Saturation Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard

Identify the four parameters

L

k

x_0

f(x) = \dfrac{200}{1+9e^{-0.4x}}

and describe the qualitative behavior of

f

Solution

1
Rewrite in standard form $\frac{L}{1+e^{-k(x-x_0)}}$ : $\frac{200}{1+9e^{-0.4x}} = \frac{200}{1+e^{\ln9}e^{-0.4x}} = \frac{200}{1+e^{-0.4(x-\ln9/0.4)}}$ .
2
So $L=200$ , $k=0.4$ , $x_0=\frac{\ln9}{0.4}=\frac{2.197}{0.4}\approx5.49$ .
3
Behavior: starts near $f(0)=200/(1+9)=20$ ; grows through inflection point at $x\approx5.49$ where $f\approx100$ ; saturates at $L=200$ .

Answer

L=200

k=0.4

x_0\approx5.49

; saturates at

200

Matching the non-standard form to the standard form requires absorbing the coefficient of

e

into

e^{-k \cdot x_0}

. The parameters control maximum (

L

), steepness (

k

), and midpoint (

x_0

) of the S-curve.

About Saturation

Saturation is the phenomenon where a growing quantity approaches a limiting value asymptotically, with the rate of growth decreasing as the limit is approached.

Learn more about Saturation →

More Saturation Examples

Example 1 medium

The logistic function [formula] models population growth. Find [formula], [formula], and [formula].

Example 3 easy

For [formula], state the saturation level and the value at [formula]. What does the saturation repre

Example 4 medium

A bacteria culture saturates at [formula] cells. At time [formula] there are [formula] cells. Write

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