÷ÈÓ°Ö±²¥

Sketch slope fields for each of the differential equations in Exercises $59-64$, and within each slope field sketch four different approximate solutions of the differential equation.

$\frac{dy}{dx}=-y$.

Suppose 100 rabbits are shipwrecked on a deserted island and their population P(t) after t years is determined by a logistic growth model, where the natural growth rate of the rabbits is k = 0.1 and the carrying capacity of the island is 1000 rabbits.

(a) Set up a differential equation describing $\frac{dp}{dt}$,and solve it to get a formula for the population P(t) of rabbits on the island in t years.

(b) Sketch a graph of the population P(t) of rabbits on the island over the next 100 years.

(c) It turns out that a population governed by a logistic model will be growing fastest when the population is equal to exactly half of the carrying capacity. In how many years will the population of rabbits be growing the fastest?

Prove the Napkin Ring Theorem: If a napkin ring is made by removing a cylinder of height h from a sphere, then the volume of the resulting shape does not depend on the radius of the sphere! Look at the figures in Exercise 62 to see why this result is surprising.

Prove the surprising fact that the arc length of the catenary curve traced out by the hyperbolic function f(x) = cosh x on any interval [a,b] is equal to the area under the same graph on [a,b].