What does the second derivative of a function represent?
The slope of the tangent line
The area under the curve
The concavity of the curve
The average rate of change
Correct Answer :
C. The concavity of the curve
Related Questions
What does the second derivative of a function represent?
The slope of the tangent line
The area under the curve
The concavity of the curve
The average rate of change
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 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 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 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
The integral ?(0 to 1) x^3 dx is equal to:
0
1/4
1/3
1/2
If h(x) = e^(2x), what is h'(x)?
2e^(2x)
e^(2x)
2e^(x)
e^(x)
What is the limit of (sin x) / x as x approaches 0?
1
0
-1
Undefined
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.
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 particular solution to the initial value problem dy/dx = 2x, y(0) = 1?
y = x^2 + 1
y = x^2
y = 2x + 1
y = 2x
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 area under the curve y = sin(x) from x = 0 to x = p?
0
1
p
2p
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 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 constant function f(x) = 7?
0
1
7
-7
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
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.
Which method is commonly used to solve separable differential equations?
Laplace transforms
Substitution
Separation of variables
Variation of parameters
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 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
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 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>
The definite integral ?(1 to 3) 2x dx is equal to:
4
8
6
10
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
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
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
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.