91Ó°ÊÓ

The position equation is a mathematical statement that describes the location of a particle at any given time. For this exercise, the position of the particle is given by the equation: \[x = 3t^{2}\]where:

• \(x\) is the position of the particle in meters.
• \(t\) is the time in seconds.

To evaluate the position, simply substitute the value of time \(t\) into the equation. For instance, at \(t = 3.00\) s, the calculation becomes:\[x = 3(3.00)^{2} = 27 \, \text{meters}\]This tells us that the particle is located 27 meters along the \(x\)-axis at 3 seconds. Similarly, to find the position at a future time \(3.00 \, s + \Delta t\), you substitute this time into the position equation:\[x = 3(3.00 + \Delta t)^{2}\]

Limits in calculus are used to understand the behavior of a function as its input approaches a specific value. They are fundamental in finding derivatives, which provide information about the rate of change. In this exercise, limits help us find the instantaneous velocity of the particle at \(t = 3.00 \, s\).To apply limits, we consider the expression for the average velocity over a small time interval \(\Delta t\):\[\frac{\Delta x}{\Delta t} = \frac{x(3.00 + \Delta t) - x(3.00)}{\Delta t}\]As \(\Delta t\) approaches zero, the average velocity approaches the instantaneous velocity. Simplifying the expression:\[x(3.00 + \Delta t) = 3(3.00 + \Delta t)^{2}\]Subtract the initial position \(x(3.00) = 27 \, \text{meters}\) to find:\[\Delta x = 3(3.00 + \Delta t)^{2} - 27\]Dividing by \(\Delta t\) and taking the limit as \(\Delta t\) approaches zero gives us the derivative, which represents the velocity at a specific moment.

Instantaneous velocity is the rate of change of position with respect to time at a single point. It is essentially the derivative of the position equation with respect to time.In this scenario, to find the instantaneous velocity at \(t = 3.00 \, s\), we take the derivative of the position equation \(x = 3t^{2}\):\[\frac{dx}{dt} = \frac{d}{dt}(3t^{2}) = 6t\]Then, substitute \(t = 3.00\) s into the derivative:\[\frac{dx}{dt} = 6(3.00) = 18 \, \text{m/s}\]This calculation shows that at \(t = 3.00\) seconds, the particle's velocity is 18 meters per second. This represents how quickly the particle is moving along the \(x\)-axis at exactly this point in time, distinguishing it from average velocity, as it reflects the speed at an exact moment rather than over a period.