If the instantaneous rate of change of a population is 50t2−100t32 (measured in individuals per year) and the initial population is 25000 then what is the population after t years?

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Ezakhenitne · Accepted Answer

To answer this question, you need to find a particular antiderivative.Let P(t) be the population t years after the initial population was counted. (So P(0)=25000)We are told that the population is changing at a rate -- stop! that means we&#039;re going to be told something about the derivative (the rate of change).The population is changing at a rate of 50t2−100t32This tells us thatP′(t)=50t2−100t32Our task is to find P(t)If f′(x)=xn, the f(x)=xn+1n+1So P(t)=50t33−100⋅25t52+CP(t)=503t3−40t52+CWe also know that P(0)=25000, so we can substitute to find C25000=503(0)3−40(0)52+C,soC=25000P(t)=503t3−40t52+25000Ithink writing all of the following clutters up the work:To find the antiderivative of t32t32+132+1=t5252=25t52(dividing is multiplying by the reciprocal.)

If the instantaneous rate of change of a population is

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