急!! exponential function

1. (A) The rate of depreciation = d[L(t)]/dt

L(t) = 0.506 (1.073)^t
(Take natural logarithm on both sides)
ln[L(t)] = ln [0.506 (1.073)^t]
______= ln 0.506 + t*ln 1.073
(Differentiate both sides w.r.t. t)
____d{ln[L(t)]}/dt = ln1.073
d[L(t)]/dt*[1/L(t)] = ln1.073
______d[L(t)]/dt = ln 1.073 * L(t)
_____________= (0.506)(ln 1.073)(.073)^t

That means : The rate of depreciation increases with a constant rate
of ln 1.073 to the number of million hectares of forest cut worldwide
for logging.

(B) In 1997, the number of hectares of forest cut worldwide for logging :
L(1997 - 2005) = 0.506 (1.073)^(1997 - 2005)
_______L(- 8) = 0.287973643 (million hectares)

Then, the percentage of the world's rain forest that was cut for
logging that year :
(0.287973643 / 647)*100% = 0.044509063%


2. a function f(x) is always decreasing <=> d[f(x)]/dx < 0 for all x in real number
So, try f(x) = ke^(-x) where d[f(x)]/dx = - ke^(-x) < 0 for all x in real number
f(0) = 12 <=> 12 = k*1 <=> k = 12
Hence, f(x) = 12e^(-x) is one of the wanted function.