Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

182 Functions of Several Variables [Ch. 6
3. Level lines and level surfaces of a function. The level line of a function
2 = f(x, y) is a line / (*, y)-C (in an *r/-plane) at the points of which
the function takes on one and the same value z C (usually labelled in
drawings).
The level surface of a function of three arguments u~f(x, y t z) is a suriace
/ (x, y, z) = C, at the points of which the function takes on a constant
value u~C.
Example
5. Construct the level lines of
the function z = x*y.
Solution. The equation of the level lines
has the form x 2
y = C or y ~ -j .
Putting C = 0, 1, i 2, .... we get a family
of level lines (Fig. 66).
1782. Express the volume V of a
Fig.
regular tetragonal pyramid as a function
of its altitude x and lateral edge y.
1783. Express the lateral surface S
of a regular hexagonal truncated pyra-
66 mid as a function of the sides x and y
of the bases and the altitude z.
1784. Find /(1/2, 3), /(I, -1), if
1785 Find f(y,x), f( x, y),
__x z
y 2
1786. Find the values assumed by
at points of the parabola y =
function
1787. Find the value of the function
at points of the circle x 2
Z rrr
1788*. Determine f(x), if
1789*. Find f(x, y) if
+y*=R 2
.
-
z
the function
1
1
f(*,y) ,
, and construct the graph of the
if