In three-dimensional space, what are the coordinates of the origin?
(0, 0, 0)
(1, 1, 1)
(0, 0)
(1, 1)
Correct Answer :
A. (0, 0, 0)
Related Questions
Which technique of integration is used to evaluate the integral of a product of two functions, often when integration by parts is required?
Trigonometric substitution
Integration by substitution
Integration by partial fractions
Integration by u-substitution
Which trigonometric substitution is commonly used to integrate rational functions involving v(a^2 - x^2)?
u = sin(x)
u = cos(x)
u = sec(x)
u = tan(x)
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
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 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(?)
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 area of the region enclosed by the polar curve r = 2sin(?)?
p square units
4 square units
2 square units
0 square units
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
In three-dimensional space, what are the coordinates of the origin?
(0, 0, 0)
(1, 1, 1)
(0, 0)
(1, 1)
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.
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
If f(x) = 3x^2 - 2x + 1, what is the value of f(4)?
25
33
41
49
The integral ?(0 to 1) x^3 dx is equal to:
0
1/4
1/3
1/2
Which test is used to determine the convergence or divergence of an infinite series involving alternating terms?
Integral Test
Comparison Test
Alternating Series Test
Divergence Test
What is the equation of the polar curve r = 2cos(?)?
A circle centered at the origin
A cardioid
A parabola
A line
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
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>
If f(x) = sin(x), what is f'(x)?
cos(x)
-cos(x)
sin(x)
-sin(x)
The definite integral ?(1 to 3) 2x dx is equal to:
4
8
6
10
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 equation of the tangent plane to the surface z = 2x^2 + 3y^2 at the point (1, -1, 5)?
z = 2x - 3y + 5
z = 2x + 3y - 5
z = 2x - 3y - 5
z = 2x + 3y + 5
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
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.
What is the particular solution to the initial value problem dy/dx = 2x, y(0) = 1?
y = x^2 + 1
y = x^2
y = 2x + 1
y = 2x
What is the area under the curve y = sin(x) from x = 0 to x = p?
0
1
p
2p
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 type of differential equation is represented by the equation dy/dx = k*y, where k is a constant?
Separable
Linear
Exact
Homogeneous
In parametric equations, what do the variables x and y depend on?
Time t
Angle ?
Radius r
Both a and b
What is the distance between the points A(2, -1, 3) and B(4, 3, -2)?
v42
v30
v29
v35