What is the derivative of ln(x) with respect to x?
1/x
1
x
0
Correct Answer :
A. 1/x
Related Questions
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(?)
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(?)
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 derivative of ln(x) with respect to x?
1/x
1
x
0
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
What is the dot product of two vectors a and b?
|a| × |b| × cos(?)
|a| × |b| × sin(?)
|a| + |b|
|a| - |b|
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 derivative of the constant function f(x) = 7?
0
1
7
-7
What is a vector?
A point in space
A scalar quantity
A quantity with magnitude and direction
A constant value
What is the limit of f(x) as x approaches 3 for the function f(x) = 2x + 1?
5
6
7
8
Which type of differential equation is represented by the equation dy/dx = k*y, where k is a constant?
Separable
Linear
Exact
Homogeneous
The integral ?(0 to 1) x^3 dx is equal to:
0
1/4
1/3
1/2
In parametric equations, what do the variables x and y depend on?
Time t
Angle ?
Radius r
Both a and b
What is the limit of (sin x) / x as x approaches 0?
1
0
-1
Undefined
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 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 average value of the function f(x) = 2x^2 on the interval [1, 3]?
4/3
8/3
16/3
32/3
If h(x) = e^(2x), what is h'(x)?
2e^(2x)
e^(2x)
2e^(x)
e^(x)
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 area under the curve y = sin(x) from x = 0 to x = p?
0
1
p
2p
Which method is commonly used to solve separable differential equations?
Laplace transforms
Substitution
Separation of variables
Variation of parameters
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
In three-dimensional space, what are the coordinates of the origin?
(0, 0, 0)
(1, 1, 1)
(0, 0)
(1, 1)
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) = sin(x), what is f'(x)?
cos(x)
-cos(x)
sin(x)
-sin(x)
What is the distance between the points A(2, -1, 3) and B(4, 3, -2)?
v42
v30
v29
v35
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.
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 integral of 3x^2 with respect to x?
x^3 + C
x^2 + C
x^3/3 + C
3x + C
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