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QUIZ 2, Version A : MATH 251, Section 505
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Write up your result, detail your calculations if necessary and BOX your final answer
1. [25pts] Find the differential dz of the function z = ex + x cos y.
2. [25pts] Use the Chain Rule to find Gx where G = ln(u2 + v 2 ) and u = x + y, v = x + y 2 .
y
.
x
(a) [25pts] Find the direction in which f increases most rapidly at the point (1, 1).
3. Let f (x, y) = x3 +
(b) [25pts] Find the directional derivative of f at the point (1, 1) in the direction < 4, 3 >.
1. We have dz = fx (x, y)dx + fy (x, y)dy, so dz = (ex + cos y)dx − x sin ydy .
2. Denote G = ln(u2 + v 2 ), where DG = R2 \ {(0, 0)}. So,
∂u ∂G ∂v ∂ G
+
,
∂x ∂u
∂x ∂v
2u
2v
= 2
+
,
u + v 2 u2 + v 2
2(x + y)
2(x + y 2 )
+
,
=
(x + y)2 + (x + y 2 )2 (x + y)2 + (x + y 2 )2
Gx =
=
4x + 2y + 2y 2
.
(x + y)2 + (x + y 2 )2
3. f increases most rapidly in the direction of the gradient, so the gradient at (1, 1) is
y
1
∇f (1, 1) = hfx (1, 1), fy (1, 1)i. We have fx (x, y) = 3x2 − 2 and fy (x, y) = on Df = {(x, y) ∈ R2 | x 6= 0}.
x
x
So, fx (1, 1) = 2 and fy (1, 1) = 1.
∇f (2, 1) = h2, 1i.
4. The directional derivative of f at (1, 1) is Du f (1, 1) = ∇f (1, 1)•
u
< 4, 3 >
8+3
11
.
= h2, 1i•
=
=
|u|
5
5
5