g ( t ) = 3000 - 2500 * e ^{-0.5*t}
a.)
f ( t ) = N0 * e ^{k*t}
f ( 0 ) = 250 = N0
f ( 1 ) = 250 * e ^{k*1} = 412
e ^{k*1} = 412 / 250 | ln
k * 1 = ln ( 412 / 250 )
k = 0.5
f ( t ) = 250 * e ^{0.5*t}
b.)
g ( 0 ) = 3000 - 2500 * e ^{-0.5*0};
g ( 0 ) = 500
lim t −> ∞ [ 3000 - 2500 * e ^{-0.5*t} ] = 3000
c.)
g ( t ) = 3000 - 2500 * e ^{-0.5*t} = 2500
3000 - 2500 * e ^{-0.5*t} = 2500
3000 - 2500 = 2500 * e ^{-0.5*t}
e ^{-0.5*t} = 0.2 | ln
-0.5 * t = ln ( 0.2 )
t = 3.22
d.)
f ´( t ) = 250 * e ^{0.5*t} * 0.5
f ´( t ) = 125 * e ^{0.5*t}
g ´( t ) = - 2500 * e ^{-0.5*t} * -0.5
g ´( t ) = 1250 * e ^{-0.5*t}
f ´( t ) = g ´( t )
125 * e ^{0.5*t} = 1250 * e ^{-0.5*t}
e ^{0.5*t} / e ^{-0.5*t} = 1250 / 125
e ^{0.5*t} * e ^{0.5*t} = 10
e ^{ 1*t } = 10 | ln
t = ln ( 10)
t = 2.3
mfg Georg