Solução_Calculo_Stewart_6e

F.
TX.10SECTION 3.10 LINEAR APPROXIMATIONS AND DIFFERENTIALS ¤ 125
(b) S =6x 2 ⇒ dS =12xdx.Whenx =30and dx =0.1, dS = 12(30)(0.1) = 36, so the maximum possible error in
computing the surface area of the cube is about 36 cm 2 .
Relative error = ∆S
S
≈ dS S = 12xdx =2 dx 0.1
6x 2 x =2 30
=0.006.
Percentage error = relative error × 100% = 0.006 × 100% = 0.6%.
35. (a)Forasphereofradiusr, the circumference is C =2πr and the surface area is S =4πr 2 ,so
r = C 2 C
2π ⇒ S =4π = C2
⇒ dS = 2 2π π
π CdC.WhenC =84and dC =0.5, dS = 2 84
(84)(0.5) =
π π ,
so the maximum error is about 84
π ≈ 27 cm2 . Relative error ≈ dS S = 84/π
84 2 /π = 1
84 ≈ 0.012
(b) V = 4 3 πr3 = 4 3 π C
2π
3
= C3
6π 2 ⇒ dV = 1
2π 2 C2 dC. WhenC =84and dC =0.5,
dV = 1
2π 2 (84)2 (0.5) = 1764
1764
, so the maximum error is about ≈ 179 cm 3 .
π2 π 2
The relative error is approximately dV V = 1764/π2
(84) 3 /(6π 2 ) = 1 56 ≈ 0.018.
37. (a) V = πr 2 h ⇒ ∆V ≈ dV =2πrh dr =2πrh ∆r
(b) The error is
∆V − dV =[π(r + ∆r) 2 h − πr 2 h] − 2πrh ∆r = πr 2 h +2πrh ∆r + π(∆r) 2 h − πr 2 h − 2πrh ∆r = π(∆r) 2 h.
39. V = RI ⇒ I = V R ⇒ dI = − V ∆I
dR. The relative error in calculating I is ≈ dI
R2 I I = −(V/R2 ) dR
= − dR V/R R .
Hence, the relative error in calculating I is approximately the same (in magnitude) as the relative error in R.
41. (a) dc = dc dx =0dx =0
dx
(c) d(u + v) = d
du
dx (u + v) dx = dx + dv
dx
(d) d(uv) = d
dx (uv) dx = u dv
dx + v du
dx
u
(e) d = d u
v du
dx = dx − u dv
dx dx =
v dx v
v 2
(f ) d (x n )= d
dx (xn ) dx = nx n−1 dx
dx = du dv
dx + dx = du + dv
dx dx
dx = u dv du
dx + v dx = udv+ vdu
dx dx
v du dv
dx − u
dx dx dx vdu− udv
=
v 2
v 2
43. (a) The graph shows that f 0 (1) = 2,soL(x) =f(1) + f 0 (1)(x − 1) = 5 + 2(x − 1) = 2x +3.
f(0.9) ≈ L(0.9) = 4.8 and f(1.1) ≈ L(1.1) = 5.2.
d
du
(b) d(cu) = (cu) dx = c dx = cdu
dx dx
(b) From the graph, we see that f 0 (x) is positive and decreasing. This means that the slopes of the tangent lines are positive,
but the tangents are becoming less steep. So the tangent lines lie above the curve. Thus, the estimates in part (a) are too
large.