EXPLANATION
Given that the arrow function is as follows:
[tex]f(t)=-16t^2+128t[/tex]
The instantaneous velocity is given by the following relationship:
[tex]\frac{f(t+\Delta t)-f(t)}{\Delta t}\text{ at t=7}[/tex]
For △t = 0.1:
[tex]Velocity=\frac{\lbrack-16(7+0.1)^2+128\cdot(7+0.1)\rbrack-\lbrack-16\cdot(7)^2+128\cdot7\rbrack}{0.1}[/tex]
Computing the powers:
[tex]Velocity=\frac{\lbrack-16\cdot50.41+128\cdot7.1\rbrack-\lbrack-16\cdot49+128\cdot7\rbrack}{0.1}[/tex]
Multiplying numbers:
[tex]Velocity_{0.1}=\frac{\lbrack-806.56+908.8\rbrack-\lbrack-784+896\rbrack}{0.1}[/tex]
Removing the parentheses and adding numbers:
[tex]Velocity_{0.1}=\frac{-9.76}{0.1}=-97.6\approx98_{\text{ }}ft/s[/tex]
For △t=0.01
[tex]Velocity=\frac{\lbrack-16(7+0.01)^2+128\cdot(7+0.01)\rbrack-\lbrack-16\cdot(7)^2+128\cdot7\rbrack}{0.01}[/tex]
Computing the powers:
[tex]Velocity=\frac{\lbrack-16\cdot49.1401+128\cdot7.01\rbrack-\lbrack-16\cdot49+128\cdot7\rbrack}{0.01}[/tex]
Multiplying numbers:
[tex]Velocity_{0.01}=\frac{\lbrack-786.2416+897.28\rbrack-\lbrack-784+896\rbrack}{0.01}[/tex]
Adding numbers:
[tex]Velocity_{0.01}=\frac{-0.9616}{0.01}=-96.16\approx-96ft/s[/tex]
For △t = 0.001
Applying the same reasoning than before, give us the following result:
[tex]Velocity_{0.001}=-96.016\text{ ft/s}\approx-96ft/s[/tex]
