Calculus Answers

. A model that describes the population p(t ) of a fishery in which harvesting takes place at

a constant rate is given by

dp/dt=kp-h

where k and h are positive constants.

(a) Solve the DE subject to P(0) = p

(b) Describe the behavior of the population P(t) for increasing time in the three cases for p=h/k , p=h/k , and 0<p<h/k

(c) Use the results from part (b) to determine whether the fish population will ever go extinct

in finite time, that is, whether there exists a time T>0 such that p(t )=0 . If the population

goes extinct, then find T

If x(t+y) = t^2 + y^2. Show that (partial x/partial t - partial x/partial y)^2 = 4[1 - partial x/partial t - partial x/partial y]

If x = t^3 + y^3 where t= a cos v , y= Vsin v. Find dx/dv

If the sum of the surface areas of a sphere and a cube is fixed, what is the ratio of

the radius of the sphere to the edge of the cube when the sum of their volumes is

least

Use differentials to approximate the volume of material needed to make a rubber ball if the

radius of the hollow inner core is 2 𝑖𝑛., and the thickness of the rubber is 1

8

𝑖𝑛.

 A differentiable function f has the property that f(5)=3

f′(5)=4 Whats the estimate for f(4.8) using the local linear approximation for f at x= 5 ?

Area Between Curves, Volumes and Average Values of Functions:

1 Calculate the area between the parabola 𝑦=𝑥2 and the line 𝑦=4

2 Derive the formula for the volume of the right-circular cone using a single integral given that 𝑦=−𝑎ℎ(𝑧−ℎ).

3. Find the average value 𝑓(𝑥)̅̅̅̅̅̅ of 𝑓(𝑥)=2𝑥2+3𝑥+3 in the interval [1,4].

What can you say about the upper bound of E(x) for numbers of the form 3^(2n) + 1?

given 1/u+1/v=1/f with f as a constant. if f=10 cm and u decrease with the rate of 2 cm/second, find the rate of v when u=40 cm