Find the Instantaneous Velocity of D(x) = x + 2cos(x) in (0,2π)

In summary, the question involves finding an x value in the interval (0,2pi) for which the instantaneous velocity equals the average velocity, using the mean value theorem and the given function d(x)=x+2cosx. After applying the formula for average velocity, the value of c is found to be either 0 or π, but since it needs to lie in the given interval, the answer is π where both the average and instantaneous velocity are equal to 1 m/s.
  • #1
indigo1
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Homework Statement


if d(x)=x+2cosx represents the distance traveled by a particle from x=0 to x= 2pi.Find an x value in (0,2pi) for which the instantaneous velocity equals the average velocity.






The Attempt at a Solution



This question is a direct application of the mean value theorem (LMVT)
D(x) = x + 2cos(x)
∴ D(0) = 0 + 2cos(0) = 2
∴ D(2pi) = 2π + 2cos(2π) = 2(π + 1)
∴ Average velocity = Total distance/Total time
= [D(2π) - D(0)]/(2π - 0)
= (2 + 2π - 2)/(2π)
= 1 m/s
Instantaneous velocity =D ' (x)
∴ v(t) = D'(x) = 1 - 2sin(x)
Given that, to find 'c' such that D ' (c) = avg velocity = 1
∴ 1 - 2sin(c) = 1
∴ sin(c) = 0
∴ c = 0 or π
But 'c' needs to lie in (0,2π)
∴ At c = π, the average velocity = instantanoeus velocity = 1

ive been at this and I am just wondering if it's right?
 
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  • #2
hi indigo1! :smile:
indigo1 said:
if d(x)=x+2cosx represents the distance traveled by a particle from x=0 to x= 2pi.Find an x value in (0,2pi) for which the instantaneous velocity equals the average velocity.


∴ Average velocity = Total distance/Total time
= [D(2π) - D(0)]/(2π - 0)
= (2 + 2π - 2)/(2π)
= 1 m/s
Instantaneous velocity =D ' (x)
∴ v(t) = D'(x) = 1 - 2sin(x)
Given that, to find 'c' such that D ' (c) = avg velocity = 1
∴ 1 - 2sin(c) = 1
∴ sin(c) = 0
∴ c = 0 or π
But 'c' needs to lie in (0,2π)
∴ At c = π, the average velocity = instantanoeus velocity = 1

looks ok :smile:
 

1. What is the definition of instantaneous velocity?

Instantaneous velocity is the rate of change of an object's displacement at a specific point in time. It is the slope of the tangent line to the curve representing the object's position as a function of time.

2. How is instantaneous velocity different from average velocity?

Instantaneous velocity is the velocity at a specific moment in time, while average velocity is the total displacement of an object divided by the total time taken. In other words, average velocity gives an overall picture of an object's motion, while instantaneous velocity gives a more specific measurement at a particular time.

3. How is instantaneous velocity calculated?

Instantaneous velocity can be calculated by taking the derivative of the object's position function with respect to time. In the case of D(x) = x + 2cos(x), the derivative would be D'(x) = 1 - 2sin(x), which represents the instantaneous velocity at any given point.

4. How do you find the instantaneous velocity of a function on a specific interval?

To find the instantaneous velocity of a function on a specific interval, you would first need to find the derivative of the function. Then, you can plug in the value of x at the desired point in the interval into the derivative to find the corresponding instantaneous velocity. For example, to find the instantaneous velocity of D(x) = x + 2cos(x) at x = π/2, you would plug in π/2 into the derivative D'(x) = 1 - 2sin(x) to get the instantaneous velocity of -1.

5. How does the graph of a function relate to its instantaneous velocity?

The graph of a function can give us a visual representation of its instantaneous velocity. The slope of the tangent line at a specific point on the graph corresponds to the instantaneous velocity at that point. A steeper slope indicates a higher instantaneous velocity, while a flatter slope indicates a lower instantaneous velocity. Additionally, the direction of the tangent line can also tell us the direction of the instantaneous velocity at that point.

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