Find the logistic function f with the given properties: f(0) = 3, f has limiting value 27, and for small values of x , f is approximately exponential and grows by 50% with every increase of 1 in x.

f(x) = ______________

Question

Find the logistic function f with the given properties: f(0) = 3, f has limiting value 27, and for small values of x , f is approximately exponential and grows by 50% with every increase of 1 in x.

f(x) = ______________

dumbcow · Answer

i believe it would take the form
$$f(x) = \frac{27}{1+9(b^{x})}$$
just need to find what b is

dumbcow · Answer

wait. oops that 9 should be an 8

dumbcow · Answer

also 0<b<1

dumbcow · Answer

when they say small values of x, do they mean close to 0?
like .01, .03 ...

anonymous · Answer

I believe so

dumbcow · Answer

in that case its saying it models the exponential function (1.5)^x for small x
i guess try x=0.1 and 0.2 and find a "b" that matches the increase from 1.5^.1 to 1.5^.2

anonymous · Answer

arent we just trying to find a function? so we are not solving it?

dumbcow · Answer

a simpler way to look at it is that approximately f(2) = 1.5f(1) and f(3) = 1.5f(2) these are still relatively small x values $$\frac{27}{1+8b^{2}} =1.5 \frac{27}{1+8b}$$ $$\rightarrow 12b^{2}-8b+0.5 = 0$$ $$b \approx 0.07or 0.6$$ however, 0.6 works better for other x_values when b=0.07, f(3)=1.036*f(2) when b=0.6, f(3)=1.42*f(2) Therefore, best estimate for f(x) is : $$f(x) = \frac{27}{1+8(0.6)^{x}}$$ here is the graph http://www.wolframalpha.com/input/?i=y+%3D+27%2F%281%2B8%28.6^x%29%29+from+x%3D-5+to+15

dumbcow · Answer

here is another graph showing how for x<5 it models the exponential function 3(1.5^x) http://www.wolframalpha.com/input/?i=plot+27%2F%281%2B8%28.6^x%29%29%2C+3*1.5^x+from+x%3D-3+to+8