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Problem 145 Easy Difficulty

Two mountain climbers start their climb at base camp, taking two different routes, one steeper than the other, and arrive at the peak at exactly the same time. Is it necessarily true that, at some point, both climbers increased in altitude at the same rate?

Answer

$\begin{aligned} f_{1 a v} &=f_{2 a v} \\ \frac{1}{b-a} \int_{a}^{b} f_{1}(t) d t &=\frac{1}{b-a} \int_{a}^{b} f_{2}(t) d t \\ f_{1}(c) &=f_{2}(c) \end{aligned}$
This means that, at some point during the mountain climbing, both climbers increased in altitude at the same rate.

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Calculus 1 / AB

Calculus 2 / BC

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The Fundamental Theorem of Calculus

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Video Transcript

So for this problem, we aren't thinking about a situation in which two mountain climbers are going to be starting at the same place at the same time, an end at the same spot the same time. But they're taking two different routes. So one of the routes is steeper and the other routes is much less steep. And we are thinking about whether it's true that their rates had to be the same at some point in time. And so this, essentially an application of the immune value here, which tells us that there's some point inner interval, other will always be some point where, uh, are our function of that point is going to be equal to the average over our interval of our graph that Lex Dex and so essentially, since we know that the mountain climber started and ended at the same points at the same time, that tells us that our average value honor interval is going to be the same. So that means that there has to be some point off of see that's going to be equal to our average for each of the mountain climbers. So for our first man climber, say they had a one of C in her second mon clamoring F to F C I'm again, since their average values were the same. That tells us that they each have to have some individual point on where their averages were the same.

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