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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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02:59

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Missouri State University

Campbell University

Oregon State University

Baylor University

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Shown is the graph of the …

01:29

Yeast population Shown is …

02:12

08:03

The graph below shows the …

The number of yeast cells …

04:47

A graph of a population o…

01:05

Modeling yeast populations…

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The number $Y$ of yeast …

The number $Y$ of yeast or…

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.

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