Thomas Calculus 13th [Solutions]

Section 14.6 Tangent Planes and Differentials 1021
y y
y y
49. Tx
( x, y)
e e and Ty ( x, y) x e e dT Tx ( x, y) dx Ty
( x, y)
dy
y y y y
e e dx x e e dy dT | (2, ln 2) 2.5 dx 3.0 dy . If | dx | 0.1 and | dy | 0.02, then the
maximum possible error in the computed value of T is (2.5)(0.1) (3.0)(0.02) 0.31 in magnitude.
50. Vr
2
rh and V 2 dV 2 rh dr r dh 2 1
h r dV V r dr V h dh ;
V 2
r h r dr h
dh now dr
r
100 1 and
dr 100 1 dV 100 2 dr (100) dh (100) 2 dr 100 dh 100 2(1) 1 3 3%
h V r h r h
2
51. dx dy
0.02, 0.03
x y
(a)
S 2 2 2
2 x 4 xy dS (4 x 4 y ) dx 4 x dy dy
4 x 4 xy dx 4 4 4 (0.02) (4 )(0.03)
x
xy y
x xy xy
2 2 2
0.04 2x 0.05(4 xy) 0.05 2x 0.05(4 xy) (0.05) 2x 4xy 0.05S
(b)
V x 2 y dV 2 2 2 2 2 2
2 xy dx x dy dy
2 x y dx
2 (0.02) (0.03) 0.07 0.07
x
x y y
x y x y x y V
52.
V 4 3 2 2 2
r r h dV 4 r 2 rh dr r dh;
r 10, h 15, dr 1 and dh
3
2
0
2 1 2 3
dV 4 (10) 2 (10)(15) (10) (0) 350 cm
2
53. Vr
2
2 2
rh and Vh r dV Vr dr Vh
dh dV 2 rh dr r dh dV 120 dr 25 dh;
(5,12)
| dr | 0.1 cm and
3
| dh| 0.1 cm dV (120 )(0.1) (25 )(0.1) 14.5 cm ;
maximum percentage error is 14.5
300
100 4.83%
2 2
54. (a) 1 1 1 1 dR 1 dR 1
2 2 1 dR
2 2 dR R dR R
R R 1 dR2
1 R2 R R R
R1 R2
1 2
2
2 1 1 2
dR R dR 1 1
1 dR2 dR (100, 400) R dR1 dR2
(b) |
2 2 2 2
R1 R2
(100) (400)
R will be more sensitive to a variation in R 1 1
1 since
2 2
(100) (400)
3
V (5,12) 300 cm
2 2
(c) From part (a), dR R
R
R
dR 1 R
dR 2 so that R changing from 20 to 20.1 ohms
1
1 2
dR 1 0.1 ohm and R 2 changing from 25 to 24.9 ohms
R
100
2
100
2
9 9
2 2
100 ohms dR| 9 (20, 25) (0.1) ( 0.1) 0.011 ohms
(20) (25)
percentage change is dR 100 0.011 100 0.1%
R (20, 25)
100
9
dR 1 1 1
2 0.1 ohms; R R R
1 2
55. ;
A xy dA x dy y dx if x y then a 1-unit change in y gives a greater change in dA than a 1-unit
change in x. Thus, pay more attention to y which is the smaller of the two dimensions.
Copyright
2014 Pearson Education, Inc.