Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 10 (a)/Solution 1

Let us work out the important features of the graph as indicated by the hints.

From the factorized right hand side of the equation, we see that ${\textstyle {\dfrac {dP}{dt}}=0}$ exactly when ${\textstyle P=0}$, ${\textstyle 1-{\dfrac {P}{T}}=0}$ or ${\textstyle 1-{\dfrac {P}{K}}=0}$, that is, when ${\textstyle P=0}$, ${\textstyle T}$ or ${\textstyle K}$.

Observe also that for ${\textstyle P>K}$, we have $P>0$, $1-{\dfrac {P}{T}}<0$, and $1-{\dfrac {P}{K}}<0$. This implies that ${\textstyle {\dfrac {dP}{dt}}<0}$ for all large ${\textstyle P}$.

We gather the above information in the following graph.

Q10a solution figure

Answer: Please see the figure.