The integral ?(0 to 1) x^3 dx is equal to:
0
1/4
1/3
1/2
Correct Answer :
C. 1/3
Related Questions
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 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 equation of the polar curve r = 2cos(?)?
A circle centered at the origin
A cardioid
A parabola
A line
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.
The definite integral ?(1 to 3) 2x dx is equal to:
4
8
6
10
What is the integral of 3x^2 with respect to x?
x^3 + C
x^2 + C
x^3/3 + C
3x + 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.
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
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
In three-dimensional space, what are the coordinates of the origin?
(0, 0, 0)
(1, 1, 1)
(0, 0)
(1, 1)
What is the line integral ?C F � dr, where C is a curve in the xy-plane and F = ?
It depends on the specific curve C.
It is always zero.
It is equal to the area enclosed by C.
It is equal to the circumference of C.
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)
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 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
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 unit vector in the direction of vector v = <4, -3>?
<1, 1>
<2, -1.5>
<4/5, -3/5>
<0, 0>
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.
What is the average value of the function f(x) = 2x^2 on the interval [1, 3]?
4/3
8/3
16/3
32/3
What is the derivative of ln(x) with respect to x?
1/x
1
x
0
What is the limit of (sin x) / x as x approaches 0?
1
0
-1
Undefined
The function P(t) represents the population of a city at time t. What does ?(a to b) P(t) dt represent?
The average population of the city on [a, b]
The change in population of the city from time a to time b
The rate of population growth at time a
The population of the city at time b
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 f(x) = x^3 + 2x^2 - 5x + 1, what is f''(x)?
3x^2 + 4x - 5
6x + 4
6x^2 + 4
3x^2 + 4
Which type of differential equation is represented by the equation dy/dx = k*y, where k is a constant?
Separable
Linear
Exact
Homogeneous
What is the dot product of two vectors a and b?
|a| × |b| × cos(?)
|a| × |b| × sin(?)
|a| + |b|
|a| - |b|
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 distance between the points A(2, -1, 3) and B(4, 3, -2)?
v42
v30
v29
v35
What is the area under the curve y = sin(x) from x = 0 to x = p?
0
1
p
2p
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>
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