The correct answer is: surface area times the rate of change of radius. This follows because the volume V of a sphere is connected to its radius r by the formula: V = (4/3) π r³ The rate of change of volume with respect to time is: dV/dt = 4 π r² (dr/dt) Here, 4 π r² is the surface area of the spherRead more
The correct answer is: surface area times the rate of change of radius.
This follows because the volume V of a sphere is connected to its radius r by the formula:
V = (4/3) π r³
The rate of change of volume with respect to time is:
dV/dt = 4 π r² (dr/dt)
Here, 4 π r² is the surface area of the sphere, and (dr/dt) is the rate of change of the radius. Therefore, the rate of change of volume is equal to the surface area times the rate of change of radius.
To find the rate at which the volume V is increasing, we need to differentiate the volume formula V = (4/3) π r³ with respect to time t. dV/dt = 4 π r² (dr/dt) Now, plug in the values: r = 10 and (dr/dt) = 0.01: dV/dt = 4 π (10)² (0.01) dV/dt = 4 π × 100 × 0.01 = 4 π Thus, the rate at which the voluRead more
To find the rate at which the volume V is increasing, we need to differentiate the volume formula V = (4/3) π r³ with respect to time t.
dV/dt = 4 π r² (dr/dt)
Now, plug in the values: r = 10 and (dr/dt) = 0.01:
dV/dt = 4 π (10)² (0.01)
dV/dt = 4 π × 100 × 0.01 = 4 π
Thus, the rate at which the volume is increasing is 4π cubic units.
The function f(x) = tan x - x increases or decreases throughout, and the only way to find this is by analyzing the derivative of f(x). Now, differentiate f(x) with respect to x: f'(x) = d/dx(tan x) - d/dx(x) = sec² x - 1 So, f'(x) = sec² x - 1 = tan² x Since tan² x is always non-negative for all reaRead more
The function f(x) = tan x – x increases or decreases throughout, and the only way to find this is by analyzing the derivative of f(x).
Now, differentiate f(x) with respect to x:
f'(x) = d/dx(tan x) – d/dx(x) = sec² x – 1
So,
f'(x) = sec² x – 1 = tan² x
Since tan² x is always non-negative for all real values of x, f'(x) ≥ 0 for all x where tan x is defined.
Therefore, f(x) is always increasing where it is defined, but has vertical asymptotes at x = (π/2) + nπ, where n is any integer.
The function f(x) = |x| is neither increasing nor decreasing throughout its domain since: - For x ≥ 0, the function is increasing (since f(x) = x for nonnegative x). - For x < 0, the function is decreasing (since f(x) = -x for negative x). The function is neither strictly increasing nor strictlyRead more
The function f(x) = |x| is neither increasing nor decreasing throughout its domain since:
– For x ≥ 0, the function is increasing (since f(x) = x for nonnegative x).
– For x < 0, the function is decreasing (since f(x) = -x for negative x).
The function is neither strictly increasing nor strictly decreasing at x = 0 because of a sharp "corner."
Hence f(x) = |x| is neither increasing nor decreasing on its entire domain.
To find the stationary points of the function f(x) = x³ - 3x² - 9x - 7, one would first have to compute the derivative of the function and set the function equal to zero. Stationary points occur at places where the derivative is equal to zero. First, differentiate f(x): f'(x) = d/dx(x³ - 3x² - 9x -Read more
To find the stationary points of the function f(x) = x³ – 3x² – 9x – 7, one would first have to compute the derivative of the function and set the function equal to zero. Stationary points occur at places where the derivative is equal to zero.
First, differentiate f(x):
f'(x) = d/dx(x³ – 3x² – 9x – 7) = 3x² – 6x – 9
Now, put f'(x) = 0 to get the stationary points:
3x² – 6x – 9 = 0
Divide the equation by 3:
x² – 2x – 3 = 0
Factor the quadratic equation:
(x – 3)(x + 1) = 0
Therefore, the solutions are x = 3 and x = -1.
Hence, the stationary points are at x = -1 and x = 3.
In order to determine the value of b for which the function f(x) = x + cos x + b is strictly decreasing over ℝ, we start by analyzing the derivative of the function. Let us differentiate f(x): f'(x) = d/dx(x + cos x + b) = 1 - sin x Hence for being strictly decreasing, the derivative must be negativRead more
In order to determine the value of b for which the function f(x) = x + cos x + b is strictly decreasing over ℝ, we start by analyzing the derivative of the function. Let us differentiate f(x):
f'(x) = d/dx(x + cos x + b) = 1 – sin x
Hence for being strictly decreasing, the derivative must be negative for all values of x; i.e., f'(x) < 0.
