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Okay, this is a video for Objective 1.14.
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It has to do with analyzing and interpreting Cartesian graph of a function in terms of its second derivative,
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which is related to something called acceleration and deceleration, which you're already familiar with,
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and the breaking point between those two areas happens at an inflection point,
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and you can tell which one is which by its concavity.
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So, if you read little summary notes here, it tells you all kinds of little facts
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that you should probably be able to understand and interpret when you look at a graph.
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So, rather than try to go through all these, you can look at these on your own,
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and you can maybe even have them in front of you when you're watching the video
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to see if I pretty much cover everything here.
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So, what I'm going to do is I'm going to bring up a graph or two,
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and I'm just going to talk about them in relation to this objective,
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and then look at a couple other interesting things.
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Okay, so let's see here.
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So, here I have a graph, and the black graph is the function.
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Say, let's call it F.
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The blue graph, excuse me, the red graph is going to be the derivative of that graph,
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and the blue graph is going to be the derivative of the derivative,
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or the second derivative of the original function.
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So, for example, if I say F prime of X, and then I prime that,
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I take the derivative, I end up with something that's called the second derivative
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of the original function F.
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Okay, so they're all related to each other.
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The question is, how are they related?
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How are they related?
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What is happening in the black graph that we can discern from the red graph and the blue graph?
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Now, you've been doing objectives where you've been trying to relate the blue,
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or excuse me, the black graph to a red graph, the function to its derivative.
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Now, we're just going to throw in one more level, which is the second derivative.
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And you already know some things about the second derivative.
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For example, you know that acceleration is when the rate of change is changing, going up,
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and when the rate of change is going down, then it's called deceleration, deceleration.
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So, let's look at the original function here and see a few things.
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Let's review a few things.
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As the original function goes through a maximum, right?
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Goes through a maximum.
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It's had a max right here.
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Then, the derivative goes through zero.
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Why? Because the tangent up here is horizontal.
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So, the derivative F goes through zero when the tangent of the original function is at zero.
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And you notice all these slopes, or tangent slopes, are positive in here.
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So, it's no surprise that the red graph is all above what?
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All above zero.
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All above zero until we get to this top here.
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So, remember what it is?
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When the function is increasing, the first derivative is positive, right?
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The first derivative is positive.
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And it remains positive until the function goes through a maximum.
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And then what?
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The first derivative becomes negative.
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So, it's all negative in here.
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And it's all negative.
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And it doesn't turn positive again until the function itself goes through a minimum.
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So, in here, all the derivatives, all the tangent line slopes are negative.
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And then all of them are positive again.
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You see that the red graph is now above zero.
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And once again, it goes through a maximum.
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So, it's zero again and so on.
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Okay? So, you should have that down already.
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That's from a different objective.
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Now, what's happening in the blue graph relative to the black graph?
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So, first of all, we notice where does the blue graph go through zero?
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Where does the blue graph go through zero?
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It happens to go through zero at exactly the same place as the original function goes through what?
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What are you going to call this thing?
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What are you going to call this point right here?
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Let's give them a different color here.
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We're going to call this guy.
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Those are called inflection points, right?
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Inflection points.
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So, whenever the original function goes through an inflection point,
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that's when the second derivative changes sign.
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Before the inflection point, in this case, it's negative and after it's positive.
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In this case, it was positive and then negative.
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Remember, positive is on this side.
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When you're above the horizontal axis, negative is down here and you're below the horizontal axis.
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So, what does that mean to be positive acceleration?
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What does it mean to be positive acceleration?
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That means that your rate of change is going up in the next instant.
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Is that true?
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What if I go back here to my original graph?
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What's happening to these tangent lines?
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They're getting what?
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They're negative and they're getting ever closer to what?
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Zero.
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So, as far as the sign is concerned, this slope is smaller than this one is smaller than zero,
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so they're actually getting bigger, right?
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So, technically, it's accelerating, right?
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The tangent slopes are getting bigger and that's exactly what we discovered.
