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What is the correct answer?

4

What is the average value of the function f(x) = 2x^2 on the interval [1, 3]?

A. 4/3

B. 8/3

C. 16/3

D. 32/3

Correct Answer :

B. 8/3


Related Questions

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4

What is the integral of 1/(x^2 + 1) with respect to x?

A. arctan(x) + C

B. ln(x^2 + 1) + C

C. 2arctan(x) + C

D. (1/2)ln(x^2 + 1) + C

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4

If f(x) = 3x^2 - 2x + 1, what is the value of f(4)?

A. 25

B. 33

C. 41

D. 49

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4

What is a vector?

A. A point in space

B. A scalar quantity

C. A quantity with magnitude and direction

D. A constant value

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4

If h(x) = e^(2x), what is h'(x)?

A. 2e^(2x)

B. e^(2x)

C. 2e^(x)

D. e^(x)

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4

The area between the curve y = x^2 and the x-axis from x = 0 to x = 2 is equal to:

A. 2

B. 4

C. 8/3

D. 16/3

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4

What is the derivative of ln(x) with respect to x?

A. 1/x

B. 1

C. x

D. 0

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4

What is the equation of the polar curve r = 2cos(?)?

A. A circle centered at the origin

B. A cardioid

C. A parabola

D. A line

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4

What is the limit of f(x) as x approaches 3 for the function f(x) = 2x + 1?

A. 5

B. 6

C. 7

D. 8

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4

What is the average value of the function f(x) = 2x^2 on the interval [1, 3]?

A. 4/3

B. 8/3

C. 16/3

D. 32/3

What is the correct answer?

4

The function P(t) represents the population of a city at time t. What does ?(a to b) P(t) dt represent?

A. The average population of the city on [a, b]

B. The change in population of the city from time a to time b

C. The rate of population growth at time a

D. The population of the city at time b

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4

What is the equation of a sphere with center (1, -2, 3) and radius 4?

A. (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 4

B. (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 16

C. (x + 1)^2 + (y - 2)^2 + (z + 3)^2 = 4

D. (x + 1)^2 + (y - 2)^2 + (z + 3)^2 = 16

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4

If the velocity of an object is given by v(t) = 3t^2, what does ?(1 to 2) v(t) dt represent?

A. The total distance traveled by the object from t = 1 to t = 2

B. The acceleration of the object at t = 1

C. The change in velocity of the object from t = 1 to t = 2

D. The displacement of the object at t = 2

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4

What does the second derivative of a function represent?

A. The slope of the tangent line

B. The area under the curve

C. The concavity of the curve

D. The average rate of change

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4

If a plane is defined by the equation 2x - 3y + z = 7, what is the normal vector to the plane?

A. <2, -3, 1>

B. <3, 2, -1>

C. <1, 3, -2>

D. <-2, 3, -1>

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4

What are the parametric equations for a circle of radius r centered at the origin in polar coordinates?

A. x = rcos(?), y = rsin(?)

B. x = rcos(t), y = rsin(t)

C. x = r?, y = r?

D. x = cos(?), y = sin(?)

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4

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>?

A. 1

B. 2

C. 3

D. 4

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4

Which test can be used to determine the convergence of a series when its terms are all positive?

A. Alternating Series Test

B. Ratio Test

C. Integral Test

D. Comparison Test

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4

What is the area of the region enclosed by the polar curve r = 2sin(?)?

A. p square units

B. 4 square units

C. 2 square units

D. 0 square units

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4

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?

A. The quantity is 10 at t = 4.

B. The quantity is 0 at t = 4.

C. The quantity is increasing at t = 4.

D. Nothing about the quantity at t = 4 can be determined.

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4

Which test is used to determine the convergence or divergence of an infinite series involving alternating terms?

A. Integral Test

B. Comparison Test

C. Alternating Series Test

D. Divergence Test

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4

In three-dimensional space, what are the coordinates of the origin?

A. (0, 0, 0)

B. (1, 1, 1)

C. (0, 0)

D. (1, 1)

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4

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)?

A. v(t) = <3, 2t, 2>

B. v(t) = <3t, t^2, 2t - 1>

C. v(t) = <3t^2, 2t, 2>

D. v(t) = <2, 2t, 2t - 1>

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4

Which of the following functions is continuous at x = 2?

A. f(x) = 1/x

B. g(x) = x^2 + 3x + 2

C. h(x) = |x - 2|

D. None of the above

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4

What is the particular solution to the initial value problem dy/dx = 2x, y(0) = 1?

A. y = x^2 + 1

B. y = x^2

C. y = 2x + 1

D. y = 2x

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4

Which method is commonly used to solve separable differential equations?

A. Laplace transforms

B. Substitution

C. Separation of variables

D. Variation of parameters

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4

What is the unit vector in the direction of vector v = <4, -3>?

A. <1, 1>

B. <2, -1.5>

C. <4/5, -3/5>

D. <0, 0>

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4

What is the integral of 3x^2 with respect to x?

A. x^3 + C

B. x^2 + C

C. x^3/3 + C

D. 3x + C

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4

Which technique of integration is used to evaluate the integral of a product of two functions, often when integration by parts is required?

A. Trigonometric substitution

B. Integration by substitution

C. Integration by partial fractions

D. Integration by u-substitution

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4

Which of the following limits does not exist?

A. lim (x -> 0) (1/x)

B. lim (x -> 2) (x^2 - 4)/(x - 2)

C. lim (x -> 1) (x^2 - 1)/(x - 1)

D. lim (x -> 3) (1/x^2)

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4

What is the integral of vx with respect to x?

A. (2/3)x^(3/2) + C

B. (1/2)x^(3/2) + C

C. 2vx + C

D. x^(3/2) + C