The position of a \(6.0-\mathrm{kg}\) shopping cart rolling down a ramp is given by \(x(t)=p+q t^{2}\), with \(p=+1.50 \mathrm{~m}\) and \(q=+2.00 \mathrm{~m} / \mathrm{s}^{2}\). What is the \(x\) component of the cart's average velocity 

(a) between \(t=2.00 \mathrm{~s}\) and \(t=3.00 \mathrm{~s}\),

(b) between \(t=2.00 \mathrm{~s}\) and \(t=2.10 \mathrm{~s}\), and \((c)\) between \(t=2.00 \mathrm{~s}\) and \(t=2.01 \mathrm{~s}\) ?

(d) Compute the limit of the average velocity between \(t_{\mathrm{i}}=2.00 \mathrm{~s}\) and \(t_{\mathrm{f}}=2.00 \mathrm{~s}+\Delta t\) as \(\Delta t\) approaches zero.

(e) Show that your result agrees with what is expected by taking the time derivative of position.