MATH 1260 - Quiz 6
Solution
2
2
3
(1) Compute
√ the directional derivative of f (x, y) = x +y −3xy at the point (1, 2) in the direction
u = h1/2, 3/2i.
√
√
→
(1,
2)
=
∇f
(1,
2)
·
u
=
h−22,
−32i
·
h1/2,
Df→
3/2i
=
−11
−
16
3.
u
→
(2) √Find the equation for the tangent plane to the surface x2 + 2y 2 + 3z 2 = 10 at the point
(1, 3, 1).√
√
∇f (1, 3, 1) = h2, 4 3, 6i, and the tangent plane is
√
2x + 4 3y + 6z = 20.
(3) Let y be a function of x satisfying F (x, y, x + y) = 0, where F (x, y, z) is a given function. Find
a formula for dy/dx.
Let f : R2 → R be the function f (x, y) = F (x, y, x + y). Then we know that dy/dx = −fx /fy .
Consider the function g : R2 → R3 defined by g(x, y) = (x, y, x + y). Then f is the composition
of g and F , i.e., f (x, y) = F (g(x, y)).
The chain rule tells us that
Df = DF · Dg,
and therefore


1 0
(fx fy ) = (Fx Fy Fz ) · 0 1 .
1 1
By working out the multiplication on the right we see that
fx = Fx + Fz ,
Therefore,
fy = Fy + Fz .
dy
fx
Fx + Fz
=− =−
.
dx
fy
Fy + Fz
1