Question

A bowling ball dropped from a height of $400$feet will be $s\left(t\right)=400-16t^2$feet from the ground after $t$seconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

After $t=2$ seconds, with $h=0.1,h=0.01h=-0.1andh=-0.01$

Short answer

Ans: When $h=0.1$instantaneous velocity is: $-65.4$

When $h=0.01$instantaneous velocity is: $-65$

When $h=-0.1$instantaneous velocity is: $-62.4$

When$h=-0.01$instantaneous velocity is:$-64$

Step-by-step solution

Step 1. Given information.

given,

A ball is dropped from a height of $400$feet and its distance from the ground after $t$seconds is, $s\left(t\right)=400-16t^2$
The objective is to estimate the instantaneous velocities for,

$t=2s,h=0.1,h=0.01,h=-0.1andh=-0.01$

Step 2. To find the instantaneous velocity for h=0.1 follows the steps:

$f\left(t\right)=f\left(2\right)=400-16{\left(2\right)}^2=336$

And

$f(t+h)=f(2.1)=329.44$

Therefore the instantaneous velocity is:

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{329.44-336}{0.1}\\ =\frac{-6.54}{0.1}\\ =-65.4\end{array}$

Step 3. To find the instantaneous velocity for h=0.01 follows the steps: 

$f\left(t\right)=f\left(2\right)=336$

And

role="math" $f(t+h)=f(2.01)=335.35$

Therefore the instantaneous velocity is:

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{335.35-336}{0.01}\\ =\frac{-0.65}{0.01}\\ =-65\end{array}$

Step 4. To find the instantaneous velocity for h=-0.1 follows the steps: 

$f\left(t\right)=f\left(2\right)=336$

And

$f(t+h)=f(1.9)=342.24$

Therefore the instantaneous velocity is:

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{342.24-336}{-0.1}\\ =\frac{6.24}{-0.1}\\ =-62.4\end{array}$

$f(t+h)=f(1.9)=324.24$

Step 5. To find the instantaneous velocity for h=-0.01 follows the steps: 

$f\left(t\right)=f\left(2\right)=336$

And, $f(t+h)=f(1.99)=336.64$

Therefore the instantaneous velocity is:

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{336.64-336}{-0.2}\\ =\frac{0.64}{-0.01}\\ =-64\end{array}$