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
Correct Answer :
B. The area between the curve y = f(x) and the x-axis on [a, b]
Related Questions
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
The definite integral ?(1 to 3) 2x dx is equal to:
4
8
6
10
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>
The integral ?(0 to 1) x^3 dx is equal to:
0
1/4
1/3
1/2
What is the dot product of two vectors a and b?
|a| × |b| × cos(?)
|a| × |b| × sin(?)
|a| + |b|
|a| - |b|
What is the 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 limit of f(x) as x approaches 3 for the function f(x) = 2x + 1?
5
6
7
8
What is the limit of (sin x) / x as x approaches 0?
1
0
-1
Undefined
If f(x) = 3x^2 - 2x + 1, what is the value of f(4)?
25
33
41
49
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
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
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)
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 integral of 3x^2 with respect to x?
x^3 + C
x^2 + C
x^3/3 + C
3x + C
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
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 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>
If h(x) = e^(2x), what is h'(x)?
2e^(2x)
e^(2x)
2e^(x)
e^(x)
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
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
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 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
In parametric equations, what do the variables x and y depend on?
Time t
Angle ?
Radius r
Both a and b
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 distance between the points A(2, -1, 3) and B(4, 3, -2)?
v42
v30
v29
v35
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
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)