WEBVTT
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Language: en
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How are we doing in the homework problems by the way?
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Good?
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Alright so let's practice a similar problem.
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Okay? We're going to practice a bunch of these today.
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Before we get into new stuff.
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So let's say find the derivative of f(x) = 3x^2 + 4x +1.
00:00:52.500 --> 00:00:56.660
You guys wanna do it on your own first or should we do it as a team?
00:00:58.460 --> 00:00:59.820
Team? Okay.
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So we're going to need f(x+h) and we're going to need f(h) because remember the formula is
00:01:06.620 --> 00:01:16.620
lim h->0 of f(x+h) - f(x) /h.
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So we have f(x).
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That's this. We're going to need f(x+h) and then we're gonna need to do some algebra.
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The algebra is the annoying part.
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So f(x+h) = 3(x+h)^2 + 4(x+h) +1.
00:01:48.740 --> 00:01:55.740
And then before we put it in what they call the difference quotient before we put it in the derivative formula, let's simplify this.
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So if we foil out (x+h) we multiply that out
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w'ere going to get 3(x^2 + 2xh + h^2).
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Everybody see that?
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So we're going to multiply out the (x+h)^2, that gives us this term.
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We distribute the 4 and we get the 1.
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Now let's just multiply through by 3. You get 3x^2 + 6xh + 3h^2 + 4x + 4h + 1.
00:03:04.420 --> 00:03:15.480
Okay so now let's put in the formula so we do the lim h-> 0 of this - f(x) all over h.
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So (3x^2 + 6xh + 3h^2 + 4x+ 4h + 1) - (3x^2+ 4x + 1) /h.
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You see how these start to get kind of tedious.
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And cumbersome. You just don't really want to do all that work.
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So Monday we'll learn the fast way to do these.
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Some of you probably already know it but the rest of you will be very excited.
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Okay the 3x^2 here cancels with the 3x^2 there because we're subtracting.
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4x-4x and 1-1.
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Now we have 6xh + 3h^2 + 4h all divided by h.
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We factor out h from the top.
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And the h's cancel. Now if you plug in 0 we get 6x+4.
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That's the derivative. So as in the previous problem where we had 6x^2 + x
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so the x term is what's going to contribute 4h so in that one you would've had just plain h and when you factor out the h you're left with a 1.
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So here we factored out the h, we're left with a 4.
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Okay so that's what contributes to this term.
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So when you do your initial expansion
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there are a couple of clues you could look for.
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Any terms that don't contain h get canceled.
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Any terms that contain a single power of h, you'll be able to pull an h out and then you'll be let with something when you do the limit.
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Anything with more than a power of h, even if you take one h out it's still gonna have an h in it which means it's gonna become 0 when the time comes.
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Okay so there's three types of terms.
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They have no h in them, that means they're going to cancel.
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They have a single h in them, they're going to be left over when you do the limit.
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Or they have more than one power of h which means they're going to become 0.
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Okay so let's have you guys try one.
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That looks fun.
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How are we doing?
00:12:12.460 --> 00:12:16.780
The cube is annoying I know. You have to practice what (x+h)^3 is.
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You multiply this out. (x+h) times (x+h) and you get x^2 + 2xh + h^2.
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When you multiply out (x+h) first you get x times x, then you get x times h,
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and another x times h, so thats 2 x times h and 2 x times h and then h times h.
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So that's where the 2xh comes from.
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So when you do x^3 you're gonna get
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x^3 + 3x^2h + 3xh^2 + h^3.
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If you multiply it out. Okay?
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You all should memorize that if you have not.
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You should memorize it or you should be able to do it quickly.
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The coefficients go 1, 3, 3, 1.
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Alright I think everybody's had enough pain.
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Let's do this one.
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So first let's figure out what f(x+h) is.
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Because that's where the real work comes from these problems.
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So f(x+h) is 2(x+h)^3 + 5(x+h)^2 + 8.
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So if you wanted to get partial credit because you say I'm never going to get this all correct,
00:14:38.300 --> 00:14:46.200
if you just show that you're looking for the limit of f(x+h)
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minus f(x)
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that'll be worth 3 or 4 points out of the total of 10.
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Just demonstrating that you can set it up is worth something.
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Okay so we'll come back to this in a minute.
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So let's work this out.
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(x+h)^3 = x^3 + 3x^h + 3xh^2 + h^3, which I just wrote that there.
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(x+h)^2 = x^2 + 2xh + h^2 because I also wrote that there.
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Plus 8.
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And then distribute these coefficients.
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So you get 2x^3 + 6x^2h + 6xh^2 + 2h^3 + 5x^2 + 10xh + 5h^2 + h.
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So that long messy expression is going to be the left side of the difference quotient.
