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Language: en
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So you're going to have some more work. Brace yourselves.
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Midterm is in a couple weeks. Same room as last time.
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Everybody's doing pretty well so I'm not very worried about everybody.
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Somebody did not do a good job cleaning this board.
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So we were talking about graphing.
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And we said well really when you're doing graphing you don't really- it's not the graph itself that's so important.
00:00:21.980 --> 00:00:27.060
Of course you should be able to draw it. Because that's just sort of a pictorial idea of what's going on.
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What's really important is to figure out the maximums and the minimums.
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So as I said last time you know now how to figure out when a curve is going up and when a curve is going down.
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But then a curve is going up, as a function is going up, it could be increasing this way, it could be increasing this way, or that way.
00:00:51.680 --> 00:00:53.600
So how do you tell them apart?
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Well if a function is just going straight up then it has a particular derivative that tells you what it's slope is.
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And that derivative is not changing.
00:01:04.080 --> 00:01:06.640
So the first derivative is a constant.
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That's a line, okay? And you know that because you can take the equation y=mx+b, the first derivative is m, the slope.
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And that doesn't change.
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Now what if the curve is doing this? Well then the first derivative is increasing.
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Because the first derivative gets steeper.
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So the second derivative will tell us how the first derivative is changing.
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And similarly, there the first derivative is getting closer to zero.
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So we can use the second derivative to figure out what's called concavity.
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And concavity is the curvature.
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So there's two types of concavity that we care about. There's concave up like that and concave down like that.
00:02:00.280 --> 00:02:03.960
Okay? So concave up the second derivative is positive.
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And concave down the second derivative is negative.
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When the second derivative is 0 then you have a straight line.
00:02:13.920 --> 00:02:16.960
Well if could be a flat spot but it's straight.
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And where the second derivative is 0 is called the point of inflection. A point of inflection is where the curve does a change in concavity.
00:02:28.160 --> 00:02:31.280
It goes either from up to down or from down to up.
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So let's take an example. Let's do one of these cubics again.
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So f(x)= x^3 -24x +10.
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How about 18? Never mind, make that 18x.
00:03:08.120 --> 00:03:11.440
It's hard to change. If you do it in pencil you just erase.
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Alright we keep learning that lesson the hard way.
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Let's take the first derivative. So the first derivative is 3x^2 -18.
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Can we do what we did before?
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Set that equal to 0. I should've stuck with 24.
00:03:32.200 --> 00:03:35.460
So you get x^2-6=0.
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x= + - √6.
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So if we went to the number line
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we can test the regions and figure out where the curve is going up and where the curve is going down.
00:03:56.940 --> 00:04:03.940
So we did this a couple of times already. You pick a number less than -√6 like say -10. This will come out positive.
00:04:05.020 --> 00:04:12.020
You take a number between those two like 0. This will come out negative and you test and you test again it's positive.
00:04:12.200 --> 00:04:15.760
So you know it's going up, then down, then up.
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So it's just like what we've had like several of these now.
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Now let's look at the second derivative.
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So the second derivative is 6x.
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Where is that 0? That's 0 at x=0.
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Now test a value at either side.
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So take a number less than 0, say -1. The second derivative comes out negative.
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So that tells you that the curve is concave down there.
00:04:55.960 --> 00:04:59.640
Now pick a number bigger than 0. Whatever you'd like. 1.
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This comes out positive so the curve is up there.
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Which kind of makes sense when you have these arrows but the concavity is not always as obvious on other types of curves.
00:05:15.960 --> 00:05:17.800
So now if you want to graph it
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We now know also that at 0 it switches from concave down to concave up.
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So let's find our y coordinates.
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f(0) is just 10.
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The √6 you get 6√6 -18√6 so you get 10-12√6. We don't care what that is it's just a negative number.
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And now you have -6√6 +18√6 so now you get 10+12√6.
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√6 is about 2.5, 2.45.
00:06:37.760 --> 00:06:42.040
So this is 35-ish. This is -25-ish.
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Or -15-ish.
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There you go so let's see we have (0, 10).
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And we've got √6 it's a negative number.
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So as I said this now we know that the concavity is concaved down all the way to this spot.
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And then it's concaved up the rest of the way.
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So it looks just like the other cubics we've drawn.
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So concavity will tell you, will help you figure out like I said, what the curvature is.
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It's very useful for finding maximums and minimums.
