Thomas Calculus 13th [Solutions]

Section 3.11 Linearization and Differentials 197
62.
4
3
8 0.06t
(1
3 2
t )
C ( t) 0.06 e , where t is measured in hours. When the time changes from 20 min to 30 min, t in
hours changes from 1 1
3
to
2
, so the differential estimate for the change in C is
1 1 1 1 1
C
3 2 3 6
C
3
0.584 mg/mL.
3
63. The relative change in V is estimated by
dV / dr 4kr 4 r
V
r 4 r r . If the radius increases by 10%, r changes to
kr
1.1r and r 0.1 r . The approximate relative increase in V is thus 4(0.1 r
r
) 0.4 or 40%.
1/2
3/2 3/2
64. (a) T 2 L dT 2 L 1 g dg Lg dg
g
2
(b) If g increases, then dg 0 dT 0. The period T decreases and the clock ticks more frequently. Both the
pendulum speed and clock speed increase.
3/2
2
2
(c) 0.001 100(980 ) dg dg 0.977 cm/sec the new g 979 cm/sec
65. (a) i. Q( a) f ( a)
implies that b 0 f ( a).
ii. Since Q ( x) b1 2 b2
( x a), Q ( a) f ( a ) implies that b 1 f ( a).
f ( a)
iii. Since Q ( x) 2 b2
, Q ( a) f ( a ) implies that b2 .
2
f ( a)
In summary, b 0 f ( a),
b 1 f ( a ), and b2 .
2
1
2 2
3 3
(b) f ( x) (1 x) ; f ( x)
1(1 x) ( 1) (1 x) ; f ( x)
2(1 x) ( 1) 2(1 x)
Since
f (0) 1, f (0) 1, and f (0) 2, the coefficients are b 2
0 1, b1 1, b 2 1. The quadratic
2
2
approximation is Q( x) 1 x x .
(c)
As one zooms in, the two graphs quickly become
indistinguishable. They appear to be identical.
(d)
1
2 3
g( x) x ; g ( x)
1 x ; g ( x) 2x
Since g(1) 1, g (1) 1, and g (1) 2, the coefficients are b 2
0 1, b1 1, b 2 1. The quadratic
2
2
approximation is Q( x) 1 ( x 1) ( x 1) .
As one zooms in, the two graphs quickly become
indistinguishable. They appear to be identical.
(e)
1/2
1/2 3/2
h( x) (1 x) ; h ( x)
1 (1 x) ; h ( x) 1 (1 x)
2 4
1
Since h(0) 1, h (0) 1 1
4
, and h (0) , the coefficients are b 1 1
2 4
0 1, b1 , b
2 2 . The quadratic
2 8
2
approximation is Q( x)
1 x x .
2 8
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