1 – sin x 1
But the function sin x can never be more than 1 for any real number x. Thus, b cannot be taken such that for all real number x, f'<0.
We find the minimum value of the function f(x) = 2 cos x + x over the closed interval [0, π/2] by determining its critical numbers and then we check the value of the function at the interval's endpoints. Compute the derivative of the function: f'(x) = d/dx(2 cos x + x) = -2 sin x + 1 Set the derivatRead more
We find the minimum value of the function f(x) = 2 cos x + x over the closed interval [0, π/2] by determining its critical numbers and then we check the value of the function at the interval’s endpoints.
Compute the derivative of the function:
f'(x) = d/dx(2 cos x + x) = -2 sin x + 1
Set the derivative equal to zero to find critical points:
-2 sin x + 1 = 0
sin x = 1/2
The solution to sin x = 1/2 in the interval [0, π/2] is x = π/6.
Evaluate f(x) at the critical point and the endpoints of the interval:
– At x = 0:
f(0) = 2 cos(0) + 0 = 2 × 1 + 0 = 2
To find the intervals where the function f(x) = x² - 4x + 6 is strictly increasing, we have to look at its derivative. Find the derivative of the function: f'(x) = d/dx(x² - 4x + 6) = 2x - 4 Determine where the derivative is positive (for strictly increasing behavior): f'(x) > 0 2x - 4 > 0 xRead more
To find the intervals where the function f(x) = x² – 4x + 6 is strictly increasing, we have to look at its derivative.
Find the derivative of the function:
f'(x) = d/dx(x² – 4x + 6) = 2x – 4
Determine where the derivative is positive (for strictly increasing behavior):
f'(x) > 0
2x – 4 > 0
x > 2
So, the function is strictly increasing when x > 2.
Conclusion:
The function is strictly increasing on the interval (2, ∞).
Three identical metal balls of the same radius are placed on a horizontal surface so that their centers form the vertices of an equilateral triangle. Because all the balls have the same mass and are symmetrically arranged, the center of mass of this system lies at the geometric center of the trianglRead more
Three identical metal balls of the same radius are placed on a horizontal surface so that their centers form the vertices of an equilateral triangle. Because all the balls have the same mass and are symmetrically arranged, the center of mass of this system lies at the geometric center of the triangle. This point is known as the centroid of the triangle.
The centroid is a point where all the medians of the triangle intersect. A median is a line segment joining a vertex to the midpoint of the opposite side. In an equilateral triangle, the centroid lies equidistant from all the three vertices and is inside the triangle. That is why it balances the system perfectly due to symmetry as well as uniformity in the mass distribution of the three balls.
Furthermore, since the balls are at rest on that horizontal surface, the center of mass will be on the surface because no part of their mass protrudes vertically either above or below the surface. The center of mass is therefore guaranteed to be in the plane containing the surface. There is an important location associated with the concept of balance and its motion-the centroid. The centroid is essentially the average position of the entire system’s mass.
A rod of length 3 meters has a mass that increases along its length, with the mass per unit length directly proportional to the distance from one end. This means that as you move further along the rod from the starting point, the density of the rod increases, making the far end heavier compared to tRead more
A rod of length 3 meters has a mass that increases along its length, with the mass per unit length directly proportional to the distance from one end. This means that as you move further along the rod from the starting point, the density of the rod increases, making the far end heavier compared to the starting end.
To find the center of gravity of this rod, we will consider the balance point where the total mass on either side is equal. The rod’s density increases with distance, so its weight is concentrated more toward the far end. Thus, the center of gravity will not be at the midpoint, which is at 1.5 meters, but will shift closer to the heavier end.
By considering the mass distribution and the balance point, it is thus determined that the center of gravity is 2.5 meters from the starting end of the rod. This position then balances the rod just right in account of its mass variability along its length.
Therefore, the center of gravity shifts more towards the denser side, thereby showing how the mass distribution impacts an object’s balancing point.
In a sphere the rate of change of volume is
The correct answer is: surface area times the rate of change of radius. This follows because the volume V of a sphere is connected to its radius r by the formula: V = (4/3) π r³ The rate of change of volume with respect to time is: dV/dt = 4 π r² (dr/dt) Here, 4 π r² is the surface area of the spherRead more
The correct answer is: surface area times the rate of change of radius.
This follows because the volume V of a sphere is connected to its radius r by the formula:
V = (4/3) π r³
The rate of change of volume with respect to time is:
dV/dt = 4 π r² (dr/dt)
Here, 4 π r² is the surface area of the sphere, and (dr/dt) is the rate of change of the radius. Therefore, the rate of change of volume is equal to the surface area times the rate of change of radius.
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If V = 4/3 πr³, at what rate in cubic units is V increasing when r = 10 and dr/dt = 0.01?
