Question

Spruce budworms are a major pest that defoliates balsam fir. They are preyed upon by birds. A model for the per capita predation rate is given by f(N)=aNk2+N2 where N denotes the density of spruce budworms and a and k are positive constants. Find f′(N), and determine when the per capita predation rate is increasing and when it is decreasing.

Answer 1

Answer:

f'(N) = a(k² - N²)/(k² + N²)

The function increases in the interval

(-k < N < k)

And the function decreases everywhere else; the intervals given as

(-∞ < N < -k) and (k < N < ∞)

Step-by-step explanation:

f(N)=aN/(k²+N²)

The derivative of this function is obrained using the quotient rule.

Then to determine the intervals where the function is increasinumber and decreasing,

The function increases in intervals where f'(N) > 0

and the function decreases in intervals where f'(N) < 0.

This inequality is evaluated and the solution obtained.

The solution is presented in the attached image.

Hope this Helps!!!

Answer 2

The model is an illustration of functions

• The value of f'(N) is $\mathbf{f'(N) = ak^2 + 2N}$
• The increasing and the decreasing intervals are: (N < ∞).

The function is given as:

$\mathbf{f(N) = aNk^2 + N^2}$

Differentiate f(N) to calculate f'(N)

$\mathbf{f'(N) = 1 \times aN^{1-1}k^2 + 2 \times N^{2-1}}$

Evaluate the exponents

$\mathbf{f'(N) = 1 \times aN^{0}k^2 + 2 \times N^{1}}$

So, we have:

$\mathbf{f'(N) = 1 \times ak^2 + 2 \times N}$

$\mathbf{f'(N) = ak^2 + 2N}$

When N increases, the function increases.

When N decreases, the function decreases.

So, the increasing and the decreasing intervals are: (N < ∞).

Read more about functions and models at:

https://brainly.com/question/20821733