007A Sample Midterm 2, Problem 4 Detailed Solution

Assume  $N(t)$  denotes the size of a population at time  $t$  and that  $N(t)$  satisfies the equation:

${\frac {dN}{dt}}=3N{\bigg (}1-{\frac {N}{20}}{\bigg )}.$

Let  $f(N)=3N{\bigg (}1-{\frac {N}{20}}{\bigg )},~N\geq 0.$  Graph  $f(N)$  as a function of  $N$  and identify all equilibria. That is, all points where  ${\frac {dN}{dt}}=0.$

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Background Information:
If  $ab=0,$  then  $a=0$  or  $b=0.$

Solution:

Step 1:
First, we start by graphing  $f(N).$
Insert graph

Step 2:
Now, we want to solve
$0=3N{\bigg (}1-{\frac {N}{20}}{\bigg )}.$
${\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {{\frac {1}{2}}(3x^{2}+5x-7)^{-{\frac {1}{2}}}(3x^{2}+5x-7)'}\\&&\\&=&\displaystyle {{\frac {1}{2}}(3x^{2}+5x-7)^{-{\frac {1}{2}}}(6x+5).}\end{array}}$

Final Answer:
See Step 1 for graph
$N=0$  and  $N=20$

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