To find the rate at which the volume V is increasing, we need to differentiate the volume formula V = (4/3) π r³ with respect to time t. dV/dt = 4 π r² (dr/dt) Now, plug in the values: r = 10 and (dr/dt) = 0.01: dV/dt = 4 π (10)² (0.01) dV/dt = 4 π × 100 × 0.01 = 4 π Thus, the rate at which the voluRead more
To find the rate at which the volume V is increasing, we need to differentiate the volume formula V = (4/3) π r³ with respect to time t.
dV/dt = 4 π r² (dr/dt)
Now, plug in the values: r = 10 and (dr/dt) = 0.01:
dV/dt = 4 π (10)² (0.01)
dV/dt = 4 π × 100 × 0.01 = 4 π
Thus, the rate at which the volume is increasing is 4π cubic units.
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The function f(x) = tan x – x
The function f(x) = tan x - x increases or decreases throughout, and the only way to find this is by analyzing the derivative of f(x). Now, differentiate f(x) with respect to x: f'(x) = d/dx(tan x) - d/dx(x) = sec² x - 1 So, f'(x) = sec² x - 1 = tan² x Since tan² x is always non-negative for all reaRead more
The function f(x) = tan x – x increases or decreases throughout, and the only way to find this is by analyzing the derivative of f(x).
Now, differentiate f(x) with respect to x:
f'(x) = d/dx(tan x) – d/dx(x) = sec² x – 1
So,
f'(x) = sec² x – 1 = tan² x
Since tan² x is always non-negative for all real values of x, f'(x) ≥ 0 for all x where tan x is defined.
Therefore, f(x) is always increasing where it is defined, but has vertical asymptotes at x = (π/2) + nπ, where n is any integer.
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For all real values of x, the function f(x) = |x| is
The function f(x) = |x| is neither increasing nor decreasing throughout its domain since: - For x ≥ 0, the function is increasing (since f(x) = x for nonnegative x). - For x < 0, the function is decreasing (since f(x) = -x for negative x). The function is neither strictly increasing nor strictlyRead more
The function f(x) = |x| is neither increasing nor decreasing throughout its domain since:
– For x ≥ 0, the function is increasing (since f(x) = x for nonnegative x).
– For x < 0, the function is decreasing (since f(x) = -x for negative x).
The function is neither strictly increasing nor strictly decreasing at x = 0 because of a sharp "corner."
Hence f(x) = |x| is neither increasing nor decreasing on its entire domain.
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See lesshttps://www.tiwariacademy.com/ncert-solutions/class-12/maths/#chapter-6
The function f(x) = x³ – 3x² – 9x – 7 has a stationary point at
To find the stationary points of the function f(x) = x³ - 3x² - 9x - 7, one would first have to compute the derivative of the function and set the function equal to zero. Stationary points occur at places where the derivative is equal to zero. First, differentiate f(x): f'(x) = d/dx(x³ - 3x² - 9x -Read more
To find the stationary points of the function f(x) = x³ – 3x² – 9x – 7, one would first have to compute the derivative of the function and set the function equal to zero. Stationary points occur at places where the derivative is equal to zero.
First, differentiate f(x):
f'(x) = d/dx(x³ – 3x² – 9x – 7) = 3x² – 6x – 9
Now, put f'(x) = 0 to get the stationary points:
3x² – 6x – 9 = 0
Divide the equation by 3:
x² – 2x – 3 = 0
Factor the quadratic equation:
(x – 3)(x + 1) = 0
Therefore, the solutions are x = 3 and x = -1.
Hence, the stationary points are at x = -1 and x = 3.
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The value of b for which the function f(x) = x + cos x + b is strictly decreasing over R is
In order to determine the value of b for which the function f(x) = x + cos x + b is strictly decreasing over ℝ, we start by analyzing the derivative of the function. Let us differentiate f(x): f'(x) = d/dx(x + cos x + b) = 1 - sin x Hence for being strictly decreasing, the derivative must be negativRead more
In order to determine the value of b for which the function f(x) = x + cos x + b is strictly decreasing over ℝ, we start by analyzing the derivative of the function. Let us differentiate f(x):
f'(x) = d/dx(x + cos x + b) = 1 – sin x
Hence for being strictly decreasing, the derivative must be negative for all values of x; i.e., f'(x) < 0.
1 – sin x 1
But the function sin x can never be more than 1 for any real number x. Thus, b cannot be taken such that for all real number x, f'<0.
