Quiz 9 Solution
Problem 1 (5 points): Find the derivative of the function f(x;y ; z) = ln(1 +
xyz)at P0(2;3;1) in the direction A=๎i+j
grad (f) = fxi+fyj+fzk=yz
1+xyz i+xz
1+xyz j+xy
1+xyz k
Thus the directional derivative in that direction Ais given by:
grad (f)๎๎i+j
j๎i+jj= grad (f)๎๎๎p2i+p2j๎=๎p2yz
1+xyz +p2xz
1+xyz
Problem 2 (5 points): Find the equation for the tangent plane on the given
surface y2๎2xy ๎z= 1 at point P0(1;4;7).
Take f(z; y ; z) = y2๎2xy ๎z๎1. Then:
fx=๎2y
fy= 2y๎2x
fz=๎1
Hence the tangent plane is given by:
fx(1;4;7) (x๎1) + fy(1;4;7) (y๎4) + fz(1;4;7) (z๎7) = 0
๎8 (x๎1) + 6 (y๎4) ๎(z๎7) = 0
1
