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

218_Functions of Several Variables_[Ch. 6
which is the equation of the tangent plane, and
XXQ _ Yy Q _ ZZQ
F' X (**, y * ) Fy (*0> 00. *0) F'z (*0. J/0. Z 0)
which are the equations of the normal.
Example 2. Write the equations of the tangent plane and the normal to
the surface 3;q/z z s = a 8 at a point for which x = 0, z/ = a.
.j/ =
Solution. Find the z-coordinate of the point of tangency, putting x = 0,
a into the equation of the surface: z* = a 8
, whence z = a. Thus, the
point of tangency is M (0, a, a).
Denoting by F (x, y, z) the left-hand side of the equation, we find the
partial derivatives and their values at the point Af:
Applying formulas (3) and (4), we get
or ^-(-z + a=:0, which is the equation of the tangent plane,
or ~r~ n =
i
x Qj/ a
' wn ^ cn are ^ ne e q ua ^ions of the normal.
1981. Write the equation of the tangent plane and the equations
of the normal to the following surfaces at the indicated
points:
a) to the paraboloid of revolution z 2 = x*+y
0- ~ 2 5 ' ) ;
b) to the cone ^ + -^ y- = at the point (4, 3, 4);
at the point
c) to the sphere x*+y* + z 2 = 2Rz at the point (ffcosa,
/?sina, /?).
1982. At what point of the ellipsoid
y2 ~2
f.2
_4. f_4.- __ 1
a 2 ^ b 2 ^ c 2 "" *
does the normal to it form equal angles with the coordinate axes?
1983. Planes perpendicular to the A:- and #-axes are drawn
through the point M (3, 4, 12) of the sphere x* + y* + z* = 169.
Write the equation of the plane passing through the tangents to
the obtained sections at their common point M.
1984. Show that the equation of the tangent plane to the
central surface (of order two)
ax 2
+ by 2
-\-cz 2 = k