# Calculus

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Could someone explain this question please.

Suppose the population (in thousands of a certain species of insect after t months is described by the function p(t)=4t+ (100t)/(t^2+400) + 500. Determine the maximum population in the first three months.

• Calculus -

when x is small (as in the interval [0,3]

100t/(t^2+400) is just basically t/4
so, p(t) looks a lot like 4t + t/4 + 500
that's just a line, so the max and min are at the ends of the interval.

• Calculus -

If I factor it out, it'll be 17t/4 + 500.

then plug in 3

17(3)/4+500=512 x 1000

=512,750.

therefore the population would be 512,750?

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