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In addition to the first derivative and sometimes you'll have an equation where it's difficult to solve, to plug in values to figure out maximums and minimums.
00:07:27.500 --> 00:07:33.080
Because it's messy so you could use a second derivative to test for maximums and minimums.
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So how can we do that?
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Well let's see you have two options.
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You take the first derivative and you have some point, c.
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f(x) is negative when x is less than c.
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And f(x) is positive when x is greater than c.
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And that's a minimum.
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Okay so what we've been doing is you look at the first derivative.
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If it's negative to the left of a point,
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If it's negative to the left of the point and positive to the right of the point and 0 then that tells you you have a minimum.
00:08:46.280 --> 00:08:53.240
And if it is positive to the left of the point and negative to the right of the point then you have a maximum.
00:08:53.240 --> 00:08:57.660
But there's another way you could check if something is a maximum or a minimum.
00:09:01.340 --> 00:09:07.240
If the second derivative, we'll call it 'a', is 0- sorry ignore that.
00:09:13.800 --> 00:09:19.960
If the first derivative at a is 0 and the second derivative at a could be positive or negative,
00:09:20.180 --> 00:09:23.300
Well if it's positive then you're concaved up.
00:09:23.300 --> 00:09:26.420
So if it's positive that means that this is a minimum point.
00:09:29.160 --> 00:09:31.960
And if it's negative it's a maximum point.
00:09:32.500 --> 00:09:37.860
So this is called the second derivative test and by the way if it's 0 it's neither.
00:09:40.000 --> 00:09:45.040
So again if you take the first derivative and you get a 0 like here we got say √6
00:09:45.360 --> 00:09:49.440
and now you take √6 and you plug it into the second derivative.
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So you plug it in you get 6√6 that's positive.
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So that tells you that's a minimum point.
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If you take the -√6 that gives you 0 for the first derivative.
00:10:00.200 --> 00:10:06.140
You plug in a second derivative you get a negative value. That tells you that your curve is concave down.
00:10:06.760 --> 00:10:08.920
And therefore that's a maximum.
00:10:08.920 --> 00:10:11.700
So why would you use the second derivative test vs. the first?
00:10:11.700 --> 00:10:13.720
We want to kind of be able to do either way.
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Because sometimes as I said with a problem it's messy to plug in numbers.
00:10:17.880 --> 00:10:21.080
You might have some complicated radical quotient kind of thing.
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And finding the second derivative's really impossible so you just plug in the first derivative.
00:10:25.620 --> 00:10:27.860
Sometimes the second derivative is trivial.
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Like this. Say the derivative is easy then you know whether something's a maximum or a minimum.
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You don't have to do the effect or plugging in numbers.
00:10:38.420 --> 00:10:40.980
Okay so let's give you guys an example.
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How are we doing so far? No one is speaking which means you either know it or you're asleep.
00:10:47.580 --> 00:10:54.380
It's good because then the audience who listens to these videos later won't have to worry about noise.
00:10:57.340 --> 00:11:02.860
I mean it is a Friday. Yesterday was a Thursday. You know what Thursday is all about.
00:11:03.960 --> 00:11:06.840
Let's say we give you something really fun.
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I'll give you something fairly hard.
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That's fun.
00:11:42.180 --> 00:11:49.180
Let's see if you can find where it's increasing, where it's decreasing, the concavity, and the maximums and the minimums.
00:11:50.640 --> 00:11:52.640
That's a nice nasty question.
00:11:52.740 --> 00:11:55.940
Of course if you can handle it that's a good sign.
00:12:18.100 --> 00:12:22.020
I recommend distributing before you take the derivative.
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x * x^1/3 is x^4/3.
00:16:23.520 --> 00:16:28.080
Alright a lot of you seem to be waiting for me so let's do this as a team.
00:16:29.100 --> 00:16:32.220
First, let's distribute that x.
00:16:35.000 --> 00:16:36.520
How do we know it's 4/3?
00:16:36.860 --> 00:16:40.460
x^1, 1+1/3 is 4/3.
00:16:41.000 --> 00:16:49.420
So first derivative 1 - 4/3x^1/3.
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And the second derivative is -4/9x^-2/3.
00:17:02.760 --> 00:17:04.920
So these are slightly annoying.
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Let's set this equal to 0.
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And you get- I'm sorry equal to 0.
