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

196_Functions of Several Variables_[Ch. 6
1876. Find the derivative of the function z = x* xy2y*
at the point P(l, 2) in the direction that produces an angle
of 60 with the x-axis.
1877. Find the derivative of the function z = x* 2x*y + xy* + 1
at the point Af(l,
point
2) in the direction from this point to the
tf (4, 6).
1878. Find the derivative of the function _
z = lnYx* + y* at
the point P(l, 1) in the direction of the bisector of the first
quadrantal angle.
1879. Find the derivative of the function u = x* 3yz + 5 at
the point Af(l, 2, 1) in the direction that forms identical
angles with all the coordinate axes.
1880. Find the derivative of the function u = xy + yz -)- zx at
the point M(2, 1, 3) in the direction from this point to the
point N(S, 5, 15).
1881. Find the derivative of the function u = \n (e* + eP + e*)
at the origin in the direction which forms with the coordinate
axes x, y, z the angles a, p, y, respectively.
1882. The point at which the derivative of a function in any
direction
Find the
is zero is
stationary
called the stationary point of this
points of the following functions:
function.
a) z-=x*
b) z = x* + y*c)
u = 2y*-{ z*xyyz
1883. Show that the derivative of the function z = taken
at any point of the ellipse 2x* + y* = C* along the normal to the
ellipse is equal to zero.
1884. Find grad z at the point (2, 1) if
1885. Find grad z at the point (5, 3) if
1886. Find grad u at the point (1, 2, 3), if u=xyz.
1887. Find the magnitude and direction of grad u at the
point (2, 2, 1) if
1888. Find the angle between the gradients of the function
*=ln-j- at the points A (1/2, 1/4) and 5(1, 1).