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I am trying to do this homework problem for calculus. It is an intro to integrals and I have no idea what I am doing wrong.

The speed of a runner is increase steadily during the first $3$ seconds of a race. Her speed at half seconds intervals is given in the table. Find lower and upper estimates for the distance she traveled during these $3$ seconds.

$$ \begin{matrix} t(\mathrm{s}) & & 0 & .5 & 1 & 1.5 & 2 & 2.5 & 3 \\ v(\mathrm{ft}/\mathrm{s}) & & 0 & 6.2 & 10.8 & 14.9 & 18.1 & 19.4 & 20.2 \end{matrix} $$

I drew two graphs of this and added the rectangles for the estimate and I got two very different looking estimates, one obviously over and one obviously lower. If I understand this right the over estimate is calculated like so $$.5(6.2) + .5(10.8) + \cdots ,$$ which seems correct to me.

The under estimate is calculated $0 \cdot 0 + \text{the rest of the series}$.

I don't see what makes these two different but neither are correct numbers and I get the same for both which I know it isn't right.

1 Answers 1

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Your under-estimate should be multiplying by the length of the intervals in the same way your over-estimate did. So you will have $$0.5(0)+0.5(6.2)+\cdots+0.5(19.4)$$ which will not be the same as your over-estimate, which is $$0.5(6.2)+0.5(10.8)+\cdots+0.5(20.2).$$

The difference is that you are using the speed at the start of the time interval rather than the (faster) speed at the end.

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    I don't undestand the difference. I am going to end up with the same numbers, just on the left interval one I start with a times 0.2011-10-31
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    On the left interval sum, you won't include the very last term. You should have the same number of terms (6) in each estimate.2011-10-31
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    So I always use xn-1? I can't visualize this. On my graphs I have 6 intervals each and they don't seem to match up with what Stewart wants me to do.2011-10-31
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    I've edited my answer to show the difference.2011-10-31
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    But I am using the speed at the start and at the end for both.2011-10-31
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    You should have six intervals for each. Based on the values you are given, you assume that the speed at the left end is the minimum speed for that interval, and the right is the maximum. So when you want to underestimate, you multiply the length of each interval (0.5), by its minimum speed (the left). To overestimate you multiply the interval length (0.5) by the maximum speed (the right). You always have six intervals, but some terms will be different in each sum.2011-10-31
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    I don't understand, so I use each interval and multiply it by the lowest speed? Stewart claims that I should multiply f(x) by delta x.2011-10-31
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    $f(x)$ is the velocity $v$ that you were given. $\Delta x$ is the length of the interval. So yes, multiply the two together and sum over all the intervals.2011-10-31
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    I don't understand, I thought I multiply all these t(s) 0, .5, 1, 1.5, 2, 2.5, 3 v(ft/s) 0, 6.2, 10.8, 14.9, 18.1, 19.4, 20.2 as in the first with the first the second with the second and so on. Also looking at my book the only difference mathematically between left hand endpoints and righthand is that right uses every point and left skips the last one. This makes no sense at all, it is not what I am seeing on a graph. When I graph the 2 one had boxes going outside the curve and the other has boxes missing inside the curve.2011-11-01
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    You never multiply by $0, 0.5, 1,$ etc. but by $(0.5-0),(1-0.5),(1.5-1),$ etc.--the lengths of the intervals, ie. $\Delta x$. If you check your book again, you should see that right uses every point *except the first* and left uses every point *except the last*. If it does not, then that is a mistake in the book. Right must use only the points on the right end of the intervals, and left only the points on the left. Your graph sounds correct.2011-11-01
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    I understand the lengths, the delta x, so I multiply by the delta x I don't understand so the last point that the left doesn't use is equal to the area missing out of the function? Or the difference between the two graph's rectangles? So the last function or the first is equal to the differences in the functions? So if I want to make the left endpoint process be equal to the right I could just add the last interval to whatever I got for the left hand and I would have the proper answer?2011-11-01
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    I just can't get this to work, I can never get the same numbers. The book says I should be able to get the same number.2011-11-01
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    The last value times the length of the interval will be the difference between your over- and under-estimates *in this problem*. This is only because your first value of $0$ essentially gets "thrown out" of the problem, your function is increasing, and your intervals are all equally spaced. Problems where this is not the case will not have such easily related sums. You should not get the same number for the left- and right-hand sums.2011-11-01
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    @Jordan Please try to keep your language polite in this forum.2011-11-01