The speed of a runner increased steadily during the first three seconds of a rac

Question

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second Intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. Step 1 We will use either L_6 or R_6 for the upper and lower estimates. Since the runner's speed is an Increasing function, then 3 L_6 will give the lower estimate, and 44.8 R_6 will give the upper estimate. Step 2 The sub-interval widths Delta t for this situation are Delta t = 0.5. The first two sub-intervals in the table are [0,0.5] and [0.5, 1.0]. When calculating L_6, we should use the function values v = and v = for these two sub-intervals, respectively.

Explanation / Answer

Step1)In the question speed vs time table is given at an interval of 0.5sec and it is clear that it is an increasing function

MISTAKE DONE:

In the question he is not asking you to find the values of lower and upper estimates he is just asking what will give you lower estimate and what will give you upper estimate.L6 will give lower estimate and R6 will give you upper estimate.For better understanding

The left estimate of the distance run for the nth interval is (v_(n-1))t , and the right estimate is (v_n)t. Since the speed increased steadily, v_(n-1) < v_n, so (v_(n-1))t < (v_n)t. Furthermore, because v is an increasing function of t, for any time t in the nth interval we have v_(n-1) < v < v(n), so (v_(n-1))t < vt < (v(n))t.

Lower estimate(L6)=(0+6.2+10.8+14.9+18.1+19.4)*0.5=(69.4)*0.5=34.7ft

upper estimate(R6)=(6.2+10.8+14.9+18.1+19.4+20.2)*0.5=44.8ft

Step3)While calculating L6 0,6.2 values are used for the first two intervals

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