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On a recent problem, I received the following scenario: An object moves back and forth on a straight track. During the time interval $0\leq{t}\leq30$ minutes, the object's position, $x$, and velocity, $v$, are continuous functions; some of their values are shown in the table (which I have reproduced below).

\begin{array}{|c|c|c|}\hline \textbf{$t$ (min)} & \textbf{$x(t)$ (feet)} & \textbf{$v(t)$ (feet/min)} \\\hline 0 & \text{12} & \text{$-20$} \\\hline 10 & \text{50} & \text{20} \\\hline 15 & \text{18} & \text{3} \\\hline 25 & \text{60} & \text{$-2$}\\\hline 30 & \text{60} & \text{10} \\\hline \end{array}

The question was, for $0<t<30$, does there exist a time $t$ when $v(t) = -22$?

I tried to apply the intermediate value theorem, and concluded that the answer was "not necessarily." I reasoned that because the values of $v(t)$ in the table ranged between $-20$ (at the beginning) and $10$ (at the end), and $-22$ wasn't between those two values, we couldn't be sure that such a $t$ exists.

The teacher gave me 3/4 points on the question. Their comment was that I should have considered the mean value theorem, too. They did not write anything about my intermediate value theorem analysis, but I still have grave doubts about my application of the intermediate value theorem ... and, as for the mean value theorem, I have no idea how to proceed.

Would anyone here be able to shed some light on how the intermediate value and mean value theorems could be used to determine whether there exists a $t$ where $v(t) = -22$? Thanks so much.

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One place your answer is lacking is that for the IVT you should consider all the known velocities. You do know there was a time when $v(t)=+15$ even though it is not between $-20$ and $+10$ because $v(10)=+20$. One place the mean value theorem would help is if the velocity at $t=11$ were $-18$ because the average velocity over that segment would be $-32$ and the average velocity over the previous segment was positive, so the velocity must have passed through $-22$. With the data you have I do not see a way to prove the velocity was ever $-22$, but it certainly could have been if there were large variations between the data points.

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  • $\begingroup$ Thanks for the answer! So my answer of "not necessarily, but possible," was correct then... (PS - if you edit your post I can upvote it - I accidentally downvoted it earlier) $\endgroup$
    – Will
    Jan 27, 2020 at 15:53
  • $\begingroup$ How do you get -32? And what do you mean by "previous segment"? $\endgroup$
    – Will
    Jan 27, 2020 at 15:55
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    $\begingroup$ I dropped a minus sign, corrected above. If the velocity were $-18$ then the average over $10-11$ is $-32$. The previous segment is $5-10$ $\endgroup$ Jan 27, 2020 at 16:05
  • $\begingroup$ How are you calculating the average - are you using $x(t)$ at all? $\endgroup$
    – Will
    Jan 27, 2020 at 18:17
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    $\begingroup$ No, I just took the difference in velocities and divided by the time to get the average velocity over the interval $\endgroup$ Jan 27, 2020 at 18:21

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