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And they're getting bigger and bigger and bigger and bigger.
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And they keep getting bigger.
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Now, they went through zero, right?
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And now, they're continuing to get bigger.
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See, the tangent slopes just went through zero.
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And they're continuing to get bigger, which means that it's still accelerating, right?
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It's still accelerating until it hits this other inflection point
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and then the slopes of the tangent lines start getting smaller and smaller and smaller and smaller and smaller and smaller and smaller.
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Okay?
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And as they do that, the function is decelerating.
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Decelerating.
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Decelerating.
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And then, apparently, apparently what?
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If I go back to the blue here, I can't really see very well because the function doesn't go on,
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but according to the second derivative graph, there's another inflection point right here.
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So, here are the...
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They're doing what?
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They're decelerating.
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Decelerating is accelerating.
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And then, what?
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They start going back up again.
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They start going back up.
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So, how are we going to distinguish quickly when we're looking at
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the original black graph, regions of acceleration from regions of deceleration?
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Well, the answer is pretty easy.
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If I just look to the left of this inflection point,
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I see that the curve is what we call a concave down.
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In other words, if you put a marble in it, if you put a marble in it, it just falls right out.
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Whereas, if I put a marble in right here, if I dropped a marble right here,
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it would sit in this little basket.
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So, we call that concave up, concave up.
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Let's get a little sloppy in here.
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Whoa, better not do that.
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That's concave up, concave up in here, and this would be concave down.
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So, what do we learn?
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Concave down is in a region in which the second derivative is negative.
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So, it's deceleration.
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And concave up is in a region where the second derivative is positive,
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and that's acceleration.
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So, I'm accelerating in here, and I'm decelerating here,
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and I'm decelerating here, and in this last little honk right here, I'm actually accelerating again.
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Okay, so, the concavity and the inflection points in the original function help me to determine
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what the second derivative graph would look like, where it will go through zero,
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and what, and whether it's going to be positive or negative.
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Positive second derivative, acceleration, negative second derivative, deceleration.
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Okay, so, let's see here, what we do in this graph here.
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This is the graph of the logistic.
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This is the graph of the logistic.
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So, what are we going to do here?
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Well, logistic has just one inflection point, right, and where is it?
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It's right about in here.
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How do I know it's right about in there?
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Well, which one of these is the first and second derivative?
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Now, don't get fool on the color.
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Don't get fool on the color.
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It's not the same color as the last one.
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Turns out what?
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Turns out the blue one is the first derivative.
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So, this one is F prime, if this function is F,
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and this one down here is F double prime, the second derivative.
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So, what does that say?
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When you go through an inflection point, the first derivative goes through a maximum rate.
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The rate is maximum.
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It's getting bigger, bigger, bigger, bigger.
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Draw your tangents in here.
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Draw your little sticky tangents, and you can see it's getting bigger.
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And as soon as it crosses there, they start flattening out, and again, that means they're getting smaller.
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So, the blue line is definitely the first derivative.
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And do you notice something?
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The inflection point and the zero of this guy happen in the same place.
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That's an indication that we're looking at the second derivative of the red line there.
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Okay.
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And it's concave up on this one.
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So, the red line is above zero, until we get to the inflection point.
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So, we're accelerating into the inflection point, and we're decelerating out of the inflection point.
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Okay.
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So, that leaves us with just a couple more things to look at here.
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What would those be?
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Well, what I want to do is think about a problem here.
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One of the problems I gave you is, let's find numerically.
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Let's numerically find this place.
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Given the function.
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So, let's pretend this function is 10, and the initial value is 2, for example.
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So, I'm going to get 1 plus, and you know, 10 divided by 2 minus 1 is what?
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4, right?
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4e to the minus.
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Let's have a 50% rate.
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X here.
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So, here's a logistic function.
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It's not exactly this one right here, but it's close.
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Okay.
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And what I want to find is, I want to find the coordinates of the inflection point.
