Solução_Calculo_Stewart_6e

F.
214 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
7.
9. f is a rational function, so it is continuous on its domain. Since f is defined at (1, 1), we use direct substitution to evaluate the
limit:
lim
(x,y)→(1,1)
2xy
x 2 +2y 2 = 2(1)(1)
1 2 +2(1) 2 = 2 3 .
T (6 + h, 4) − T (6, 4)
11. (a) T x (6, 4) = lim
, so we can approximate T x (6, 4) by considering h = ±2 and
h→0 h
using the values given in the table: T x (6, 4) ≈
T x(6, 4) ≈
T (4, 4) − T (6, 4)
−2
=
72 − 80
−2
T (8, 4) − T (6, 4)
2
=
86 − 80
2
=3,
=4. Averaging these values, we estimate T x(6, 4) to be approximately
3.5 ◦ T (6, 4+h) − T (6, 4)
C/m. Similarly, T y (6, 4) = lim
, which we can approximate with h = ±2:
h→0 h
T y (6, 4) ≈
T (6, 6) − T (6, 4)
2
=
75 − 80
2
= −2.5, T y (6, 4) ≈
values, we estimate T y (6, 4) to be approximately −3.0 ◦ C/m.
(b) Here u =
√
1
2
,
T (6, 2) − T (6, 4)
−2
=
87 − 80
−2
= −3.5. Averagingthese
√
1
2
, so by Equation 15.6.9 [ ET 14.6.9], D u T (6, 4) = ∇T (6, 4) · u = T x(6, 4) √ 1
2
+ T y(6, 4) √ 1 2
.
Using our estimates from part (a), we have D u T (6, 4) ≈ (3.5) √ 1
2
+(−3.0) √ 1
2
= 1
2 √ ≈ 0.35. This means that as we
2
move through the point (6, 4) in the direction of u, the temperature increases at a rate of approximately 0.35 ◦ C/m.
T 6+h √ 1
2
, 4+h √ 1
2
− T (6, 4)
Alternatively, we can use Definition 15.6.2 [ ET 14.6.2]: D u T (6, 4) = lim
,
h→0 h
which we can estimate with h = ±2 √ 2.ThenD u T (6, 4) ≈
D u T (6, 4) ≈
(c) T xy(x, y) = ∂ ∂y
T (4, 2) − T (6, 4)
−2 √ 2
=
[Tx(x, y)] = lim
h→0
T (8, 6) − T (6, 4)
2 √ 2
=
80 − 80
2 √ 2
=0,
74 − 80
−2 √ 2 = 3 √
2
. Averaging these values, we have D u T (6, 4) ≈ 3
2 √ 2 ≈ 1.1◦ C/m.
T x(x, y + h) − T x(x, y)
T x(6, 4+h) − T x(6, 4)
,soT xy(6, 4) = lim
which we can
h
h→0 h
estimate with h = ±2. WehaveT x(6, 4) ≈ 3.5 from part (a), but we will also need values for T x(6, 6) and T x(6, 2). Ifwe
use h = ±2 and the values given in the table, we have
T x(6, 6) ≈
T (8, 6) − T (6, 6)
2
=
80 − 75
2
=2.5, T x(6, 6) ≈
Averaging these values, we estimate T x(6, 6) ≈ 3.0. Similarly,
T (4, 6) − T (6, 6)
−2
=
68 − 75
−2
=3.5.