What is the distance between the points A(2, -1, 3) and B(4, 3, -2)?
v42
v30
v29
v35
Correct Answer :
C. v29
Related Questions
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 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
If the velocity of an object is given by v(t) = 3t^2, what does ?(1 to 2) v(t) dt represent?
The total distance traveled by the object from t = 1 to t = 2
The acceleration of the object at t = 1
The change in velocity of the object from t = 1 to t = 2
The displacement of the object at t = 2
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
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 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 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 f(x) = x^3 + 2x^2 - 5x + 1, what is f''(x)?
3x^2 + 4x - 5
6x + 4
6x^2 + 4
3x^2 + 4
What is the volume of the region bounded by the surfaces z = x^2 + y^2 and z = 4 - x^2 - y^2?
4p
8p
16p
32p
What is the equation of the polar curve r = 2cos(?)?
A circle centered at the origin
A cardioid
A parabola
A line
If f(x) = 3x^2 - 2x + 1, what is the value of f(4)?
25
33
41
49
What does the definite integral ?(a to b) f(x) dx represent geometrically?
The average value of f(x) on [a, b]
The area between the curve y = f(x) and the x-axis on [a, b]
The derivative of f(x) at x = a
The slope of the tangent line to the curve y = f(x) at x = b
What is the integral of e^(3x) with respect to x?
(1/3)e^(3x) + C
3e^(x) + C
(1/3)e^(x) + C
e^(3x) + C
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 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
In three-dimensional space, what are the coordinates of the origin?
(0, 0, 0)
(1, 1, 1)
(0, 0)
(1, 1)
What is the unit vector in the direction of vector v = <4, -3>?
<1, 1>
<2, -1.5>
<4/5, -3/5>
<0, 0>
The definite integral ?(1 to 3) 2x dx is equal to:
4
8
6
10
In parametric equations, what do the variables x and y depend on?
Time t
Angle ?
Radius r
Both a and b
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
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.
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
If h(x) = e^(2x), what is h'(x)?
2e^(2x)
e^(2x)
2e^(x)
e^(x)
What is the derivative of ln(x) with respect to x?
1/x
1
x
0
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 derivative of the constant function f(x) = 7?
0
1
7
-7
What is the area under the curve y = sin(x) from x = 0 to x = p?
0
1
p
2p
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
If the rate of change of a quantity is given by f'(t) and ?(0 to 4) f'(t) dt = 10, what can you conclude about the quantity at t = 4?
The quantity is 10 at t = 4.
The quantity is 0 at t = 4.
The quantity is increasing at t = 4.
Nothing about the quantity at t = 4 can be determined.