00:17:12.980 --> 00:17:17.520
You get 1= 4/3x^1/3.
00:17:17.760 --> 00:17:19.520
Remember 1/3 is cube root.
00:17:23.320 --> 00:17:29.120
So put the fraction over to the other side you get 3/x = x^1/3.
00:17:29.620 --> 00:17:34.800
And if you cube that you get 27/64 for x.
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It's an annoying number.
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I wouldn't worry too much about exact values.
00:17:42.020 --> 00:17:46.060
It's important just to figure out positives and negatives and about how big things are.
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It's a little less than half.
00:17:49.580 --> 00:17:57.900
And here this is the same as -4/9 3√x ^2. That's what x^-2/3 means.
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That's never 0.
00:18:00.140 --> 00:18:03.920
Why is that never 0 because the numerator can't be 0.
00:18:04.320 --> 00:18:09.420
However, the denominator still has a problem when x is 0.
00:18:09.780 --> 00:18:13.000
So it's possible concavity will change.
00:18:14.540 --> 00:18:17.180
So if you have to do a hard graph like this.
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So now you say to yourself so I want to test the numbers from the first derivative?
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Or do I want to test the numbers in the second derivative?
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Because do I really want to plug 27/64^2 and put it in that?
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Not particularly.
00:18:37.700 --> 00:18:40.980
So what do we do? Well we use the first derivative.
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Make a number line.
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27/64.
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Pick a number less than that like 0.
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Cube root of 0 is just 0 so this is positive.
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The curve is going up.
00:19:00.060 --> 00:19:04.560
Pick a number bigger than 27/64 like 1.
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1-4/3 is negative so it's now going down so this is a maximum.
00:19:12.980 --> 00:19:14.400
The y coordinate will be messy.
00:19:17.800 --> 00:19:18.600
Not really.
00:19:20.120 --> 00:19:24.680
Alright the second derivative we'll do the second derivative test.
00:19:25.360 --> 00:19:31.520
We don't have a place where the numerator is 0 so we're not going to have a point of inflection.
00:19:32.400 --> 00:19:38.400
Something interesting goes on at the second derivative when you plug in 0 it's undefined.
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So it's possible you will change concavity there.
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Pick a number less than 0 say -1.
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You get, well this is squared so it's always going to be positive.
00:19:49.140 --> 00:19:51.780
Cube root is positive. This thing's always going to be negative.
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So this curve is always concave down.
00:19:55.780 --> 00:19:58.260
But first it's concave down going up.
00:19:58.640 --> 00:20:01.680
Then it's concave down going down.
00:20:08.300 --> 00:20:10.220
So what would that look like?
00:20:10.220 --> 00:20:12.880
Well let's see if we can figure out how to graph that.
00:20:13.240 --> 00:20:17.180
So if we plug in- hang on- well we could do it here.
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f(27/64)
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You'll get 27/64(1- the cube root of that. The cube root of that is 3/4).
00:20:30.800 --> 00:20:36.480
So you get 27/256 which is a very small positive number.
00:20:39.680 --> 00:20:40.800
That looks good.
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When x is 0 you get 0.
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And we know that the curve is alway going up but it's concave down like that.
00:20:55.360 --> 00:21:00.100
Then what happens after? Let's see.
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It's going down and concave down.
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It can't be that simple.
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What else do we know?
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That's probably about it so yeah it kinda goes like that. I take it back.
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Easier than it looks. It's not a parabola however.
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Don't make that mistake.
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Were any of you able to pull this off? Graph it?
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Do the curves? No? Too hard?
00:21:28.240 --> 00:21:30.640
Alright I'll give you an easier one.
00:21:30.640 --> 00:21:34.940
One person solved it. The rest of you just sort of stared straight ahead.
00:21:49.480 --> 00:21:52.040
We'll give you an easier one let's see.
00:22:28.200 --> 00:22:32.920
Yeah I like that one. Alright how about something that looks like that?
00:22:36.900 --> 00:22:38.980
I'll give you a couple minutes.
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Okay here's a fun one.
00:30:20.180 --> 00:30:24.660
So your attack when you do these problems should always be the same.
00:30:24.940 --> 00:30:28.380
First, find the derivatives and set them equal to 0.
00:30:28.620 --> 00:30:32.300
Then sign test them and then do the best you can to graph.
