Page 135 - Schaum's Outline of Theory and Problems of Advanced Calculus

126 PARTIAL DERIVATIVES [CHAP. 6

1
f ðx; y þ kÞ f ðx; yÞ 3 2 3 2
ðcÞ ¼ f½x 2xðy þ kÞþ 3ðy þ kÞ Š ½x 2xy þ 3y Šg
k k
1 3 2 2 2 2
k
¼ ðx 2xy 2kx þ 3y þ 6ky þ 3k x þ 2xy 3y Þ
1
2
¼ ð 2kx þ 6ky þ 3k Þ¼ 2x þ 6y þ 3k:
k
6.2. Give the domain of definition for which each of the following functions are defined and real, and
indicate this domain graphically.
2 2 2 2
(a) f ðx; yÞ¼ lnfð16 x y Þðx þ y 4Þg
The function is defined and real for all points ðx; yÞ such that
2
2
2
2
2
2
ð16 x y Þðx þ y 4Þ > 0; i.e., 4 < x þ y < 16
which is the required domain of definition. This point set consists of all points interior to the circle of
radius 4 with center at the origin and exterior to the circle of radius 2 with center at the origin, as in the
figure. The corresponding region, shown shaded in Fig. 6-5 below, is an open region.

Fig. 6-5 Fig. 6-6

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(b) f ðx; yÞ¼ 6 ð2x þ 3yÞ
The function is defined and real for all points ðx; yÞ such that 2x þ 3y @ 6, which is the required
domain of definition.
The corresponding (unbounded) region of the xy plane is shown shaded in Fig. 6-6 above.

6.3. Sketch and name the surface in three-dimensional space represented by each of the following.
What are the traces on the coordinate planes?
(a)2x þ 4y þ 3z ¼ 12.
Trace on xy plane ðz ¼ 0Þ is the straight line x þ 2y ¼ 6, z ¼ 0:
Trace on yz plane ðx ¼ 0Þ is the straight line 4y þ 3z ¼ 12, x ¼ 0.
Trace on xz plane ðy ¼ 0Þ is the straight line 2x þ 3z ¼ 12, y ¼ 0.
These are represented by AB, BC; and AC in Fig. 6-7.
The surface is a plane intersecting the x-, y-, and z-axes in the
points Að6; 0; 0Þ, Bð0; 3; 0Þ, Cð0; 0; 4Þ. The lengths OA ¼ 6, OB ¼ 3,
OC ¼ 4are called the x, y, and z intercepts, respectively.
x 2 y 2 z 2
¼ 1
a b c
ðbÞ 2 þ 2 2
x 2 y 2
Trace on xy plane ðz ¼ 0Þ is the ellipse þ ¼ 1, z ¼ 0.
a 2 b 2
y 2 z 2
Trace on yz plane ðx ¼ 0Þ is the hyperbola ¼ 1, x ¼ 0. Fig. 6-7
b 2 c 2