WEBVTT
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Language: en
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The practice problems. So we're practicing chapters zero and one.
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I don't have a way to project the computer up there.
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But for those of you who have my math lab I don't know if it shows up on your phone you might have to have it on your computer.
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That I'm not sure about.
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But under study plan it'll give you lots of practice problems on every chapter. You should be able to access those.
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So I'll give you guys a minute to see if you can.
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I mean it might come up on the phone. I don't know.
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We'll find out. Is it coming up for you?
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Tablets are the same as phones.
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Yes it should be study plan which should be, there's assignments that's where your homework is.
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And there should be something that says study plan. See my view is not necessarily the same as your view.
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So if you can't see that then I have to unlock it somehow.
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Which I thought I had done but maybe not.
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You see homework but you don't see study plan?
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Alright so I'll have to find out how to unlock that.
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Although frankly what we're doing in class should be enough review.
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I'll be honest but- I'm sorry?
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You have study plan? And you have study plan?
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So look more carefully. It's on the left bar.
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Right so you should have course home, assignments, then study plan after that.
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And then you're going to want to click where it says all chapters.
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I don't know where to project this. And then you can do chapters zero and one.
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We are in chapter one up to sum and difference.
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The derivative using sum and difference which is...
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So the exam will cover chapter zero or chapter R, excuse me, up through 1.5.
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So anything in chapter R for review.
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And anything in chapter 1.1 through 1.5.
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Are you guys finding it?
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Yes? Yes? Did you find it? No?
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Good okay so it's there.
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That makes me happy.
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But in the mean time let's do some more review.
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So we did derivative stuff, the definition of derivative stuff, did we do remaining? Yes?
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So everything in chapter R, and then up to 1.5.
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So chapter 1 through 1.5.
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We don't get very far in the course. We get somewhere into chapter 5.
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We go nice and slowly just to make everybody happy.
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You all know the midterm is tomorrow night.
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Correct? And you all know where to go?
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You go to Harriman right?
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Room 137.
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The exam will end at approximately 10:15 at night.
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I know this is going to make some of you very upset and I don't normally do this but we're not going to have class on Wednesday.
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Now who was not listening when I said that?
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Maybe you should've paid attention. We'll see you Wednesday then.
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Yes okay so who was not listening when I said that?
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Good okay then we'll see you.
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Alright so we did definition of derivative, we did some derivatives, we did um we did domains?
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We did not do domains. Let's practice a couple of domain problems.
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We will have class on Friday however.
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I know that'll excite you.
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So the domain is what you're allowed to plug into a function.
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I'll move this back up a little bit.
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Domain is what you're allowed to plug into a function for x.
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So it's not what you give out, that's range. It's what you put in.
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That's domain. So let's see.
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Closed f(x) = √ 6x-1.
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What is the domain?
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This shouldn't take you very long.
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Remember how to find the domain. What are you allowed to plug in that radical?
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What are you allowed to plug into the radical?
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Okay that should be all the time you need.
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So when you take the square root you can't take, you are not permitted to take the square root of a negative number in reals.
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So this cannot be negative.
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So 6x-1 has to be ≥ 0.
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So if you solve that you get 6x ≥ 1 and x ≥ 1/6.
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So that's the domain.
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You don't have to do any fancy notation anything like that.
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Okay the domain is what's under the radical must be ≥ 0.
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Okay? What if instead I gave you...
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What's the domain?
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So you look at this and you say that's a fraction.
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You can put whatever you want in a fraction. You can have anything divided by anything.
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Except you can't have 0 in the denominator.
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Because a number divided by 0 doesn't make any sense.
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So the only thing you cannot have is x-2 ≠ 0 so x ≠ 2.
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So far so good?
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Alright let's put it together.
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Suppose I said f(x), what is I gave you that?
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Alright you do not need to use interval notation for the exam.
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Okay? You need interval notation to put the problems into mymathlab but you do not need it for the exam.
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You can write it just like that.
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Interval notation isn't better it's just compact.
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Okay let's do the third one.
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So what's changed?
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Well we still have a radical. So the radical, you still cannot take the square root of a negative number
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so 2x-1 still cannot be a negative number.
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And then you look and you say but wait.
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This told me I can't have 0 in the denominator of a fraction.
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But can't have equal to either.
00:10:06.740 --> 00:10:10.980
So here I could plug in 1/6 because I'm going to get √ 0.
00:10:11.360 --> 00:10:15.220
Here this is x is going to have to be > 1/2.
00:10:15.640 --> 00:10:20.840
I cannot plug in 1/2 here because then the denominator would be 0.
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So far so good?
00:10:24.840 --> 00:10:28.340
Okay what if I gave you
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that?