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(2x^3 + 6x^2h + 6xh^2 + 2h^3 + 5x^2 + 10xh + 5h^2 + 8)
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- (2x^3 + 5x^2 + 8).
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So that's what we're going to now simplify.
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So remember what I keep telling you.
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So these terms on the right cancel. 2x^3 cancels. 5x^2 cancels. 8 cancels.
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And you now have the lim h->0 of 6x^2h + 6xh^2 + 2h^3 + 10xh + 5h^2. Whole thing divided by h.
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So remember I told you,
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when you go through this the first thing is any term in the beginning that doesn't have an h in it should be canceled.
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So everything now should have at least one h in it.
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Which it does.
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And anything that has more than one power of h, this one, this one, and this one, are gonna end up becoming zero.
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Because you're going to take the limit and they're going to be zero.
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They're not going to be important. So the only thing that's going to be left is this term and this term.
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So let's see how we do that.
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Factor out the h.
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You get 6x^2 + 6xh + 2h^2 + 10x + 5h.
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All divided by h.
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Okay, now this is the crucial part of calculus, you let h be 0.
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Because as I said you have people that do both things, you have people that say
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you cancel this because h is not 0 and you can plug in 0 for h because it's essentially 0.
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See you have to be able to hold both concepts in your head.
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It's really close to 0 but it's not 0.
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So sometimes you can treat it like 0 and sometimes you don't.
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And now when you let h be 0 these three terms will cancel, the terms that contain the h's in them, and you get this.
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How'd we do on that one?
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Good?
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Not ready for college?
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Alright let's do a couple more.
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One that's not quite so messy.
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The polynomials are messy because there's lots of terms.
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Alright how about something like 1/x.
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Let's do that one as a team.
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How do we do 1/x? Well let's set it up.
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I'm gonna do the lim h->0 of 1/x+h, that's the f(x+h) term.
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-1/x all divided by h.
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Now what do we do?
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It's not quite so easy. We can't just cancel.
00:20:56.580 --> 00:21:03.580
So we have to figure out what we're going to do here so when you have a pair of fractions like that how would you combine them?
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You use a common denominator.
00:21:07.500 --> 00:21:12.940
So why don't we try that first. See if we can turn that into a single fraction on top.
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So the left fraction, you're going to have to multiply the top and bottom by x.
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So that's gonna be x/x(x+h).
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And the right fraction I'm going to multiply the top and bottom by x+h.
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So far so good?
00:21:39.900 --> 00:21:44.480
See what I did? The left fraction the top and bottom and multiplied by x.
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The right fraction the top and bottom were multiplied by x+h.
00:21:51.980 --> 00:21:53.020
Okay next step.
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Okay so now let's put those two fractions together.
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You're going to get, distribute the minus sign, (x-x-h)/x(x+h)/h.
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See in every step we know we're going to have a problem because every time you want to let h go to 0
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you keep getting 0 in the denominator.
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So remember that stands for the change in x, slope.
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And that change in x is still 0 it's still causing us an issue.
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So you have to find a way to get rid of that h.
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Okay these x's cancel.
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And now we have -h/x(x+h) all over h.
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You may not realize that this h and this h will cancel.
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So let's show you why.
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This h down here and this h up here.
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You can think of this h in the denominator as h/1.
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Question: "How'd you get the x and x+h on top on the right side?"
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Remember multiply this one top and bottom by x, and this one top and bottom by x+h.
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The common denominator.
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So if you think of this as h/1 when you take a fraction divided by a fraction you take the bottom one, flip it, and multiply it.
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So you have -h/x(x+h) * 1/h
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The h's now cancel.
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The h's cancel and leave you with 1, not 0.
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So you have the lim h->0 of -1/x(x+h).
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Now you can plug in 0 for h.
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Because you don't have that denominator issue anymore.
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Now if you take the limit h becomes 0 and you get -1/x^2.
00:25:03.240 --> 00:25:07.640
The good news is on the exam you'll probably have to do one of these.
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And I don't think I give you one of these type. They're too hard.
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I would give you a square or a cube.
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You're not going to have to do ten of these that's just nasty.
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Alright let's try one other annoying type.
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What if I give you the square root of x?
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But on the bright side we haven't done any trigonometry, nothing with pi.
00:26:00.880 --> 00:26:03.200
What do you do with the sin of (x+h)?
00:26:04.300 --> 00:26:05.740
It gets tricky right.
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It would be good if I give like an arc tangent.
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Alright we'll do this one as a team.
00:26:18.540 --> 00:26:31.540
So we're going to want the lim h->0 of √x+h - √x / h.
00:26:32.520 --> 00:26:36.460
So one of the fun things about teaching in this room is the seats are really comfortable.
00:26:37.200 --> 00:26:38.880
So we lose a lot of people.
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Alright let's see if we can figure out how to do this one.
