Shown is the graph of the population function $ P(t) $ for yeast cells in a laboratory culture. Use the method of Example 1 to graph the derivative $ P'(t) $. What does the graph $ P' $ tell us about the yeast population?
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So in this problem we're given this graph on the population of the number of the cells in a culture. And that functions called PFT. And were asked to graph The rescues example one. Right? & graft the derivative P. Prime of T. So down here we'll put P. Prime of T. And we'll have axes down here. We're still T. Is in hours. Okay. This is five. This is 10 15. Okay. And what do we know? We know that P prime of T. Is the slope uh tangent lines for P. Of tea. So in other words, if we were to draw tangent lines through here right Like this and then like this and then like this. Okay. And of course something like this. Okay. All we would do is graph their slopes. So what do we see? We'll see first of all that we start out at near zero. And the tangent line. The slope is increasing as we move along and it hits a maximum right here, doesn't it gets the steepest right there because that's the inflection point where we're going from concave up to concave down on this curve. Then the slope starts to decrease. Still remaining positive. But it is decreasing as we work our way on up the curve all the way back to zero again, doesn't it? So what does that mean? That means we have a graph that does this, don't we like that? Or my slope a graph the slopes then they increase increase increase increase and they hit a maximum right here, don't they? Okay. At the inflection point and then they begin to decrease from there still positive, but they're decreasing back to zero. So therefore we now have the graph of the derivative PFT.