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The graph of a function $p(t)$, where $p$ is the population of a group of foxes after $t$ months is shown above.

a. Estimate $p(20)$ and example what this quantity represents in the context of this problem.

b. For $h=10$, estimate the value of $\frac{p(20+h)-p(20)}{h}$ and explain what this quantity represents in the context of this problem.

c. Estimate the average rate of change of the fox population during the first $20$ months. Include units to your answer.

d. What can you say about the growth rate of the fox population during the first $80$ months?

For a., I know that the value is approximately $340$, and it basically means that after $20$ months, there are about $340$ foxes in the population.

For d., I think that the explanation goes somewhere along the lines of the following: The fox population increases until the rate flattens out at $1000$.

My actual question: Can someone please explain to me all the parts of this question (especially parts b. and c.)?

I really appreciate the efforts of anyone who tries to help me understand this problem overall. Thanks! :)


Your answer for part (a) is wholly correct, so I'll not delve into it.

For part (b): $p(20)=340$, $p(20+h)=p(30)=540$, $\frac{p(20+h)-p(20)}{h}=\frac{540-340}{10}=20$. Intuitively, this means that at t = 20 months the fox population is increasing by 20 per month (which is rather nice if you're a conservationist, eh?)

For part (c), take the endpoints of the interval in question and calculate the slope between them. $$\frac{p(20)-p(0)}{20-0}=\frac{340-100}{20-0}=12\text{ foxes/month}$$

For part (d), the fox population follows a logistic growth model: it starts out growing exponentially until t = 30 months, then starts saturating (slowing down) until it finally plateaus out at 1000 foxes, the carrying capacity of the ecosystem the foxes live in.

  • $\begingroup$ Cool, thanks a lot! @ParclyTaxel $\endgroup$ – 关一骏 Aug 19 '16 at 19:37

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