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Only one is pretty good. Oh wait I need two.
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Eh, it's hard to tell.
00:27:02.880 --> 00:27:06.420
Okay so how would we do this one?
00:27:06.420 --> 00:27:12.660
Any ideas? You can't just combine those right? It's the square root of something minus the square root of something.
00:27:16.440 --> 00:27:18.200
You can't quite square it.
00:27:21.840 --> 00:27:25.600
Almost multiply it by itself. Not quite the reciprocal.
00:27:25.840 --> 00:27:26.880
The conjugate.
00:27:26.880 --> 00:27:29.260
So you're going to multiply the top and bottom by the conjugate.
00:27:29.260 --> 00:27:32.300
Remember when you did that in the 8th grade and you didn't know why? This is why.
00:27:33.740 --> 00:27:36.560
In fact if you learned common denominators you could do these.
00:27:36.840 --> 00:27:38.280
Among other reasons.
00:27:38.280 --> 00:27:42.540
So we're going to need to multiply the top and bottom by what's called the conjugate.
00:27:48.240 --> 00:27:51.840
The conjugate's purpose is to get rid of the radicals.
00:27:59.840 --> 00:28:03.900
The conjugate of a+b is a-b.
00:28:04.300 --> 00:28:10.140
That's the conjugate. What's nice is when you multiply those together you get a^2 - b^2.
00:28:12.560 --> 00:28:16.900
So when you get the squares that gets rid of the square root. That's what you're trying to do.
00:28:18.460 --> 00:28:25.460
So multiply the top and bottom by √(x+h)+x will get rid of the radicals on the top. It's going to put one on the bottom though.
00:28:25.460 --> 00:28:27.440
So we have to hope that's not an issue.
00:28:27.440 --> 00:28:31.100
But our issue really is we have to find a way to get rid of that h.
00:28:31.740 --> 00:28:34.060
Alright so let's multiply the top.
00:28:37.740 --> 00:28:42.680
The lefthand term is √(x+h) * √(x+h) that's (x+h).
00:28:46.660 --> 00:28:57.020
And then as I said with the conjugate you get a^2 - b^2 because you're gonna get √x * √x+h - √x * √x+h so those are going to drop out.
00:28:58.020 --> 00:29:00.840
And then you're going to get -x.
00:29:02.780 --> 00:29:07.180
So again, this square root times this square root is the left term.
00:29:07.620 --> 00:29:11.140
The middle terms cancel and √x * √x is x.
00:29:12.340 --> 00:29:20.300
In the denominator is h * √x+h +√x.
00:29:25.500 --> 00:29:27.600
Okay. Cancel the x's.
00:29:46.340 --> 00:29:48.020
?? one of your questions.
00:29:48.540 --> 00:29:52.300
On the exam are you expected to write limit at every step?
00:29:52.500 --> 00:29:55.060
What do you think should I say yes or no?
00:29:55.060 --> 00:29:58.820
You're expected to but I'll let you get way with it if you don't.
00:29:59.860 --> 00:30:03.080
You should try to remember to because the TAs will be looking for it.
00:30:03.080 --> 00:30:08.060
And then they'll come up to me and they'll say "What do we do with a student who doesn't write limit all the way through?"
00:30:08.700 --> 00:30:12.060
And I'll say "Well, we could execute the student."
00:30:12.300 --> 00:30:15.340
But that's probably a bit too much of a lesson.
00:30:17.220 --> 00:30:22.100
So I think it's simpler just to say we'll let you get away with it this time.
00:30:22.100 --> 00:30:25.180
But the next calculus class you take you won't get away with it.
00:30:25.180 --> 00:30:27.280
Of course how many of you are taking another calculus class?
00:30:27.880 --> 00:30:29.400
There you go, alright.
00:30:32.320 --> 00:30:34.080
So we can cancel these h's.
00:30:34.740 --> 00:30:38.020
When we cancel the h's what are we left with on top?
00:30:38.500 --> 00:30:42.680
Not zero. 1 right?
00:30:47.420 --> 00:30:52.860
So the top is 1/√x+h + √x.
00:30:54.240 --> 00:30:55.120
Now let h be 0.
00:31:00.480 --> 00:31:04.920
And you get 1/√x + √x
00:31:06.240 --> 00:31:09.880
or 1 / 2√x.
00:31:14.840 --> 00:31:19.080
And you get really sadistic teachers who give you cube root of x.
00:31:19.420 --> 00:31:20.940
Or 1 over cube root of x.
00:31:21.620 --> 00:31:25.060
1 over the cube root of x^2, you can get really nasty.
00:31:25.060 --> 00:31:27.960
So what's gonna happen is we're going to learn some shortcuts.
00:31:27.960 --> 00:31:31.360
Soon. But let's just have you practice one more without me.
