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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ab=0,}   then  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a=0}   or  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b=0.}

Solution:

Step 1:
First, we start by graphing  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(N).}
Insert graph

Step 2:
Now, we want to solve
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0=3N{\bigg (}1-{\frac {N}{20}}{\bigg )}.}
Then, we have  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3N=0}   or  $1-{\frac {N}{20}}=0.$
This means either  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N=0}   or  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N=20.}
Therefore, the equilibria occur at
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N=0}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N=20.}

Final Answer:
See Step 1 for graph
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N=0}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N=20}

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