The solution to the logistic differential equation dy/dx = ky(1 - y) is often used to model population growth. What happens to the population when y = 1?
The population is increasing.
The population is decreasing.
The population is zero.
The population remains constant.
Correct Answer :
D. The population remains constant.
Related Questions
What is the limit of f(x) as x approaches 3 for the function f(x) = 2x + 1?
5
6
7
8
What is the antiderivative of e^(2x) with respect to x?
e^(2x) + C
(1/2)e^(2x) + C
(1/2)e^(2x^2) + C
(1/4)e^(2x) + C
What is the area of the region enclosed by the polar curve r = 2sin(?)?
p square units
4 square units
2 square units
0 square units
What is the general solution to the differential equation dy/dx = 3x^2?
y = x^3 + C
y = 3x + C
y = x^3/3 + C
y = 3x^3 + C
What is the derivative of f(x) = 4x^3 - 2x^2 + 5x - 1 with respect to x?
12x^2 - 4x + 5
12x^2 - 4x - 5
12x^2 - 2x + 5
12x^2 - 2x - 5
Which of the following functions is continuous at x = 2?
f(x) = 1/x
g(x) = x^2 + 3x + 2
h(x) = |x - 2|
None of the above
In parametric equations, what do the variables x and y depend on?
Time t
Angle ?
Radius r
Both a and b
If a plane is defined by the equation 2x - 3y + z = 7, what is the normal vector to the plane?
<2, -3, 1>
<3, 2, -1>
<1, 3, -2>
<-2, 3, -1>
A ball is thrown into the air, and its height (in meters) above the ground at time t (in seconds) is given by h(t) = 4.9t^2 + 15t + 2. What is its initial velocity?
4.9 m/s
15 m/s
2 m/s
0 m/s
Which of the following is an exact differential equation?
dy/dx = 2xy
dy/dx = x^2 + y^2
dy/dx = e^x
dy/dx = y^2 - x^2
What is the directional derivative of the function f(x, y) = x^2 + y^2 at the point (1, 2) in the direction of the vector v = <3, 4>?
1
2
3
4
The area between the curve y = x^2 and the x-axis from x = 0 to x = 2 is equal to:
2
4
8/3
16/3
What is the integral of 3x^2 with respect to x?
x^3 + C
x^2 + C
x^3/3 + C
3x + C
What is the unit vector in the direction of vector v = <4, -3>?
<1, 1>
<2, -1.5>
<4/5, -3/5>
<0, 0>
What is the distance between the points A(2, -1, 3) and B(4, 3, -2)?
v42
v30
v29
v35
What is a vector?
A point in space
A scalar quantity
A quantity with magnitude and direction
A constant value
If a car's speed is decreasing over time, what can you conclude about its acceleration?
The acceleration is positive.
The acceleration is negative.
The acceleration is zero.
Nothing about the acceleration can be determined.
What is the integral of 1/(x^2 + 1) with respect to x?
arctan(x) + C
ln(x^2 + 1) + C
2arctan(x) + C
(1/2)ln(x^2 + 1) + C
If a point has polar coordinates (r, ?), what are its Cartesian coordinates?
x = rcos(?), y = rsin(?)
x = r?, y = r?
x = cos(?), y = sin(?)
x = rsin(?), y = rcos(?)
What is the derivative of the vector-valued function r(t) = <2t, t^2, 3t + 1> with respect to t?
<2, 2t, 3>
<2, 2t, 3t + 1>
<2t, 2t^2, 3>
<0, 0, 0>
Which of the following limits does not exist?
lim (x -> 0) (1/x)
lim (x -> 2) (x^2 - 4)/(x - 2)
lim (x -> 1) (x^2 - 1)/(x - 1)
lim (x -> 3) (1/x^2)
In three-dimensional space, what are the coordinates of the origin?
(0, 0, 0)
(1, 1, 1)
(0, 0)
(1, 1)
Which test can be used to determine the convergence of a series when its terms are all positive?
Alternating Series Test
Ratio Test
Integral Test
Comparison Test
What is the derivative of ln(x) with respect to x?
1/x
1
x
0
What is the equation of a sphere with center (1, -2, 3) and radius 4?
(x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 4
(x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 16
(x + 1)^2 + (y - 2)^2 + (z + 3)^2 = 4
(x + 1)^2 + (y - 2)^2 + (z + 3)^2 = 16
What is the integral of vx with respect to x?
(2/3)x^(3/2) + C
(1/2)x^(3/2) + C
2vx + C
x^(3/2) + C
If a particle's position at time t is given by r(t) = <3t, t^2, 2t - 1>, what is its velocity vector v(t)?
v(t) = <3, 2t, 2>
v(t) = <3t, t^2, 2t - 1>
v(t) = <3t^2, 2t, 2>
v(t) = <2, 2t, 2t - 1>
What does the second derivative of a function represent?
The slope of the tangent line
The area under the curve
The concavity of the curve
The average rate of change
What are the parametric equations for a circle of radius r centered at the origin in polar coordinates?
x = rcos(?), y = rsin(?)
x = rcos(t), y = rsin(t)
x = r?, y = r?
x = cos(?), y = sin(?)
The integral ?(0 to 1) x^3 dx is equal to:
0
1/4
1/3
1/2