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Problem 65 Medium Difficulty
A graph of a population of yeast cells in a new laboratory culture as a function of time is shown.
(a) Describe how the rate of population increase varies.
(b) When is this rate highest?
(c) On what intervals is the population function concave upward or downward?
(d) Estimate the coordinates of the inflection point.
Answer
$f(x)=\frac{x^{4}+x^{3}+1}{\sqrt{x^{2}+x+1}}$, In Maple, we define $f$ and then use the command
$\text { plot (diff (diff }(f, x), x), x=-2 \ldots 2) ;$ In Mathematica, we define $f$
and then use $\operatorname{Plot}[\mathrm{Dt}[\mathrm{Dt}[\mathrm{f}, \mathrm{x}], \mathrm{x}],\{\mathrm{x},-2,2\}] .$ We see that $f^{\prime \prime}>0$ for
$x<-0.6$ and $x>0.0[\approx 0.03]$ and $f^{\prime \prime}<0$ for $-0.6<x<0.0 .$ So $f$ is $\mathrm{CU}$
on $(-\infty,-0.6)$ and $(0.0, \infty)$ and $\mathrm{CD}$ on (-0.6,0.0)
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Video Transcript
So in this problem we're given this graph of number of the cells over time in a new laboratory and were asked a series of questions. First question is described how the rate of this population varies with time. Well, if we look at the first part of the graph over here, what do we see? We see that the rate of increase of the population is very slow or very low, isn't it? Is initially very we'll call we'll say very small about okay then what happens in this area? Well then we get a large rate of increase. Don't we have a very steep slope? Then I read a growth gives large until it reaches a maximum somewhere around eight hours. Okay. And then what then decreases, doesn't it? Because what's the rate up here? Well, that's like nearly zero. So it decreases toward zero. Okay. They were asked next. Were asked what is the rate the highest? Well, we just kind of answer this question, didn't we? You could say the rate is the highest at t equals eight hours. Hey then we ask when is the current draft? Concave upper, Concave down? Remember concave up? Is the bent wire going upwards or the smiley face? Okay. So we are con Hey dave ah in this section, aren't we? If uh so we could say from T equals 0, 2 t equals eight hours. All right then. What then we're asked when we when are we concave down? Well, came down just the opposite, isn't it? And where khan cave down here, aren't we? Okay. We could say from T equals eight two T equals 16 hours. All right. They were asked the coordinates of the inflection point. Remember the inflection point is when we go from concave up to concave down or concave down to concave up and others would change can cavity. Well that means the inflection point is here, isn't it? And what do we see? We see that that is that approximately eight and spending on how you read this graph? It could be 350 or 400. Let's call it 350 cause we're just approximating it anyway or estimating it. And so there were There we go. We have all four questions now answered.
Oklahoma State University
Top Calculus 2 / BC Educators
Catherine R.
Missouri State University
Kristen K.
University of Michigan - Ann Arbor
Samuel H.
University of Nottingham
Michael J.
Idaho State University