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Find the difference between the upper and lower estimates of the distance traveled at velocity $f(t)=34-t^{2}$ on the interval $1 \leq t \leq 4$, for $n=500$ subdivisions.

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The difference between the upper and lower estimates is

Step 1 ：Divide the interval \(1 \leq t \leq 4\) into \(n=500\) subdivisions.

Step 2 ：Calculate the width of each subdivision: \(\Delta t = \frac{4-1}{500} = \frac{3}{500}\)

Step 3 ：Calculate the upper and lower estimates of the distance traveled:

Step 4 ：For the upper estimate, calculate the maximum value of \(f(t)\) in each subdivision and multiply it by the width of the subdivision.

Step 5 ：For the lower estimate, calculate the minimum value of \(f(t)\) in each subdivision and multiply it by the width of the subdivision.

Step 6 ：Calculate the difference between the upper and lower estimates:

Step 7 ：Difference = Upper estimate - Lower estimate

Step 8 ：Calculate the upper estimate:

Step 9 ：Upper estimate = \(\Delta t \cdot (f(1) + f(1+\Delta t) + f(1+2\Delta t) + ... + f(4-\Delta t) + f(4))\)

Step 10 ：Calculate the lower estimate:

Step 11 ：Lower estimate = \(\Delta t \cdot (f(1) + f(1+\Delta t) + f(1+2\Delta t) + ... + f(4-\Delta t) + f(4))\)

Step 12 ：Calculate the difference between the upper and lower estimates:

Step 13 ：Difference = Upper estimate - Lower estimate