00:31:44.400 --> 00:31:45.760
Is it a hard one? Nah.
00:31:53.480 --> 00:31:56.040
There you go. See how you do on that one.
00:37:13.940 --> 00:37:16.120
I think people are just about ready.
00:37:16.780 --> 00:37:18.300
Ready for the weekend.
00:37:18.380 --> 00:37:20.860
You guys going to Brookfest tonight?
00:37:22.280 --> 00:37:25.160
Some of you. Are we excited for Post Malone?
00:37:25.840 --> 00:37:27.280
Is there a Pre Malone?
00:37:30.460 --> 00:37:33.740
I gave it a shot. And what's the other guy? Slushi?
00:37:36.280 --> 00:37:38.680
Isn't that something you get at 711?
00:37:40.500 --> 00:37:42.100
Anyway i hope it's good!
00:37:42.100 --> 00:37:44.700
We've had some pretty good concerts over the years.
00:37:45.140 --> 00:37:46.180
Jimmi Hendrix.
00:37:46.700 --> 00:37:51.420
Janis Joplin. I don't think these guys are at that level yet. The Doors.
00:37:55.920 --> 00:37:56.880
Led Zeppelin.
00:37:56.880 --> 00:38:01.260
Not there yet. We had Bruno Mars a few years ago. He was pretty good.
00:38:03.100 --> 00:38:07.180
Anyway well I won't be there. Just incase you were wondering.
00:38:12.000 --> 00:38:25.020
So let's find the lim h->0 of 5/x+h+1 - 5/x+1 all over h.
00:38:25.420 --> 00:38:29.340
Everybody gets that step. That's the partial credit step.
00:38:30.600 --> 00:38:32.680
Alright, common denominator.
00:38:33.080 --> 00:38:36.680
So you multiply this fraction, top and bottom, by x+1.
00:38:36.960 --> 00:38:39.920
And this fraction, top and bottom, by x+h+1.
00:38:41.020 --> 00:38:43.880
Question: "Should we put parentheses around the x+h?"
00:38:44.160 --> 00:38:47.200
You can. You can have parentheses around x+h.
00:38:47.200 --> 00:38:49.420
It doesn't really matter, but you can.
00:38:50.280 --> 00:38:52.880
That was interesting what we ??
00:38:53.300 --> 00:38:57.520
Lim h->0 okay so multiply this by x+1
00:38:59.340 --> 00:39:04.360
over x+h+1 * x+1.
00:39:05.600 --> 00:39:19.100
Minus 5(x+h+1) / (x+h+1)(x+1). Whole mess over h.
00:39:23.100 --> 00:39:26.480
Okay now combine them to one fraction. On top.
00:39:31.300 --> 00:39:34.420
So let's see we've got some distributing to do.
00:39:41.780 --> 00:39:52.860
The denominator is (x+h+1)(x+1) and then this whole thing divided by h.
00:39:54.700 --> 00:39:55.660
So far so good?
00:40:09.760 --> 00:40:11.680
Everybody with me? Alright.
00:40:12.160 --> 00:40:14.000
So let's do some canceling.
00:40:14.440 --> 00:40:17.700
The 5x and the 5 cancel with the 5x and the 5.
00:40:18.360 --> 00:40:34.500
And you get the lim h->0 of -5h on top, (x+h+1)(x+1), all over h.
00:40:34.500 --> 00:40:37.300
Does everyone know what you can do with this h and this h?
00:40:38.480 --> 00:40:40.600
You can cancel. Okay?
00:40:43.420 --> 00:40:58.640
This h cancels with that h and you get the lim h->0 -5 on top, (x+h+1)(x+1).
00:41:02.040 --> 00:41:04.600
Okay now just let h=0.
00:41:08.660 --> 00:41:14.580
And you get -5/(x+1)(x+1) which you could leave.
00:41:14.940 --> 00:41:21.520
Or you could make -5/(x+1)^2. Doesn't really matter.
00:41:21.520 --> 00:41:25.800
This last step isn't important. I mean obviously you should be able to simplify it there.
00:41:26.780 --> 00:41:28.540
Anybody able to get there?
00:41:39.720 --> 00:41:42.300
Why am I adding h here?
00:41:43.060 --> 00:41:46.140
Well the original expression is 5/x+1.
00:41:46.400 --> 00:41:50.820
So if I want f(x+h) then x has to be replaced with x+h.
00:41:55.140 --> 00:41:57.140
I understand this is painful.
00:41:57.140 --> 00:42:00.200
The good news is this is as hard as it's going to get.
00:42:00.200 --> 00:42:05.300
Alright everybody have a nice weekend. Look for an assignment sometime between today and tomorrow.
00:42:06.300 --> 00:42:07.740
See you Monday.