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Alright that's long enough.
00:12:23.440 --> 00:12:25.300
So now we're combining.
00:12:26.880 --> 00:12:30.140
So here, still can't take the square root of a negative number.
00:12:30.440 --> 00:12:35.800
So 3x-4 ≥ 0. Is it allowed to be 0?
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Yes because you could have 0 in the numerator of a fraction.
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And here x ≠ 5 right? Because x-5 can't be 0.
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So here you have two parts to the domain.
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x has to be ≥ 4/3 and x ≠ 5.
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Again, you don't need to use interval notation.
00:13:06.760 --> 00:13:18.140
Now what if I flipped that the other way. What if I said f(x) is x-5/√ 3x-4?
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Well it's the same numbers.
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But now notice, can I have 0 in the numerator of a fraction?
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Yes, so x can be 5 now.
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And check by the way we plug in 5 that's the square root of a positive number.
00:13:32.960 --> 00:13:36.960
So I only have to worry about the radical in the denominator.
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So 3x-4 has to be greater than 0.
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It can't be equal to 0 because you can't have 0 in the denominator.
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So the domain would just be x > 4/3.
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Everybody got the domain stuff down?
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We good?
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Alright let's do some linear stuff.
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Okay.
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A plane is flying from New York to London. One hour after takeoff it is 300 miles from New York.
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4 hours later it is 15 miles from New York.
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Assuming constant velocity find an equation of the plane's distance from New York t hours after takeoff.
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And then if London is 3,000 miles from New York, how long is the flight?
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This is as good as my handwriting gets folks.
00:20:16.960 --> 00:20:20.000
You want to think about a pair of coordinates.
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??
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I'll give you guys one more minute.
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See when we edit the videos we can cut all this stuff out.
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Alright that's long enough.
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I put the chalk somewhere.
00:23:01.060 --> 00:23:04.280
Walk with the chalk. It's not over there.
00:23:06.700 --> 00:23:08.700
Alright how about this chalk.
00:23:09.100 --> 00:23:13.280
Alright so we want to find the equation of a line.
00:23:13.760 --> 00:23:22.460
So the equation of a line y-y1= m(x-x1) although here instead of x we use t but it really doesn't matter.
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Okay? If you used x we wouldn't take off points.
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So one hour after takeoff the plane is 300 miles from New York.
00:23:31.700 --> 00:23:40.940
So at time 1 it's location is 300 miles, 4 hours later, so at time 5 it's location is 1,500 miles.
00:23:42.360 --> 00:23:45.240
Okay? Because it's 4 hours later not time 4.
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Okay?
00:23:51.760 --> 00:23:53.360
So let's find the slope.
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So m, that's 1,500-300/5-1 which is 1,200/4 which is 300.
00:24:09.500 --> 00:24:10.000
Okay?
00:24:12.660 --> 00:24:18.260
And I think you could say that you could use either point so it doesn't really matter.
00:24:18.660 --> 00:24:28.260
So y-300 is the slope times x-1 except instead of x we should use t.
00:24:31.560 --> 00:24:33.640
And if you want to simplify that
00:24:39.160 --> 00:24:42.440
the 300's cancel and you get y= 300t.
00:24:42.960 --> 00:24:46.640
Now some of you wanted to just use rate * time = distance.
00:24:46.640 --> 00:24:52.640
On this problem that would come out the same but I can give you a different type of problem where it wouldn't.
00:24:52.640 --> 00:24:55.420
So be careful about just assuming that formula.
00:24:58.060 --> 00:25:00.260
Okay this wasn't very hard.
00:25:00.440 --> 00:25:06.040
So if London is 3,000 miles from New York, that's about right, how long is the flight?
00:25:06.360 --> 00:25:15.780
That's a. This is b. 3,000 = 300t. That's a terrible t.
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So t = 10.
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I don't know how long it actually takes the fly to London. Yes.
00:25:32.020 --> 00:25:34.340
Oh sure you could say time equals x.
00:25:34.340 --> 00:25:36.980
I mean I don't care if you use t or x honestly.
00:25:37.260 --> 00:25:43.700
Okay? You should really do d(t) but that's not important. What's important is understanding the relationship and how to set it up properly.
00:25:44.620 --> 00:25:45.400
Okay?
00:25:47.060 --> 00:25:50.420
So we could do a slightly different one.
00:26:07.880 --> 00:26:11.000
Okay you're given a choice of telephone plans.
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Plan number one you pay a flat rate of $40 plus $0.10 per minute of using the phone.
00:26:39.280 --> 00:26:42.400
Because it's some sort of burner phone I guess.
00:26:42.760 --> 00:27:02.200
Plan number two you pay a flat rate of $25 but it's $0.12 per minute.