00:30:32.300 --> 00:30:39.020
Lots of you will have trouble with finding zero's, plugging in those things to find the y coordinates,
00:30:39.020 --> 00:30:43.420
and maybe with the graph but you should be able to do the other half effectively.
00:30:43.420 --> 00:30:46.500
And if you practice this you'll get good at the first types.
00:30:46.500 --> 00:30:48.060
So let's take the first derivative.
00:30:48.900 --> 00:30:55.400
You get f'(x)= 4x^3 -16x.
00:30:57.560 --> 00:30:58.920
And set it equal to 0.
00:31:01.440 --> 00:31:07.580
Factor out 4x. 4x(x^2 -4)=0.
00:31:08.160 --> 00:31:16.580
Then you get 4x(x+2)(x-2)=0. Were we able to do that?
00:31:17.540 --> 00:31:20.760
So 3 points. You have 0, 2, and -2.
00:31:24.920 --> 00:31:28.760
Alright let's do the second derivative while we're here.
00:31:33.940 --> 00:31:40.280
f''(x)= 12x^2 -16.
00:31:40.280 --> 00:31:46.840
By the way usually if the first derivative is easy numbers the second derivative is easy numbers or the other way around.
00:31:46.840 --> 00:31:52.740
It's very hard to write a problem where they come out integers twice in a row other than a trivial graph.
00:31:54.180 --> 00:32:00.500
So here you get 12x^2 =16.
00:32:01.100 --> 00:32:05.380
So that means x^2 =4/3.
00:32:05.700 --> 00:32:09.960
x= +-√4/3
00:32:11.320 --> 00:32:19.960
So the number line would the be -√4/3, √4/3.
00:32:20.440 --> 00:32:24.200
4/3 is a little bit bigger than 1 so the √4/3 will be 1-ish.
00:32:27.460 --> 00:32:29.340
How'd we do on that?
00:32:38.520 --> 00:32:40.040
That's an equals sign.
00:32:41.380 --> 00:32:43.940
I could. You mean factor a 4 out of here?
00:32:46.180 --> 00:32:48.180
You'll end up at the same spot.
00:32:50.020 --> 00:32:54.440
And in fact √4/3 is 2/√3.
00:32:54.440 --> 00:32:56.960
So some of you may reduce that one more step.
00:32:56.960 --> 00:32:59.220
Doesn't really matter much either way.
00:32:59.220 --> 00:33:01.140
Alright I'm going to cover this for a second.
00:33:01.360 --> 00:33:04.240
So I pick a number less than -2 like -3.
00:33:04.620 --> 00:33:11.220
And I plug in. I get a negative value, negative value, negative value.
00:33:11.220 --> 00:33:15.180
3 negatives makes a negative value so the curve is going down.
00:33:17.500 --> 00:33:20.220
I plug in a number between 0 and -2 like -1.
00:33:21.280 --> 00:33:27.600
Negative, positive because -1+2= -1, negatives. That's 2 negatives, 2 negatives a positive.
00:33:28.080 --> 00:33:31.520
I mean a negative times a negative makes a positive.
00:33:31.520 --> 00:33:35.660
You only have to count the number of negative signs by the way. You never care about the positive signs.
00:33:35.660 --> 00:33:40.440
The number of negatives multiplied together tells you whether something is positive or negative.
00:33:40.440 --> 00:33:45.460
If it's an odd number of negatives you get negative. If it's an even number of negatives you get a positive.
00:33:47.320 --> 00:33:49.640
Pick a number between 0 and 2 like 1.
00:33:50.260 --> 00:33:54.420
Positive, positive, negative. We only have one negative now.
00:33:57.660 --> 00:33:58.460
Going down.
00:33:59.160 --> 00:34:04.760
So this tells us right the minimum, maximum, minimum.
00:34:06.520 --> 00:34:09.080
And I pick a number bigger than 2 like 3.
00:34:09.400 --> 00:34:13.400
And when you plug in everything's positive so it's going up.
00:34:13.660 --> 00:34:20.300
So if this were a test question or a homework question because you're going to have homework coming,
00:34:20.300 --> 00:34:23.720
we might ask where is this increasing and where is it decreasing?
00:34:24.180 --> 00:34:32.840
So you'd say it's increasing when x is between -2 and 0,
00:34:35.660 --> 00:34:38.080
or when x > 2.
00:34:38.080 --> 00:34:40.520
And you could do that in interval notation if you want.
