Question

The population density of a destructive caterpillar can be approximated by the equation D(t)=t^2/90+t/3 caterpillars per plant, where t represents the number of days since the initial infestation.

a. Using the correct units find and explain the meaning of D(30)

b. Using the correct units, explain the meaning of D'(30)

Would really like a thorough explanation of the problem, I want to understand it, not just get the answer. Thank you very much

Answer 1

a. 30 days after the initial infestation, the density of destructive catepillars per plant is
D(30)=30^2/90+30/3=20 catepillars per plant

Answer 2

The first derivative D'(x) would be D'(x)=t/45 + 1/3. When you evaluate the derivative at an x-value, one interpretation is whether the function is increasing, decreasing or neither. In this case D'(30)=30/45+15/45=1, which is positive. This implies that the given density curve is increasing 30 days after initial infestation. That means that the catepillar population is still growing.

Answer 3

@mengheng: are you certain that the function D(x) did not have a negative t^2 term?

Answer 4

I am sure, thank you for your reply. I'm going to try and work this one out again.

Answer 5

The first derivative can also be interpreted as the instantaneous rate of change of caterpillar growth at 30 days.

Answer 6

One more quick question, how would you take the derivative of D(t)=t^2/90+t/3
I know I'm suppose to use f'(x)=(X+h)-f(x)/h

Answer 7

You are very early in your calculus class; the method I used was "simple power rule." If your function is\[y=ax^n\] then the derivative is\[y'=nax ^{n-1}\]

Answer 8

Yup, I remembered my power rule from last year and it worked out :) thank you very much for your time!

Answer 9

You can apply this to each of the polynomial terms in your original function D(t).\[d'(t)=\frac{2}{90}t ^{2-1}+(1)\frac{1}{3}t ^{1-1}=\frac{t}{45}+\frac{1}{3}\]