Question

the population p, in thousands, of a small city is given by the following function, where t is time in years. answer parts a) through c). p(t)=\frac{500t}{2t^{2}+9} a) find the growth rate. the growth rate is

Explanation:

Step1: Apply quotient - rule for differentiation

Quotient rule: If $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$, where $u = 500t$, $u'=500$, $v = 2t^{2}+9$, $v' = 4t$.

Step2: Calculate the derivative

$P'(t)=\frac{500(2t^{2}+9)-500t\times4t}{(2t^{2}+9)^{2}}=\frac{4500 - 1000t^{2}}{(2t^{2}+9)^{2}}$

Answer:

$\frac{4500 - 1000t^{2}}{(2t^{2}+9)^{2}}$