2
r
where p is a constant. Prove that the maximum directional derivative at any point is
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
p sin þ 4cos
r 3
m
ð 1 x x n m þ 1
8.74. Prove that dx ¼ ln if m > 0; n > 0. Can you extend the result to the case
0 ln x n þ 1
m > 1; n > 1?
2
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
2
2
8.75. (a)If b 4ac < 0 and a > 0; c > 0, prove that the area of the ellipse ax þ bxy þ cy ¼ 1is 2 = 4ac b .
2
2
2
2
[Hint: Find the maximum and minimum values of x þ y subject to the constraint ax þ bxy þ cy ¼ 1.]
8.76. Prove that the maximum and minimum distances from the origin to the curve of intersection defined by
2
2
2
2
2
2
x =a þ y =b þ z =c ¼ 1 and Ax þ By þ Cz ¼ 0can be obtained by solving for d the equation
2 2
2 2
2 2
A a B b C c
¼ 0
2
2
2
a d 2 þ b d 2 þ c d 2
2
2
8.77. Prove that the last equation in the preceding problem always has two real solutions d 1 and d 2 for any real
non-zero constants a; b; c and any real constants A; B; C (not all zero). Discuss the geometrical significance
of this.
ð M dx 1 1 M M
8.78. (a)Prove that I M ¼ ¼ tan þ
2 2 2 2 3 2 2 2
0 ðx þ Þ 2 ð þ M Þ
ð x dx
ðbÞ Find lim I M : This can be denoted by :
2 2 2
M!1
0 ðx þ Þ
d ð M dx d ð M dx
ðcÞ Is lim lim ?
2 2 2 ¼ 2 2 2
d M!1
M!1 d 0 ðx þ Þ
0 ðx þ Þ
2
2
8.79. Find the point on the paraboloid z ¼ x þ y which is closest to the point ð3; 6; 4Þ.
Ans. ð1; 2; 5Þ
2
2
2
8.80. Investigate the maxima and minima of f ðx; yÞ¼ ðx 2x þ 4y 8yÞ .
Ans. minimum value ¼ 0
ð =2 cos xdx ln
8.81. (a)Prove that ¼ 2 2 :
0 cos x þ sin x 2ð þ 1Þ þ 1
=2 cos xdx 3 þ 5 8ln 2
ð 2
ðbÞ Use ðaÞ to prove that 2 ¼ :
0 ð2cos x þ sin xÞ 50
8.82. (a)Find sufficient conditions for a relative maximum or minimum of w ¼ f ðx; y; zÞ.
2
2
2
(b) Examine w ¼ x þ y þ z 6xy þ 8xz 10yz for maxima and minima.