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Calculus helppppp ??

A photo of MariamGuled MariamGuled
Hi guys, I was wondering if anyone could help me with these questions?

The number of rabbits in a forest at time, 't', in months is : R(t) = 500(10+e^-t/10)

(a) determine the rate of change of the number of rabbits at time 't'
(b) how fast is the number of rabbits changing over one year
(c) determine the largest number of rabbits in the forest during the first three years

any help will be appreciated!
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2 replies
A photo of Ba Ba Blue Ba Ba Blue
a) This is just simple derivative. Time 't' is just referring to the fact that you won't have to substitute a value once you find the derivative. Keep it as t.

b) "Over one year" tells you that it is not instantaneous. A period of 1 year stretches from t=0 to t=12 since t is represented in months. Instead of using the derivative, you should use the slope formula to find the slope of the line passing through R(0) and R(12).

c) What you want to do here is maximize R(t) because that's their population. To do this, simply take your result from part (a) and set it equal to 0 (because this will tell you at which points you have a local minimum and a local maximum. You should check to see if the derivative is positive and negative on either side of the value. Positive to the right and negative to the left means its a minimum (NOT what you want), whereas the opposite is also true (positive on left and negative on right is a maximum). Remember, you've been given a closed interval so you have to check the values of R(t) at t=0 and t=12 also regardless of what the derivative is at those points to make sure the end points aren't the maximum. Also, if you get t=37 that is WRONG, as is t=-1. The question is clearly saying t is between 0 and 12 or is equal to those values.
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A photo of MariamGuled MariamGuled
okay thanks that helped alottt ! :)
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