A Particle Moves Along The Y Y Y -axis So That At Time T ≥ 0 T \geq 0 T ≥ 0 Its Position Is Given By Y ( T ) = T 3 − 6 T 2 + 9 T Y(t) = T^3 - 6t^2 + 9t Y ( T ) = T 3 − 6 T 2 + 9 T . Over The Time Interval 0 \textless T \textless 4 0 \ \textless \ T \ \textless \ 4 0 \textless T \textless 4 , For What Values Of T T T Is The

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Introduction

In mathematics, the study of motion and position functions is a fundamental concept in understanding various physical phenomena. In this article, we will delve into the motion of a particle along the y-axis, with its position given by the function y(t) = t^3 - 6t^2 + 9t. We will analyze the position function over the time interval 0 < t < 4 and determine the values of t for which the particle's position is critical.

The Position Function

The position function y(t) = t^3 - 6t^2 + 9t represents the particle's position at time t. To understand the behavior of the particle, we need to analyze the function's critical points, which are the values of t where the function's derivative is equal to zero or undefined.

Critical Points

To find the critical points, we need to find the derivative of the position function. Using the power rule of differentiation, we get:

y'(t) = 3t^2 - 12t + 9

Now, we set the derivative equal to zero and solve for t:

3t^2 - 12t + 9 = 0

We can solve this quadratic equation using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 3, b = -12, and c = 9. Plugging these values into the formula, we get:

t = (12 ± √((-12)^2 - 4(3)(9))) / (2(3)) t = (12 ± √(144 - 108)) / 6 t = (12 ± √36) / 6 t = (12 ± 6) / 6

Simplifying, we get two possible values for t:

t = (12 + 6) / 6 = 18/6 = 3 t = (12 - 6) / 6 = 6/6 = 1

These are the critical points of the position function.

Second Derivative Test

To determine whether these critical points correspond to a maximum, minimum, or saddle point, we need to use the second derivative test. We take the derivative of the first derivative:

y''(t) = 6t - 12

Now, we evaluate the second derivative at the critical points:

y''(1) = 6(1) - 12 = -6 y''(3) = 6(3) - 12 = 6

Since the second derivative is negative at t = 1, this critical point corresponds to a maximum. Since the second derivative is positive at t = 3, this critical point corresponds to a minimum.

Conclusion

In conclusion, the particle's position function y(t) = t^3 - 6t^2 + 9t has two critical points: t = 1 and t = 3. The critical point t = 1 corresponds to a maximum, while the critical point t = 3 corresponds to a minimum. These critical points are the values of t for which the particle's position is critical over the time interval 0 < t < 4.

Mathematical Analysis

To further analyze the position function, we can use various mathematical techniques, such as graphing the function, finding the inflection points, and using numerical methods to approximate the critical points.

Graphing the Function

We can graph the position function y(t) = t^3 - 6t^2 + 9t to visualize its behavior. The graph will show the particle's position at different times, with the critical points corresponding to the maximum and minimum values of the function.

Inflection Points

To find the inflection points, we need to find the values of t where the second derivative is equal to zero or undefined. We take the derivative of the second derivative:

y'''(t) = 6

Since the third derivative is a constant, the second derivative is never equal to zero. Therefore, the inflection points are not critical points.

Numerical Methods

To approximate the critical points, we can use numerical methods, such as the Newton-Raphson method or the bisection method. These methods involve iteratively refining an initial guess for the critical point until the desired accuracy is achieved.

Conclusion

In conclusion, the particle's position function y(t) = t^3 - 6t^2 + 9t has two critical points: t = 1 and t = 3. The critical point t = 1 corresponds to a maximum, while the critical point t = 3 corresponds to a minimum. These critical points are the values of t for which the particle's position is critical over the time interval 0 < t < 4.

Final Thoughts

The analysis of the particle's position function y(t) = t^3 - 6t^2 + 9t provides valuable insights into the behavior of the particle over the time interval 0 < t < 4. The critical points correspond to the maximum and minimum values of the function, which are the values of t for which the particle's position is critical. This analysis can be applied to various physical phenomena, such as the motion of objects in a gravitational field or the behavior of electrical circuits.

References

  • [1] Calculus: Early Transcendentals, James Stewart, 8th edition.
  • [2] Differential Equations and Dynamical Systems, Lawrence Perko, 3rd edition.
  • [3] Mathematical Methods for Physicists, George Arfken, 7th edition.

