Page 202 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 8] APPLICATIONS OF PARTIAL DERIVATIVES 193

y
8.9. (a) Find the envelope of the family x sin þ y cos ¼ 1. (b) Illus-
trate the results geometrically.
(a)ByProblem 8 the envelope, if it exists, is obtained by solving simulta-
neously the equations ðx; y; Þ¼ x sin þ y cos 1 ¼ 0 and
ðx; y; Þ¼ x cos y cos ¼ 0. From these equations we find x
2
2
x ¼ sin ; y ¼ cos or x þ y ¼ 1.
(b) The given family is a family of straight lines, some members of which
2
2
are indicated in Fig. 8-5. The envelope is the circle x þ y ¼ 1.
Fig. 8-5
2
8.10. Find the envelope of the family of surfaces z ¼ 2 x a y.
By a generalization of Problem 8.8 the required envelope, if it exists, is obtained by solving simulta-
neously the equations
2
¼ 2 x y z ¼ 0 and ¼ 2x 2 y ¼ 0
ð1Þ ð2Þ
2
From (2) ¼ x=y. Then substitution in (1)yields x ¼ yz,the required envelope.

8.11. Find the envelope of the two-parameter family of surfaces z ¼ x þ y .
The envelope of the family Fðx; y; z; ; Þ¼ 0, if it exists, is obtained by eliminating and between the
equations F ¼ 0; F ¼ 0; F ¼ 0 (see Problem, 8.43). Now
F ¼ z x y þ ¼ 0; F ¼ x þ ¼ 0; F ¼ y þ ¼ 0
Then ¼ x, ¼ y; and we have z ¼ xy.

DIRECTIONAL DERIVATIVES

u
3
2
8.12. Find the directional derivative of F ¼ x yz along the curve x ¼ e , y ¼ 2 sin u þ 1, z ¼ u cos u
at the point P where u ¼ 0.
The point P corresponding to u ¼ 0is ð1; 1; 1Þ. Then
2 3
2
2
3
rF ¼ 2xyz i þ x z j þ 3x yz k ¼ 2i j þ 3k at P
Atangent vector to the curve is
dr d u
du ¼ du fe i þð2 sin u þ 1Þj þðu cos uÞkg
u
¼ e i þ 2cos uj þð1 þ sin uÞk ¼ i þ 2j þ k at P
i þ 2j þ k
:
and the unit tangent vector in this direction is T 0 ¼ p ffiffiffi
6
Then

i þ 2j þ k 3 1 p ffiffiffi
6:
ffiffiffi
Directional derivative ¼rF T 0 ¼ð 2i j þ 3kÞ p ffiffiffi ¼ p ¼
6 6 2
Since this is positive, F is increasing in this direction.
8.13. Prove that the greatest rate of change of F, i.e., the maximum directional derivative, takes place in
the direction of, and has the magnitude of, the vector rF.