1. ## Differentiation

4. Consider the logistic equation for a population at time t given by

N(t) = 500/(1+9e^(-5t))

(a) Determine the initial size of the population at time
t = 0.

(b) Show that N(t) is strictly increasing.

(c) Determine the carrying capacity of this population (i.e.limt→∞ N(t).

(d) Determine the point in time at which the population reaches half of its carrying capacity. Show thatN(t) has an inflection point at this point.

(e) From the information determined in this question, graphthe function N(t).

a) 50

For b, would I find the first derivative, find the critical points and use an interval chart to show that the function is always increasing? When the derivative is equated to 0 it cannot be solved, to isolate t and get rid of e the ln of both sides is taken which results in an error b/c ln 0 doesn't exist.

derivative is: 22500e^5t/(1+9e^5t)^2

I know how to do c.

For d, to find the time do I use the first equation and let it equal to half the carrying capacity found in part c and do I just need to find the second derivative to show that there is an inflection point?

The sheet doesn't provide any answers so I just want to make sure that I am doing this question right.

2. b. WAY too complicated! Just look at the derivative.

----- $t \ge 0$ by definition.

----- Denominator > 0 by inspection.

----- Done! The 1st Derivative is ALWAYS non-negative. It is zero only for t = 0.

Don't get caught up into thinking there is only one way to proceed. Unique solutions don't care how you achieve them!

3. Originally Posted by as_xoxo
4. Consider the logistic equation for a population at time t given by

N(t) = 500/(1+9e^(-5t))

(d) Determine the point in time at which the population reaches half of its carrying capacity. Show that N(t) has an inflection point at this point.

For d, to find the time do I use the first equation and let it equal to half the carrying capacity found in part c and do I just need to find the second derivative to show that there is an inflection point?
Yes, that is the plan. To find when N(t) is half, set the denominator to 2.

I think you have a typo when you showed the first derivative .. should be e^(-5t) in two places.

To find when the 2nd derivative is zero, you only have to look at its numerator.