Aa(t) = e^{- a·t}·(t + 7.5) + 7.5 ; a > 0 und t > 0
a)
Aa'(t) = e^{- a·t}·(1 - a·(t + 7.5))
Aa''(t) = e^{- a·t}·(a^2·(t + 7.5) - 2·a)
Extrempunkt Aa'(t) = 0
e^{- a·t}·(1 - a·(t + 7.5)) = 0 --> t = 1/a - 7.5
Aa(1/a - 7.5) = e^{7.5·a - 1}/a + 7.5
Wendepunkt Aa''(t) = 0
e^{- a·t}·(a^2·(t + 7.5) - 2·a) = 0 --> t = 2/a - 7.5
b)
Ortskurve der Extrempunkte
1 - a·(t + 7.5) = 0 --> a = 1/(t + 7.5)
y = e^{- t/(t + 7.5)}·(t + 7.5) + 7.5
c)
lim (x --> ∞) e^{- a·t}·(t + 7.5) + 7.5 = e^{- a·∞}·(∞ + 7.5) + 7.5 = 0 + 7.5 = 7.5
