Thomas Calculus 13th [Solutions]

56 Chapter 1 Functions
21. (a) y = 100,000 10,000x, 0 x 10
(b) y 55,000
100,000 10,000x
55,000
10,000x
x
55,000
4.5
The value is $55,000 after 4.5 years.
22. (a) f(0) = 90 units
(b) f(2) = 90 52 ln 3 32.8722 units
(c)
23.
t t 5000 10 t 10 10 ln(10/3)
1500(1.08) 5000 1.08 ln(1.08) ln t ln1.08 ln t 15.6439
1500 3 3 3 ln1.08
It will take about 15.6439 years. (If the bank only pays interest at the end of the year, it will take 16 years.)
rt
rt rt
24. A( t) A0
e ; A( t) 2A0 2A0 A0
e e 2 rt = ln 2 t ln 2 t 0.7 70 70
r r 100 r ( r%)
25.
x
( x ) x
x x x 2
ln x x ln x and ln( x ) x ln( x ) x ln x ; then,
Therefore,
x
( x ) x x
x ( x ) when x = 2.
x 2 x 2
x ln x x ln x x x x ln x 2ln x x 2.
26. (a) No, there are two intersections: one at x = 2
and the other at x = 4.
(b) Yes, because there is only one intersection.
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