Population growth Consider the following population functions.
e.Use a graphing utility to graph the population and its growth rate.
p(t) = 600 (t²+3/t²+9)

Verified step by step guidance

1. Step 1: Identify the function for the population, which is given as \( p(t) = 600 \left( \frac{t^2 + 3}{t^2 + 9} \right) \). This function represents the population at time \( t \).
2. Step 2: To find the growth rate of the population, we need to compute the derivative of \( p(t) \) with respect to \( t \). This involves using the quotient rule for derivatives, which is \( \frac{d}{dt} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \), where \( u = t^2 + 3 \) and \( v = t^2 + 9 \).
3. Step 3: Calculate the derivatives \( u' \) and \( v' \). For \( u = t^2 + 3 \), \( u' = 2t \). For \( v = t^2 + 9 \), \( v' = 2t \).
4. Step 4: Substitute \( u, v, u', \) and \( v' \) into the quotient rule formula to find \( p'(t) \), the growth rate of the population. This gives \( p'(t) = 600 \left( \frac{(2t)(t^2 + 9) - (t^2 + 3)(2t)}{(t^2 + 9)^2} \right) \).
5. Step 5: Simplify the expression for \( p'(t) \) to obtain a more manageable form for graphing. This involves expanding and combining like terms in the numerator, and then simplifying the entire expression.