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
Correct Answer :
B. Ratio Test
Related Questions
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
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 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
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 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
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
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.
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
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 dot product of two vectors a and b?
|a| × |b| × cos(?)
|a| × |b| × sin(?)
|a| + |b|
|a| - |b|
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
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
What is a vector?
A point in space
A scalar quantity
A quantity with magnitude and direction
A constant value
Which method is commonly used to solve separable differential equations?
Laplace transforms
Substitution
Separation of variables
Variation of parameters
The integral ?(0 to 1) x^3 dx is equal to:
0
1/4
1/3
1/2
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
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
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 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(?)
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 definite integral ?(1 to 3) 2x dx is equal to:
4
8
6
10
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 volume of the region bounded by the surfaces z = x^2 + y^2 and z = 4 - x^2 - y^2?
4p
8p
16p
32p
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)
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
In parametric equations, what do the variables x and y depend on?
Time t
Angle ?
Radius r
Both a and 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
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)