# Euler Car Depreciation Calculus

• Apr 10th 2014, 09:32 PM
bardstavern
Euler Car Depreciation Calculus
Say a car's value drops 20% annually. In ﬁve years estimate
the percentage of the original vehicle value that is retained.

Assume you bought this vehicle at the time t = 0 (assume
the mileage of x = 0 km) for \$ 20,000. Assume you drive about
15,000 km annualy (assume uniform driving habits over the year).
Let the input variable x be mileage of the car in km and the
output variable y = f(x) be the value of the car in dollars.

i. Find an expression for the function f(x) in terms of x.

The both equations need to be exponential, but I am having a difficult time figuring it out
• Apr 11th 2014, 01:47 AM
romsek
Re: Euler Car Depreciation Calculus
well the underlying variable that relates x and f(x) is time.

let \$t\$ be the years owned

\$val(t)=20000(1-0.2)^t=20000(0.8)^t\$

\$mileage(t)=15000 t\$

let \$x=mileage(t)\$

\$x=15000 t\$

\$t = \dfrac{x}{15000}\$

so

\$f(x) = val(t(x))=20000(0.8)^{x/15000}\$