Keywords

  • Position function
  • Critical points
  • Second derivative test
  • Inflection points
  • Numerical methods
  • Particle motion
  • Gravitational field
  • Electrical circuits

Categories

  • Mathematics
  • Physics
  • Engineering

Tags

  • Calculus
  • Differential equations
  • Dynamical systems
  • Mathematical analysis
  • Numerical methods
  • Particle motion
  • Position function
  • Critical points
  • Second derivative test
  • Inflection points

Introduction

In our previous article, we analyzed the motion of a particle along the y-axis, with its position given by the function y(t) = t^3 - 6t^2 + 9t. We found the critical points of the position function and determined the values of t for which the particle's position is critical. In this article, we will answer some frequently asked questions about the particle's motion.

Q: What is the initial position of the particle?

A: The initial position of the particle is given by the value of the position function at t = 0. We can find this value by plugging t = 0 into the position function:

y(0) = (0)^3 - 6(0)^2 + 9(0) = 0

So, the initial position of the particle is 0.

Q: What is the maximum position of the particle?

A: The maximum position of the particle is given by the value of the position function at the critical point t = 1. We can find this value by plugging t = 1 into the position function:

y(1) = (1)^3 - 6(1)^2 + 9(1) = 4

So, the maximum position of the particle is 4.

Q: What is the minimum position of the particle?

A: The minimum position of the particle is given by the value of the position function at the critical point t = 3. We can find this value by plugging t = 3 into the position function:

y(3) = (3)^3 - 6(3)^2 + 9(3) = 0

So, the minimum position of the particle is 0.

Q: What is the velocity of the particle at t = 1?

A: The velocity of the particle is given by the derivative of the position function. We can find the velocity by taking the derivative of the position function:

y'(t) = 3t^2 - 12t + 9

Now, we can plug t = 1 into the velocity function:

y'(1) = 3(1)^2 - 12(1) + 9 = -6

So, the velocity of the particle at t = 1 is -6.

Q: What is the acceleration of the particle at t = 1?

A: The acceleration of the particle is given by the derivative of the velocity function. We can find the acceleration by taking the derivative of the velocity function:

y''(t) = 6t - 12

Now, we can plug t = 1 into the acceleration function:

y''(1) = 6(1) - 12 = -6

So, the acceleration of the particle at t = 1 is -6.

Q: What is the inflection point of the position function?

A: The inflection point of the position function is given by the value of t where the second derivative is equal to zero. We can find this value by setting the second derivative equal to zero and solving for t:

y''(t) = 6t - 12 = 0

Now, we can solve for t:

6t = 12 t = 2

So, the inflection point of the position function is t = 2.

Q: What is the physical significance of the inflection point?

A: The inflection point represents a change in the concavity of the position function. At t = 2, the position function changes from being concave up to being concave down.

Q: Can you provide more information about the particle's motion?

A: Yes, we can provide more information about the particle's motion. The particle's motion is described by the position function y(t) = t^3 - 6t^2 + 9t. The particle's velocity is given by the derivative of the position function, and the particle's acceleration is given by the derivative of the velocity function. The particle's motion is critical at the points t = 1 and t = 3, where the position function has a maximum and minimum, respectively.

Q: Can you provide more information about the mathematical techniques used to analyze the particle's motion?

A: Yes, we can provide more information about the mathematical techniques used to analyze the particle's motion. The mathematical techniques used include calculus, specifically the derivative and second derivative of the position function, as well as the second derivative test to determine the nature of the critical points.

Q: Can you provide more information about the physical significance of the particle's motion?

A: Yes, we can provide more information about the physical significance of the particle's motion. The particle's motion represents a physical system where the particle's position is given by the position function y(t) = t^3 - 6t^2 + 9t. The particle's velocity and acceleration are given by the derivative and second derivative of the position function, respectively. The particle's motion is critical at the points t = 1 and t = 3, where the position function has a maximum and minimum, respectively.

Conclusion

In conclusion, the particle's motion along the y-axis is described by the position function y(t) = t^3 - 6t^2 + 9t. The particle's velocity and acceleration are given by the derivative and second derivative of the position function, respectively. The particle's motion is critical at the points t = 1 and t = 3, where the position function has a maximum and minimum, respectively. The inflection point of the position function is t = 2, where the position function changes from being concave up to being concave down.

References

  • [1] Calculus: Early Transcendentals, James Stewart, 8th edition.
  • [2] Differential Equations and Dynamical Systems, Lawrence Perko, 3rd edition.
  • [3] Mathematical Methods for Physicists, George Arfken, 7th edition.

Keywords

  • Position function
  • Critical points
  • Second derivative test
  • Inflection points
  • Numerical methods
  • Particle motion
  • Gravitational field
  • Electrical circuits

Categories

  • Mathematics
  • Physics
  • Engineering

Tags

  • Calculus
  • Differential equations
  • Dynamical systems
  • Mathematical analysis
  • Numerical methods
  • Particle motion
  • Position function
  • Critical points
  • Second derivative test
  • Inflection points