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I want to find the coordinates of this inflection point in this function here,
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which is not the same one on the graph.
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Obviously, if I ask that question, you'd answer 5, so that's not very interesting.
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How can we do this without, with the tools that we have available to us right now?
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How can we find the coordinates of that inflection point?
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Okay. Well, we know something, right?
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We know that we can find the maximum of the first derivative,
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or the root of the second derivative.
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We can do this.
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We find the root of the second derivative.
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So, what about it if I just use the maximum feature in the calculator,
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get the first derivative plotted, use the maximum feature,
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and I can find it that way, right?
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I can find it that way.
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And then I can plug that value in that I get.
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I can plug it back into the original function and figure out its corresponding output.
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So, I'm just going to find the input now.
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How am I going to do that?
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Well, here I'll show you how I did it in the calculator.
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What I did is I typed in the function in Y1,
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then as you saw in a previous video, I took the numerical derivative,
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previous objective, right?
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We talked about how to take a numerical derivative.
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So, here I'm going to use a numerical derivative,
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and it's here in the math key.
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And I'm not going to go into the details here,
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but this is the syntax of how you get it to go.
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So, I'm plotting both of them, the function, and its derivative on my graph.
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And I know something about the logistic, right?
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So, it wasn't too hard for me to put a window together here.
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And then I'm able to graph it, right?
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I'm able to graph it.
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And you can see it looks very similar to the one that I have behind me here.
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So, I'm trying to find this point here,
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which is sitting as same as this maximum.
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So, if I just use the Calc feature, maximum,
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and look, it's on the wrong curve.
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I want it to be on this curve, so if I push the down arrow,
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now it's on the second curve, the derivative.
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So, here I am.
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Whoops, I've got to be to the right.
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So, I'm just using the max feature,
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like it was any other old function, right?
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So, I'm to the left of the maximum.
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I'm to the up.
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I did it wrong.
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Better try that again.
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Well, let's see, I'm just going to ignore this.
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Let's try one more time there.
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So, I'm going to graph it.
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And I'm going to go second, Calc, and maximum again.
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Now, I've got to get it on the right side,
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down to get to the other graph.
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Okay, now I'm on the left side.
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Now, I'm going to go to the right side of the maximum.
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And then, I guess, anywhere in between,
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and lo and behold, what do I find out?
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But the maximum occurs at 2.772, right?
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So, I'll evaluate, for example, Y1 on the Homescreen,
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2.773.
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What second quit?
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So, if I put Y1 on the Homescreen,
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and evaluated it at 2.773, it was 2.88, remember?
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I could get the corresponding value of 5, okay, on that graph.
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So, the coordinates of the inflection point are 2.77 and 5.
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Okay, so I just want to show you something else.
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That's not the only technology that can do this.
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Here's another piece of technology that can do this pretty groovy.
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If you just know the syntax, how to type it in,
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Wolfram Alpha, maybe you should visit there.
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A lot in this class.
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We'll see.
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So, what do I type in?
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Well, it's hard to see here, but I'm typing in what?
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I'm typing in my logistic function here,
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and then this is the way you tell Wolfram Alpha
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to take a derivative.
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Diff stands for derivative with respect to x.
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So, it's taking the derivative with respect to x,
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and then I have it in parentheses with the word maximize outside.
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And so, it's going to maximize the derivative
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of the logistic function.
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So, here's the derivative of the logistic function right here.
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And now, it's asking to maximize it,
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and lo and behold, it finds the same place, 2.772.
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Then I could actually get Wolfram Alpha to evaluate the original function too.
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It would do it if I asked it, but not in this video.
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Okay, so, there you go.
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There's, hopefully, enough of a video example
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to get you rolling on dealing with the idea of concavity,
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and the second derivative, excuse me,
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concavity in the original function,
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and how that is related to what?
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How that is related to the sign of the second derivative.
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Remember, concave up, positive, second derivative,
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concave down, negative, second derivative.
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Object of 1.14.
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To the final derivative,