00:34:40.820 --> 00:34:42.900
Notice you never use an equals.
00:34:43.420 --> 00:34:48.320
Because at 0, at -2, it's not increasing or decreasing.
00:34:48.460 --> 00:34:52.140
So these will always be just less than and greater than.
00:34:52.140 --> 00:34:55.480
Okay there won't be less than or equal to or greater than or equal to.
00:34:58.860 --> 00:35:10.380
Decreasing, well it's decreasing when x < -2 or when x is between 0 and 2.
00:35:16.160 --> 00:35:17.120
So far so good?
00:35:18.740 --> 00:35:21.480
Painful? It's a lot.
00:35:24.680 --> 00:35:26.680
Okay second derivative test.
00:35:27.860 --> 00:35:30.340
Pick a number less than -√4/3 like -2.
00:35:30.980 --> 00:35:35.220
This will come out positive, bigger than 16 so it'd be positive.
00:35:35.880 --> 00:35:37.800
So it's concave up.
00:35:38.140 --> 00:35:40.620
Which kind of agrees with that shape.
00:35:41.980 --> 00:35:45.580
Between -√4/3 and √4/3 like 0.
00:35:46.080 --> 00:35:48.560
This is negative because you get -16.
00:35:48.560 --> 00:35:51.020
So it's negative so it's concave down,
00:35:52.340 --> 00:35:56.500
And bigger than √4/3 like 2.
00:35:56.900 --> 00:36:00.480
Positive because 48 > 16.
00:36:02.580 --> 00:36:16.740
So it is concave up when x < -√4/3 or when x > √4/3.
00:36:24.200 --> 00:36:34.820
And it is concave down when -√4/3 < x < √4/3.
00:36:36.620 --> 00:36:39.520
Okay now let's graph it.
00:37:09.260 --> 00:37:11.580
Okay so we need some y coordinates.
00:37:12.680 --> 00:37:27.160
We've got f(0), f(2), f(-2), f(√4/3), and f(-√4/3).
00:37:29.040 --> 00:37:30.880
Okay f(0) is easy that's 10.
00:37:32.720 --> 00:37:37.200
f(2)= -6.
00:37:39.900 --> 00:37:41.980
f(-2)= -6.
00:37:45.480 --> 00:37:48.560
√4/3 let's see.
00:37:55.080 --> 00:37:57.620
It's -22/9 I think.
00:38:00.540 --> 00:38:01.740
And so is that one.
00:38:03.620 --> 00:38:05.060
I think that's right.
00:38:06.580 --> 00:38:07.700
Doesn't matter.
00:38:10.440 --> 00:38:11.800
Okay let's graph it.
00:38:12.600 --> 00:38:14.280
This curve is symmetric.
00:38:14.440 --> 00:38:19.320
Symmetric because these are matching, these are matching. So let's see.
00:38:19.800 --> 00:38:22.360
We have (0, 10).
00:38:23.340 --> 00:38:24.700
(2, -6)
00:38:24.960 --> 00:38:26.700
(-2, -6)
00:38:27.260 --> 00:38:29.500
Then a couple of values like that.
00:38:30.080 --> 00:38:32.080
Okay this is a maximum.
00:38:33.040 --> 00:38:34.480
This one's a minimum.
00:38:34.840 --> 00:38:40.040
And then we switch right there from concave up to concave down.
00:38:40.040 --> 00:38:43.140
We switch right there from concave down to concave up.
00:38:43.140 --> 00:38:46.040
So these are called the points of inflection.
00:38:53.540 --> 00:39:05.660
So you have a point of inflection at (+-√4/3, -22/9).
00:39:10.980 --> 00:39:12.260
How'd we do on this?
00:39:15.540 --> 00:39:17.780
Eh? Well fourth degrees are hard.
00:39:18.440 --> 00:39:20.280
I can give you a 5th you know.
00:39:24.760 --> 00:39:27.160
Alright let's just do a little more.
00:39:27.420 --> 00:39:29.980
Make sure everybody gets the concept.
00:39:33.100 --> 00:39:38.060
And as I said notice I'm sticking to polynomials. That won't last forever.
00:39:46.620 --> 00:39:51.980
So let's do another where I'm not going to graph them I'm just going to analyze it.
00:40:18.900 --> 00:40:22.800
That says x^5 -80x +1.
