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.
Correct Answer :
B. It is always zero.
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(?)
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>
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 f(x) as x approaches 3 for the function f(x) = 2x + 1?
5
6
7
8
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 integral ?(0 to 1) x^3 dx is equal to:
0
1/4
1/3
1/2
What is the distance between the points A(2, -1, 3) and B(4, 3, -2)?
v42
v30
v29
v35
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
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>
What is the integral of 3x^2 with respect to x?
x^3 + C
x^2 + C
x^3/3 + C
3x + C
What is the area under the curve y = sin(x) from x = 0 to x = p?
0
1
p
2p
What is the unit vector in the direction of vector v = <4, -3>?
<1, 1>
<2, -1.5>
<4/5, -3/5>
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What is the derivative of ln(x) with respect to x?
1/x
1
x
0
What is the equation of the polar curve r = 2cos(?)?
A circle centered at the origin
A cardioid
A parabola
A line
The definite integral ?(1 to 3) 2x dx is equal to:
4
8
6
10
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 derivative of the constant function f(x) = 7?
0
1
7
-7
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 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)
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
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
If h(x) = e^(2x), what is h'(x)?
2e^(2x)
e^(2x)
2e^(x)
e^(x)
In three-dimensional space, what are the coordinates of the origin?
(0, 0, 0)
(1, 1, 1)
(0, 0)
(1, 1)
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 average value of the function f(x) = 2x^2 on the interval [1, 3]?
4/3
8/3
16/3
32/3
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 method is commonly used to solve separable differential equations?
Laplace transforms
Substitution
Separation of variables
Variation of parameters
The cost C(x) of producing x units of a product is given by C(x) = 3x^2 + 2x + 7. What is the marginal cost when x = 5?
47
37
27
17
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)
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.