Using the following formula, I have to calculate the rate of growth of a population of rabbits after 3 years. The formula is
P(t)=500/1+e^-t
there are no brackets
I know I can't use my calculator to solve for e when the exponent is negative so I am guessing I have to some how alter the equation in order to solve it. If anyone can give me an idea of what to do to solve for e, I would really appreciate it. Thank you
I'm still not clear on what you're trying to do, but the fact is that you can use negative exponents in what you just wrote. Do you have an e button on your calculator? If not, use the approximation I gave you in my first post.
Just do the arithmetic. Comes out to about 525. Does that make sense?
I get the impression you're being asked to find . From what I can gather the question would read
In this case sub 3 for t:
.
This can be evaluated directly on your calculator or with google/wolfram.
I get an answer of
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is a constant, much like or . It is the base of the natural logarithm ( ). It's unique in calculus for being the only function equal to it's derivative.
As is a constant the normal rules of exponents apply - .
what the question is asking for is the rate of growth, so in order for me to find that i need to first find the derivative which I believe can be found if I alter the equation to become P(t)=500(1+e^-t)^-1 , the thing is I don't really know how to find the derivative for this equation since its exponential but it also contains e which makes it tricky, I know I can use a combination of the chain rule for exponential functions as well as the product rule, but I can't really figure it out using e.
I agree. The rate of population growth is the derivative of the population equation:
can be rewritten as
. Using the chain rule we get
, and simplifying gives
or
.
Plugging 3 in makes the resultant growth rate 22.588.
The rules for differentiating are very simple, since (which means is its own derivative and all you have to worry about is the chain rule). For example, the derivative of is and the derivative of is .
One comment though - shouldn't this be in the calculus forum and not the precalc forum?
Actually, is a standard form for logistic growth, not exponential growth. The function behaves exponentially initially, but growth eventually slows and becomes asymptotic. This is the way many populations actually grow since increasing growth rates associated with true exponential growth usually cannot be sustained (i.e. some environmental factor like lack of food, water, or space will eventually limit growth).