00:27:07.820 --> 00:27:12.380
Okay so what's the question I'm going to ask?
00:27:15.660 --> 00:27:19.020
Give it a shot what's the question I'm going to ask?
00:27:19.020 --> 00:27:22.700
What's the better plan? And you say well it depends on how many minutes you use right?
00:27:22.700 --> 00:27:26.480
So what's the sport where we don't care which plan we're at?
00:27:26.480 --> 00:27:30.940
Okay which would be called the break even point or here it would be called the indifference point.
00:27:43.860 --> 00:27:46.900
Indifference. I don't care which plan I'm on.
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Okay? If I'm only going to use one minute I clearly want plan 2.
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If I'm going to use a billion minutes I clearly want plan 1.
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That ends up cheaper. So somewhere I don't care. That's indifferent. So what's the indifference point?
00:30:30.660 --> 00:30:33.860
Alright let's set this up as a pair of equations.
00:30:39.420 --> 00:30:51.180
So the plan 1 equation, your cost, is going to be 40+.10t.
00:30:51.180 --> 00:30:56.440
Because it's 10 cents. Or you could make it 4,000 cents instead of $40. It's really up to you.
00:30:57.340 --> 00:30:58.100
Plan 2.
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Is 25+.12t.
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And then you want to set those equal to each other.
00:31:10.660 --> 00:31:19.000
So 40+ .1t = 25+ .12t.
00:31:21.200 --> 00:31:25.820
And you get 15= .02t.
00:31:28.080 --> 00:31:31.400
So you should get 750 minutes.
00:31:33.180 --> 00:31:37.900
So if you wrote 10 instead of .1 I think you got 7.5 minutes or something.
00:31:38.060 --> 00:31:41.740
That's because you weren't paying attention to units.
00:31:43.520 --> 00:31:45.760
Okay how do we feel about this one?
00:31:46.540 --> 00:31:49.900
So we could do linear stuff. Or we could do domains.
00:32:05.980 --> 00:32:09.980
Alyssa is asking if we could to the vertical line test stuff.
00:32:09.980 --> 00:32:12.880
No. Did we do finding the inverse of a function?
00:32:12.880 --> 00:32:17.380
Did I show you guys how to find the inverse of a function? I think I should do those.
00:32:18.140 --> 00:32:20.860
No we don't want quadratics.
00:32:21.420 --> 00:32:24.940
But we didn't really do anything with quadratics so.
00:32:26.260 --> 00:32:28.980
We did polynomials. Did we do any limits?
00:32:29.540 --> 00:32:31.300
Let's review some limits.
00:32:46.760 --> 00:32:48.760
There's not a lot on this exam.
00:32:50.080 --> 00:32:52.320
So very few chances to trick you up.
00:32:58.880 --> 00:33:00.640
Okay let's do some limits.
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There's 4. See how you do on those.
00:37:08.860 --> 00:37:10.280
You got this?
00:37:11.020 --> 00:37:12.700
You need another minute?
00:37:13.260 --> 00:37:16.840
There's got to be a difference between these two limits right?
00:37:19.900 --> 00:37:21.740
One's squared one's cubed.
00:37:35.000 --> 00:37:37.160
Alright let's do these as a team.
00:37:41.340 --> 00:37:46.760
So the top one. The lim x-> ∞. You look at the power of the top.
00:37:46.760 --> 00:37:49.320
The degree of the top, the highest power.
00:37:49.360 --> 00:37:53.200
It's cubed. And you look at the bottom and it's also cubed.
00:37:53.560 --> 00:38:01.620
Okay? So the limit is just going to be the coefficient of the highest term on top divided by the coefficient of the highest term on bottom.
00:38:01.820 --> 00:38:03.160
Which is 4.
00:38:03.240 --> 00:38:06.120
You don't have to reduce 8/2 but you should.
00:38:07.860 --> 00:38:14.740
So again you look at the powers. Since the powers match it's just the coefficient over the coefficient.
00:38:16.740 --> 00:38:20.660
Now when you do the second one when you plug in 3- oh yes, yes?
00:38:21.580 --> 00:38:24.140
Question: Is it only for infinity you can do that?
00:38:24.760 --> 00:38:26.360
It's only for infinity.
00:38:28.680 --> 00:38:34.000
Now we look at the second one. When you plug in 3 on top you get 0.
00:38:34.580 --> 00:38:37.700
You plug in 3 on the bottom you also get 0.
00:38:38.360 --> 00:38:43.520
So does that mean the limit does not exist? You have to factor it.
00:38:45.700 --> 00:38:47.560
So let's factor this.
00:38:50.320 --> 00:38:54.740
We get (x-3)(x-4) on top.
