I have no clue how to even begin this problem..
Okay, so the function:
satisfies this logistic model:
For which value of t is P'(t) maximum? What is the value of P for this t?
For this problem, this function is used to model the spread of a disease where P is the proportion of people infected. Each person infects, on average, k other people.
Now, I understand that the first function is the integral of the second. But I'm so confused when it comes to understanding what exactly I need to derive. Can someone please help me walk through this?
Thank you so much! Could you explain to me what happens to the negative sign from the in the second derivative?
Could someone help me with graphing P(t) and P'(t)?
If I take A = 50 and k = 2, would my domain be [0, infinity) for P(t) and P'(t)?
=======P(t)========
For intercepts I got none because for
it would be is that correct?
Also, I realize that eventually all values of t go to P(t) = 1, but I don't know how to express that because it's not quite an asymptote, is it?
For the intervals of increase/decrease, I set
but I'm not able to calculate any actual values. I know it's increasing. I can't really find the max/min either...
For concavity and inflection, the value I got for the concave down interval, (0, 1.96), does not match up with the graph on my calculator.
=========P'(t)========
Domain: [0, infinity]
Intercepts: none
Asymptotes: as clueless as I am for P(t)
Intervals Inc/Dec: decrease (0, difficulties calculating)
Max/min: min(??,0) max(.5,1)
Concavity/inflection: do I need to calculate P'''(t) to find this one?
...ahh. help please.
I got the graph for P(t) off my calculator.. but I have to show my professor that I calculated everything out by hand for both, and that's where I'm running in to many of my problems. I have to show the exact values for:
domain
intercepts
symmetry
asymptotes
intervals of incr/decreasing
local max/min
concavity/inflection
I know how to do these with most functions, but I'm having a hard time calculating them for P(t) and P'(t).
For example, if the function can achieve the value of P(t) = 1, then 1 can't be an asymptote, can it?
I am also having trouble with the calculations of incr/decr, max/min, conc/infl pts.
Oh, and the same values for k and A. k = 2, A = 50.