Question

step 1
we will use either ( l_6 ) or ( r_6 ) for the upper and lower estimates.
since the runners speed is an increasing function, then ( l_6 ) ( l_6 ) will give the lower estimate, and ( r_6 ) ( r_6 ) will give the upper estimate.
step 2
the sub - interval widths ( delta t ) for this situation are ( delta t=)
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Explanation:

Step1: Recall the formula for sub - interval width

The formula for the width of a sub - interval \(\Delta t\) when using \(n\) sub - intervals over an interval \([a,b]\) is \(\Delta t=\frac{b - a}{n}\). Assuming the interval for time is \([0,6]\) (since \(n = 6\) sub - intervals are used, a common interval for \(n=6\) is from \(t = 0\) to \(t=6\)).

Step2: Calculate \(\Delta t\)

Using \(a = 0\), \(b = 6\) and \(n=6\), we have \(\Delta t=\frac{6 - 0}{6}\).

$$ \Delta t = 1 $$

Answer:

\(\Delta t=1\)