00:38:55.480 --> 00:38:59.480
(x-3)(x+3) on the bottom.
00:38:59.780 --> 00:39:05.540
So our problem was the (x-3) term. You plug in 3 here you get 0, you plug in 3 here you get 0.
00:39:05.820 --> 00:39:15.760
Now you cancel and now if you plug in 3, and that's -1/6.
00:39:26.180 --> 00:39:27.720
So far so good?
00:39:28.900 --> 00:39:31.880
Okay now for these 2 on the bottom.
00:39:35.140 --> 00:39:41.220
So remember when you plug in a number and you get 0 in the bottom it's going to be some kind of ∞.
00:39:41.660 --> 00:39:45.020
The problem is you always have to check the left and the right side for these.
00:39:45.420 --> 00:39:54.240
If you check the left side you do the lim x->2 from the minus side you get 8/x-2.
00:39:55.440 --> 00:39:59.260
Okay well, (x-2)^2 sorry.
00:39:59.600 --> 00:40:03.920
You'll get 8/0 this will be positive so you get positive ∞.
00:40:04.360 --> 00:40:16.360
If you do the limit when x->2 from the plus side you get (x-2)^2, that's going to be positive again so you're going to get plus ∞.
00:40:17.360 --> 00:40:21.380
Therefore this limit is ∞.
00:40:21.380 --> 00:40:26.780
You check the left side and the right side. They both get you positive ∞ so the limit is ∞.
00:40:27.480 --> 00:40:33.240
And it's always positive because 8 is always positive and (x-2)^2 is always positive.
00:40:34.240 --> 00:40:36.800
If you look at the bottom one, however.
00:40:44.220 --> 00:40:47.580
When you cube a negative number you get a negative.
00:40:47.960 --> 00:40:51.080
So this will approach negative infinity.
00:40:51.860 --> 00:40:59.540
Okay? Because if x is just a little bit less than 2 at 1.99, -2 is a negative number.
00:40:59.800 --> 00:41:03.440
When you cube it it's still a negative so this comes out negative infinity.
00:41:03.900 --> 00:41:05.100
On the other hand,
00:41:14.500 --> 00:41:20.740
if you plug in a number greater than 2, x-2 is positive so this will come out positive infinity.
00:41:21.180 --> 00:41:24.920
Because they don't match the limit does not exist.
00:41:30.240 --> 00:41:32.000
How do we feel about those?
00:41:35.560 --> 00:41:38.520
Let's see what other limits I could give you.
00:41:42.100 --> 00:41:45.020
Oh yes one other kind. So far so good?
00:41:48.020 --> 00:41:52.020
Well you sort of are plugging in 2, when you plug in 2 you get 0.
00:41:52.180 --> 00:41:57.620
Okay so remember 8/0 is nonsense. So 8 over a very tiny number is a very big number.
00:41:58.560 --> 00:42:03.160
Because you have a fraction when you plug the fraction in you get a very large number.
00:42:03.520 --> 00:42:07.600
So if you have 8/1/1,000,000,000,000 it becomes 8 trillion.
00:42:09.740 --> 00:42:12.420
Alright one other type of limit.
00:43:00.460 --> 00:43:03.420
Okay how about this one.
00:44:47.620 --> 00:44:50.740
Okay so if you want to do the lim x->2 for this one.
00:44:53.440 --> 00:44:58.880
The lim x->2 from the minus side, that's a number less than 2.
00:44:59.180 --> 00:45:01.000
You use the top branch.
00:45:01.640 --> 00:45:06.780
That would be 16-3=13.
00:45:07.340 --> 00:45:12.260
And if you go the lim x->2 from the other side
00:45:15.620 --> 00:45:21.260
you use the bottom piece which is 7*2 is 14-1 which is also 13.
00:45:21.860 --> 00:45:24.700
So therefore the limit is 13.
00:45:27.380 --> 00:45:28.100
Okay?
00:45:32.160 --> 00:45:34.100
What if I took this away?
00:45:35.940 --> 00:45:38.020
Would that effect the problem?
00:45:39.000 --> 00:45:40.820
Why does it not matter?
00:45:42.500 --> 00:45:45.220
Because you don't care if it's equal to 2.
00:45:45.220 --> 00:45:49.580
You're doing the limit. You're getting very close to 2 but you're not exactly at 2 so it wouldn't effect it.
00:45:49.580 --> 00:45:52.500
It would affect continuity but I didn't ask about continuity.
00:45:52.720 --> 00:45:54.640
It's not on the test.
00:45:56.320 --> 00:46:03.320
Alright so look you guys are in good shape. Do some more studying. The tes is very straightforward there's not that much on it.