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Numerade Educator

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Problem 69 Hard Difficulty

Let $ K(t) $ be a measure of the knowledge you gain by studying for a test for $ t $ hours. Which do you think is larger, $ K(8) - K(7) $ or $ K(3) - K(2) $? Is the graph of $ K $ concave upward or concave downward? Why?

Answer

Most students learn more in the third hour of studying than in the eighth hour, so $K(3)-K(2)$ is larger than $K(8)-K(7)$
In other words, as you begin studying for a test, the rate of knowledge gain is large and then starts to taper off, so $K^{\prime}(t)$
decreases and the graph of $K$ is concave downward.

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Video Transcript

But we're told that KFT tells us the knowledge gained after tea hours of studying and we want to figure out if K. Of eight minus K. Of seven. So the knowledge gained after eight hours minus the knowledge gained after seven hours. If this is greater than or equal to the knowledge gained after three hours minus the knowledge gained after two hours. So the way that we want to think about this is when you're studying it seems like we get at the beginning we're studying and we're increasing in terms of our knowledge. Our knowledge is increasing at a higher rate after three hours versus after eight hours after eight hours of studying you're going to be increasing your knowledge but it's going to be less and less of a big increase than after three hours. So after three hours minus the knowledge after two hours, this is actually gonna be bigger than the increase from eight to seven hours. Since the increase in knowledge studying from three hours to two hours is a lot bigger than the increase from seven hours to eight hours. So this one K over three minus or K. Of three minus cave to is going to be bigger than cave eight -K. of seven. And now we want to figure out if we're concave up or concave down, well, we know that we are increasing our knowledge at a decreasing rate. So our first derivative K. Prime of T. That's still going to be positive since we're increasing however, were increasing at a decreasing rate. So our second derivative is going to be negative, which means that we are going to be concave down. So this graph cave cave T. Is going to be concave down. All right.