Question

1. the number of fish in a lake is modeled by the function f that satisfies the logistic differential equation \\( \frac{df}{dt}=0.04f\left(1 - \frac{f}{5000}\

ight) \\), where t is the time in months and \\( f(0)=2000 \\). what is \\( \lim_{t\to\infty}f(t) \\)?
(a) 10,000
(b) 5000
(c) 2500
(d) 2000

1. a curve is defined by the parametric equations \\( x(t)=t^{2}+3 \\) and \\( y(t)=\sin(t^{2}) \\). which of the following is an expression for \\( \frac{d^{2}y}{dx^{2}} \\) in terms of t?

(a) \\( -\sin(t^{2}) \\)
(b) \\( -2t\sin(t^{2}) \\)
(c) \\( \cos(t^{2})-2t^{2}\sin(t^{2}) \\)
(d) \\( 2\cos(t^{2})-4t^{2}\sin(t^{2}) \\)

Explanation:

Step1: Recall logistic differential equation properties

The general form of a logistic differential equation is \(\frac{dP}{dt}=kP(1 - \frac{P}{L})\), where \(L\) is the carrying capacity. As \(t
ightarrow\infty\), \(P(t)
ightarrow L\) (the population approaches the carrying capacity).

Step2: Identify the carrying capacity

In the given logistic differential equation \(\frac{dF}{dt}=0.04F(1-\frac{F}{5000})\), by comparing with \(\frac{dP}{dt}=kP(1 - \frac{P}{L})\), we can see that \(L = 5000\).

Answer:

B. 5000