Solução_Calculo_Stewart_6e

F.
TX.10
CHAPTER 15 REVIEW ET CHAPTER 14 ¤ 213
f(a, b + h) − f(a, b)
1. True. f y(a, b) = lim
from Equation 15.3.3 [ET 14.3.3]. Let h = y − b. Ash → 0, y → b. Thenby
h→0 h
f(a, y) − f(a, b)
substituting, we get f y(a, b) =lim
.
y→b y − b
3. False. f xy = ∂2 f
∂y ∂x .
5. False. See Example 15.2.3 [ET 14.2.3].
7. True. If f has a local minimum and f is differentiable at (a, b) then by Theorem 15.7.2 [ET 14.7.2], f x (a, b) =0and
f y (a, b) =0,so∇f(a, b) =hf x (a, b),f y (a, b)i = h0, 0i = 0.
9. False. ∇f(x, y) =h0, 1/yi.
11. True. ∇f = hcos x, cos yi, so|∇f| = cos 2 x +cos 2 y.But|cos θ| ≤ 1, so|∇f| ≤ √ 2.Now
D u f (x, y) =∇f · u = |∇f||u| cos θ,butu is a unit vector, so |D u f(x, y)| ≤ √ 2 · 1 · 1= √ 2.
1. ln(x + y +1)is defined only when x + y +1> 0 ⇒ y>−x − 1,
so the domain of f is {(x, y) | y>−x − 1}, all those points above the
line y = −x − 1.
3. z = f(x, y) =1− y 2 , a parabolic cylinder 5. The level curves are 4x 2 + y 2 = k or 4x 2 + y 2 = k 2 ,
k ≥ 0, a family of ellipses.