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The least value of the function f(x) = 2cos x + x in the closed interval [0, π/2] is
We find the minimum value of the function f(x) = 2 cos x + x over the closed interval [0, π/2] by determining its critical numbers and then we check the value of the function at the interval's endpoints. Compute the derivative of the function: f'(x) = d/dx(2 cos x + x) = -2 sin x + 1 Set the derivatRead more
We find the minimum value of the function f(x) = 2 cos x + x over the closed interval [0, π/2] by determining its critical numbers and then we check the value of the function at the interval’s endpoints.
Compute the derivative of the function:
f'(x) = d/dx(2 cos x + x) = -2 sin x + 1
Set the derivative equal to zero to find critical points:
-2 sin x + 1 = 0
sin x = 1/2
The solution to sin x = 1/2 in the interval [0, π/2] is x = π/6.
Evaluate f(x) at the critical point and the endpoints of the interval:
– At x = 0:
f(0) = 2 cos(0) + 0 = 2 × 1 + 0 = 2
– At x = π/6:
f(π/6) = 2 cos(π/6) + π/6 = 2 × (√3/2) + π/6 = √3 + π/6
– At x = π/2:
f(π/2) = 2 cos(π/2) + π/2 = 2 × 0 + π/2 = π/2
Compare the values:
– f(0) = 2
– f(π/6) = √3 + π/6 ≈ 1.732 + 0.523 = 2.255
– f(π/2) = π/2 ≈ 1.570
The minimum value of the function in the interval [0, π/2] is attained at f(0) = 2.
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Find the intervals in which the function f given by f(x) = x² – 4x + 6 is strictly increasing:
To find the intervals where the function f(x) = x² - 4x + 6 is strictly increasing, we have to look at its derivative. Find the derivative of the function: f'(x) = d/dx(x² - 4x + 6) = 2x - 4 Determine where the derivative is positive (for strictly increasing behavior): f'(x) > 0 2x - 4 > 0 xRead more
To find the intervals where the function f(x) = x² – 4x + 6 is strictly increasing, we have to look at its derivative.
Find the derivative of the function:
f'(x) = d/dx(x² – 4x + 6) = 2x – 4
Determine where the derivative is positive (for strictly increasing behavior):
f'(x) > 0
2x – 4 > 0
x > 2
So, the function is strictly increasing when x > 2.
Conclusion:
The function is strictly increasing on the interval (2, ∞).
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Three identical metal balls, each of the radius r are placed touching each other on a horizontal surface such that an equilateral triangle is formed surface such that an equilateral triangle is formed when centres of three balls are joined. The centre of the mass of the system is located at
Three identical metal balls of the same radius are placed on a horizontal surface so that their centers form the vertices of an equilateral triangle. Because all the balls have the same mass and are symmetrically arranged, the center of mass of this system lies at the geometric center of the trianglRead more
Three identical metal balls of the same radius are placed on a horizontal surface so that their centers form the vertices of an equilateral triangle. Because all the balls have the same mass and are symmetrically arranged, the center of mass of this system lies at the geometric center of the triangle. This point is known as the centroid of the triangle.
The centroid is a point where all the medians of the triangle intersect. A median is a line segment joining a vertex to the midpoint of the opposite side. In an equilateral triangle, the centroid lies equidistant from all the three vertices and is inside the triangle. That is why it balances the system perfectly due to symmetry as well as uniformity in the mass distribution of the three balls.
Furthermore, since the balls are at rest on that horizontal surface, the center of mass will be on the surface because no part of their mass protrudes vertically either above or below the surface. The center of mass is therefore guaranteed to be in the plane containing the surface. There is an important location associated with the concept of balance and its motion-the centroid. The centroid is essentially the average position of the entire system’s mass.
Click here for more: – https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See lessA rod has length 3 m and its mass acting per unit length is directly proportional to distance x from one of its end, then its centre of gravity from that end will be at
A rod of length 3 meters has a mass that increases along its length, with the mass per unit length directly proportional to the distance from one end. This means that as you move further along the rod from the starting point, the density of the rod increases, making the far end heavier compared to tRead more
A rod of length 3 meters has a mass that increases along its length, with the mass per unit length directly proportional to the distance from one end. This means that as you move further along the rod from the starting point, the density of the rod increases, making the far end heavier compared to the starting end.
To find the center of gravity of this rod, we will consider the balance point where the total mass on either side is equal. The rod’s density increases with distance, so its weight is concentrated more toward the far end. Thus, the center of gravity will not be at the midpoint, which is at 1.5 meters, but will shift closer to the heavier end.
By considering the mass distribution and the balance point, it is thus determined that the center of gravity is 2.5 meters from the starting end of the rod. This position then balances the rod just right in account of its mass variability along its length.
Therefore, the center of gravity shifts more towards the denser side, thereby showing how the mass distribution impacts an object’s balancing point.
Click here for more:- https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See less