WEBVTT
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derivatives and second derivatives are very powerful. Ultimately,
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allow us to, um, determine the shapes of
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graphs. That's one really special use of them.
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And that's what we're going to see in this problem
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. The fact that we can take derivative, we
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could take the second derivative and were able thio see
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minimums and maximums, con cavity and a lot of
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other really important details within graphs. Okay, so
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this is the graph that we're dealing with for this
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problem in particular. Um, and we first want
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to determine intervals of increase and decrease. So we
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see the graph is decreasing from negative infinity to zero
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. It increases from 0 to 1 and then decreases
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from one to infinity. Then we want to determine
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local minimums and maximums. We see that there's a
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local minimum right here in 00 I'm Interestingly, if
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we look at the derivative craft, we see that
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, um, it's undefined at this 0.0, which
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also indicates that it's critical point on a local minimum
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. Then we see there's a local maximum at one
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, and it's this value here 13 With that,
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we now want to move on to Concha Vitti.
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We see that the graph is con cave up until
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negative one half and then it becomes con cave,
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um down and it's undefined zero. Then it goes
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con cave down. Um, it remains con cave
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down for the rest of the graph. I mean
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, when we have the sharp turns and make things
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really interesting because the graph is not actually differential at
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that point, then we have, um Now,
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since we've determined con cavity, we can also determine
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inflection points. We see that this is where the
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second derivative graph equals. Zero spot tells us that
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, um, negative 0.5 is an inflection point of
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this graph.