Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

l
CHAPTER 14 REVIEW 0 237
EXERCISES
1. ln(x + y + 1) is defined only when x + y + 1 > 0 ~ y > - x - 1,
so the domain off is {(x, y) I y > - x - 1}, all those points above the
line y = -x - 1.
',
' ' ' ' ' ' '
y
3. z = f(x, y) = 1 - y 2 , a parabolic cylinder 5. The level curves are J 4x2 + y2 = k or 4x 2 + y 2 = k 2 ,
k ::2::: 0, a family of e ll ipses.
y
X
7. y 9. f is a rational function, so it is continuous on its domain.
2
I
(©J (©J
(©J (©J
0 I 2 X
Since f is defined at (1, 1), we use direct substitution to
1 th I" . lim 2x y 2(1)(1) 2
eva uate e 1m1t:
2
2 2 = 12
+ ( ) 2 1 2 = - .
(z,v)-(1 ,1) X + y
3
11 . (a) T.,(6 , 4) = lim T(B + h , 4 ~ - T (B, 4 ), so we can approximateT:c (6, 4) by considering h = ± 2 and
h- o
using the values given in the table: Tx(6, 4) ~ T( 8 , 4 ) ; T ( 6 , 4 ) = 86 ;
80
= 3,
. T.,(6, 4) ~ T( 4 , 4 ) ~ 2 T(B, 4 ) = 72 ~ 2 80 = 4. Averaging these values, we estimate Tx(6, 4) to ~e approximately
3.'5°C/ m . Similarly, Tv (6, 4} = lim T(B, 4 + h~ - T ( 6 , 4 ) , which we can approximate with h = ±2:
h - 0
,.., (B ) _ T(6, 6} - T(6, 4) _ 75 80 _ _ ,.,., ( ) _ T(6, 2) - T(6, 4) _ 87- 80 _
4 2 5 6 4 3 5 A . th
.Lv ' - 2 - - - 2
- - · ' .L v , - _ 2
- ~- - .. veragmg ese
values, we estimate Tv(6, 4) to be approximately - 3.0°C/ m.
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