An object moves along the $x$ axis according to the equation $x=2\mathrm{.}65t^2-2\mathrm{.}00t+3\mathrm{.}00,$ where $x$ is in meters and $t$ is in seconds.

$($a$)$ Determine the average speed between $t=1\mathrm{.}50s$ and $t=2\mathrm{.}60s.$

The average speed is the distance traveled divided by the time. Is the distance traveled equal to the displacement in this case? $\frac{m}{s}$

$($b$)$ Determine the instantaneous speed at $t=1\mathrm{.}50s.$

This is the average velocity between $0$ and $1\mathrm{.}50$ seconds. $\frac{m}{s}$

Determine the instantaneous speed at $t=2\mathrm{.}60s.$

This is the average velocity between $0$ and $2\mathrm{.}60$ seconds. $\frac{m}{s}$

$($c$)$ Determine the average acceleration between $t=1\mathrm{.}50s$ and $t=2\mathrm{.}60s.$

$\frac{m}{s^2}$

$($d$)$ Determine the instantaneous acceleration at $t=1\mathrm{.}50s.$

The instantaneous acceleration is determined by taking the second time derivative of the position versus time function. $\frac{m}{s^2}$

Determine the instantaneous acceleration at $t=2\mathrm{.}60s.$

The instantaneous acceleration is determined by taking the second time derivative of the position versus time function. $\frac{m}{s^2}$

$($e$)$ At what time is the object at rest?

The object is moving at $t=0s.$ You can demonstrate this by taking the derivative of $x(t)$ and