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

SECTION 14.3 PARTIAL DERIVATIVES 0 195
sin(xy)
43. f(x, y) = xy
{
1
if (x, y) :f. (0, 0)
if (x, y) = (0, 0)
From the graph, it appears that f is continuous everywhere. We know
xy is continuous on R 2 .and sin t is continuous everywhere, so
.
( )
R2 d sin(:cy) . . R2
sin xy is contmuous on an --- IS contmuous on
x y
-1
except possibly where x y = 0. To show that f is continuous at those points, consider any point (a, b) in R 2 where ab = 0.
Because x y is continuous, x y --+ ab == 6 as {x, y ) --+ (a, b). If we lett= x y, then t--+ 0 as (x, y) --+ (a, b) and
lim sin(xy). = Jim sin(t) = l by Equation 2.4.2 [ET 3.3.2). Thus lim f (x, y) = f(a, b) and f is continuous
(x,y)-(a,b) x y t-o t (:z:,y)-(a,b)
on R 2 •
45. since Jx - aJ 2 = lxl 2 + Jal 2 . - 2JxJ ial cos O ~ Jxl 2 + Jal 2 - 2Jx.JJal = (lxJ -Ial) 2 , we have llxl - lall $ lx- aJ. Let
t > 0 be given and set o = t. Then ifO < lx - al < o, llxJ- IaJI $ Jx - aJ < o = t . Hence limx-a Jxl = Jal and
f (x) = Jxl is continuous on R".
14.3 Partial Derivatives
1. (a) 8T I 8x represents the rate of change ofT when we fi x y and t and consider T as a function of the single variable x, which
describes how quickly the temperature changes when longitude changes but latitude and time are constant. 8T 18y
represents the rate of change of T when we fix x and t and consider T as a function of.y, which describes how quickly the
temperature changes when latitude changes but longitude and time are constant. 8T I 8t represents the rate of change ofT
when we fix x andy and consider T as a function oft , which describes how quickly the temperature changes over time for
a constant longitude and latitude.
(b) f .,(158, 21, 9t represents the rate of change of temperature at longitude 158°W, latitude 21 °N at 9:00AM when only
longitude varies. Since the air is warmer to the west than to the east, increasing longitude results in an increased air
temperature, so we would expect .f:r: (158, 21, 9) to be positive. / 11 (158, 21 , 9) represents the rate of change oftemperature
at the same time and location when only latitude varies. Since the air is warmer to the south and cooler to the north,
increasing latitude results in a decreased air temperature, so we would expect / 11 (158, 21, 9) to be negative. ft(158, 21, 9)
represents the rate of change of temperature at the same time and location when onJy time varies. Since typically air
temperature increases from the morning to the afternoon as the sun warms it, we would expect f t(158, 21, 9) to be
positive.
3. a By De tlon
. 4
,
f ( 15 30) lim f(- 15 + h,30) - f(- 15,30) h' h . b 'd . h
T - , = h , w 1c we can approx1mate y cons1 ermg = 5
() fini
h-0
and h = -5 and using the values given in the table:
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