Populations starting out in closed emironments grow slowly at first when there a
ID: 2854745 • Letter: P
Question
Populations starting out in closed emironments grow slowly at first when there are relatively few members, then more rapidly as the number of reproducing individuals increases and resources are still abundant then slowly again as the population reaches the carrying capacity of the environment Answer pans a and b below. Use graphical techniques to graph the derivative of the fruit fly population, using the given graph of the population. Choose the correct graph of the derivative. During what days does the population seem to be increasing fastest? Slowest? Fill in the blanks. The population seems to increase fastest during daysExplanation / Answer
a) The answer for the a part is the option A.
As we know that the derivative of a function represents the slope of tangent line to that function. Using this principle if we draw a tangent line at the starting point of the graph it will be more or less a horizontal line for which the slope will be close to zero. So, the graph of our derivative should begin from near about zero. This conclusion give us the opportunity to neglect the graphs given in options (B) & (C) as they are not starting from anywhere near zero.
We can also see from our given graph that the slope of tangent line to the graph will first increase lets say from 5 to 35 days and then it will decrease from 35 to 50 days, that means the slope will be highest at around 35 days, which is approximately equal to 3 (Slope=tan(x), if x is approximately equals to 71 degrees at 35 days time then Slope=3). So, this means the highest value of slope can only be approximately equals to 3 which leaves us only with the option (A) as the most suitable graph of the derivative of given graph.
b) The days during which the polpulation is increasing fastest are from 20 to 35 because during these day the slope of tangent is highest and days from 5 to 20 & 35 to 50 are the days of slowest population increase.