00:40:25.300 --> 00:40:28.420
The 80 looks scary but it should make it easier.
00:40:30.220 --> 00:40:37.220
So we're just going to find where is this increasing, decreasing, where is the concave up, where is the concave down?
00:40:37.220 --> 00:40:39.240
I won't worry about the picture.
00:40:40.200 --> 00:40:46.120
So first derivative you get 5x^4 -80.
00:40:51.820 --> 00:40:53.900
Set that equal to 0 so let's see.
00:40:54.100 --> 00:40:57.600
5x^4 =80.
00:40:58.120 --> 00:41:03.440
x^4= 16. x= +-2. That's not so bad.
00:41:21.680 --> 00:41:24.000
Let's take the second derivative.
00:41:26.340 --> 00:41:27.460
It's just 20x^3.
00:41:28.740 --> 00:41:30.820
So that equals 0 when x equals 0.
00:41:38.220 --> 00:41:40.620
Okay now let's do a little analysis.
00:41:43.140 --> 00:41:47.300
Don't get your finger caught in there. I know from experience.
00:41:57.900 --> 00:42:00.580
Okay so number line.
00:42:00.580 --> 00:42:03.020
So our number line for the first derivative.
00:42:05.620 --> 00:42:08.400
We've got -2 and 2.
00:42:08.400 --> 00:42:15.220
We're missing a couple by the way because notice it's x^4 =16. We're missing the two imaginary complex roots.
00:42:15.220 --> 00:42:19.120
But that's okay don't worry about that. That never effects this class.
00:42:23.540 --> 00:42:29.040
So at -3. -3^4 is 81 times 5 is going to come out bigger than 80.
00:42:29.380 --> 00:42:33.700
So you get a positive first derivative there so it's increasing.
00:42:34.760 --> 00:42:40.440
So plug in 0. 5(0)^4 is 0 so it's negative. And then this will go positive again.
00:42:40.720 --> 00:42:43.200
These alternate a lot but not always.
00:42:45.560 --> 00:42:47.720
And we do the second derivative.
00:42:51.220 --> 00:42:52.180
We just have 0.
00:42:52.180 --> 00:42:58.560
And since it's 20x^3 and a negative number will make that negative and a positive number will make tat positive.
00:42:58.940 --> 00:43:04.240
So the curve is going up, down, up. Concave down, concave up.
00:43:06.380 --> 00:43:22.360
So if I say on what intervals is this increasing it is increasing when x < -2 or x > 2.
00:43:23.000 --> 00:43:31.520
And then it's decreasing between them.
00:43:32.660 --> 00:43:35.660
It has a maximum at x= -2.
00:43:40.240 --> 00:43:43.200
I won't worry about the y coordinate for now.
00:43:44.120 --> 00:43:45.160
It's findable.
00:43:46.880 --> 00:43:52.180
And it has a minimum at x=2.
00:43:57.340 --> 00:44:04.340
As I said you can use interval notation it doesn't matter. The TAs probably like it but it's not necessary.
00:44:04.340 --> 00:44:06.880
Certainly won't be necessary on the midterm.
00:44:11.700 --> 00:44:17.660
It's concave up when x>0 and concave down when x<0.
00:44:18.120 --> 00:44:28.940
And there's a point of inflection, well we could find the coordinate (0, 1).
00:44:28.940 --> 00:44:35.280
Point of inflection- the concavity changes goes from negative to positive and has a zero there.
00:44:40.120 --> 00:44:41.800
How do we feel about that?
00:44:41.800 --> 00:44:44.780
So you'll have some MyMathLab stuff of this to practice.
00:44:44.780 --> 00:44:47.580
Plus on the paper homework there's two of these.
00:44:48.060 --> 00:44:50.960
An easy one and a not so easy one.
00:44:50.960 --> 00:44:56.480
Remember you could use your calculator so feel free to graph it in your calculator for your homework.
00:44:56.480 --> 00:44:57.740
You just can't use it on the exam.
00:44:57.740 --> 00:45:02.140
But you can feel free to put it in your calculator so you can get a picture to know what the graph looks like.
00:45:02.140 --> 00:45:04.140
And then start to interpret it from there.
00:45:04.440 --> 00:45:09.000
After all if I told you couldn't you could anyway. I can't enforce it.
00:45:10.520 --> 00:45:12.760
Alright see everybody on Monday.