Question

1. the velocity in feet per seconds of a car is given below.

(i) use the left - endpoint approximation to estimate the total distance traveled from ( t = 0 ) seconds to ( t = 10 ) seconds.

(ii) repeat (i) using the rig - endpoint approximation.

Explanation:

Step1: Determine the width of each sub - interval

The interval is from \(t = 0\) to \(t=10\), and \(n = 5\) sub - intervals. The width of each sub - interval \(\Delta t=\frac{10 - 0}{5}=2\)

Step2: Left - Endpoint Approximation

The left - endpoint formula for the sum \(L=\sum_{i = 0}^{n-1}v(t_i)\Delta t\)
Here, \(t_0 = 0\), \(t_1=2\), \(t_2 = 4\), \(t_3=6\), \(t_4 = 8\)
\(L=(v(0)+v(2)+v(4)+v(6)+v(8))\times\Delta t\)
Substitute \(v(0) = 0\), \(v(2)=10\), \(v(4)=14\), \(v(6)=16\), \(v(8)=18\) and \(\Delta t = 2\)
\(L=(0 + 10+14+16+18)\times2=(58)\times2 = 116\)

Step3: Right - Endpoint Approximation

The right - endpoint formula for the sum \(R=\sum_{i = 1}^{n}v(t_i)\Delta t\)
Here, \(t_1 = 2\), \(t_2=4\), \(t_3 = 6\), \(t_4=8\), \(t_5 = 10\)
\(R=(v(2)+v(4)+v(6)+v(8)+v(10))\times\Delta t\)
Substitute \(v(2)=10\), \(v(4)=14\), \(v(6)=16\), \(v(8)=18\), \(v(10)=19\) and \(\Delta t = 2\)
\(R=(10 + 14+16+18+19)\times2=(77)\times2=154\)

Answer:

(i) The left - endpoint approximation of the distance is \(116\) feet.
(ii) The right - endpoint approximation of the distance is \(154\) feet.