Expert: Abe Mantell - 8/11/2011

Question
I don't understand how to do problems like this:

q(t)=B-Ae^-kt

I have the notes from class so I know that

lim q(t)
t-> infinity =B

q'(t)= -Ae^-kt (-k) = KAe^-kt which is >0

But I don't understand why or how to these kinds of problems! How are we getting the derivative of variables that are not x. . .I'm just so lost and don't understand any of this, and my frustration is keeping me from wrapping my head around the problem!

Answer
Hello Brittney,

Q(t)=B-Ae^(-kt) is a standard exponential model
"t" is the independent variable while A, B, and k are constants which
will depend on the initial information for the particular problem.

As for the calculus aspect...
the limit as t-->infinity is B, as long as k>0, so that e^(-kt) becomes
exponentially smaller and approaches zero.  See now?

Q'(t)=(B)'-(Ae^(-kt))'
= 0 - Ae^(-kt)*(-kt)', from the chain-rule and treating A, B, and k as constants
= - Ae^(-kt)*(-k)
= kAe^(-kt), which is greather than zero when k>0 and A>0.

Does that help?

TTYL, Abe