| Start | New class.
As you know I don't care about attendance. Show up or don't show up it's really, you're an adult. You yell at your mom and say treat me like an adult so I treat you like an adult, however. Your reward for coming to class is I say things like there's a new homework assignment in the assignments folder. And there's a new my math lab that you should be working on. I don't feel the need to post that on Blackboard, after all if you're in class you'll know. If you're not in class well maybe you should check. Right? So everybody knows there's a my math lab and there's a paper homework. |
| 0:31 | Paper homework the first problem has a typo in it. I will fix it later.
But I'm not at my computer to fix it. The first problem there's a 'z' in there, the 'z' is supposed to be an 'x'. It's obviously supposed to be an 'x'. Why would you think it's anything else? Come on. So it's obviously a typo. If you know how to type x is next to the z. So I apologize I will fix it. But there you go. Alright how are we doing on this whole chain rule thing? |
| 1:00 | Do we like the chain rule?
Power rule, quotient rule, Ja Rule? He's my favorite. Did you check out Eminem? His rant on Donald Trump? Kinda funny. Alright let's do a little more chain rule. There you go. Let's do this one. |
| 1:46 | Okay. I like that they cleaned the board for me. It's really nice.
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| 2:18 | Okay work on that for two minutes. Let's see how you did.
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| 6:53 | How are we doing on this? You should be able to do the derivative.
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| 7:08 | Okay so equation of the tangent line. It's just like the other ones we did it's just a slightly harder derivative.
So when you see the square root you want to rewrite this as to the 1/2 power. That way it's usually easier to think of because you're going to use the power rule twice. So for the derivative you do the derivative of the outer function. |
| 7:40 | Which is 1/2, leaving the inside alone, to the -1/2.
Times the derivative of what's on the inside. And now don't simplify. You're plugging in 1. Why would you simplify before you plug in a number? It's much easier after you plug in a number. |
| 8:03 | So dy/dx at x=1 is 1/2, 1+3 is (4)^-1/2 times 5.
Okay what is 4^ -1/2? 1/2 right because 4 to the positive 1/2 is the √4 is 2. So 4 to the -1/2 is 1/2. So 1/2 * 1/2 = 1/4 times 5 is 5/4. |
| 8:38 | So now we have the slope.
Now need the y-coordinate. So we plug in 1 we're going to get y=√1+3 which is √4 which is 2. So the equation becomes y-2= 5/4(x-3). |
| 9:05 | Alright let's give you another one of these.
What did I do wrong? Oh x-1, sorry. ? It's just you know it's Alzheimer's is starting up now. That was a long time ago I already forgot. I shouldn't joke because one day that's going to happen in the class and then they're going to come in and take me out and that'll be it. |
| 9:33 | We're both hanging in there for a few more years.
Alright are we good? Alright so let's do another one of these. You guys should be wearing red for homecoming. It's very disappointing. A few of you are representing. Your socks are red, okay? Soothing. Alright let's try another one of these. How about... |
| 10:10 | So find the equation of the tangent line at x=25.
No I was right. Okay so another equation of a tangent line. That's a 4 if you can't see that. |
| 13:55 | Alright let's do this one.
So the square root inside the square root looks scary but it's really not. |
| 14:01 | You want to rewrite this as (4+√x) ^1/2 and you could rewrite √x as x^1/2 if you want.
So the derivative. So you have 1/2 (4+√x) ^-1/2 and what's the derivative of 4+√x? Well the derivative of 4 is 0 and the derivative of √x is 1/2√x. |
| 14:35 | Okay?
And now we're doing this at x=25. Aright √25 is 5, 5 plus 4 is 9. 9 to the minus 1/2 is? Not quite 3. 1/3. |
| 15:01 | Because 9^1/2 is 3 so 9^-1/2 is 1/3.
Plug in 25 here you get 5. 2*5 is 10 so that's 1/10. Put the whole thing together you get 1/60. How'd we do on that? We found that one? Okay. And when you plug in 25 here you get y= √4+√25 The √25 is 5 so the √9 is 3. |
| 15:33 | Didn't that come out nicely?
My problems will almost always work out well so if you get some weird number you probably made a mistake. Probably. Okay so y-3= 1/60(x-25). So far so good? So how do we feel about the whole chain rule? We're good with that? We're not going to get into anything much harder. |
| 16:02 | Alright so one other thing that's important for us to learn, and you started to learn all of your technique of integration and differentiation,
is something that sounds fancy but it's not. It's called high order derivatives.
Also known as the second derivative, the third derivative, and so on so let's see. We have just we'll take a basic function. f(x)=x^5 |
| 16:32 | Then I see the function, take the derivative.
5x^4. Okay. Now let's take the derivative of that. That's called the second derivative so that's what they mean by higher. Derivatives. Higher order derivatives. It's just taking the derivative again. The second derivative which tells you how fast the derivative is changing. It's actually very useful to know. |
| 17:04 | And you can take the derivative again.
Three little lines. You can keep going but you see it doesn't go very far. The forth derivative, which if you're old school like me, you used to write IV but now we do (4). Parentheses like that. |
| 17:31 | The forth root is 120x.
The fifth derivative is 120. So what's the sixth derivative? Zero. 7th would be 0, 8th would be 0, so every derivative after the fifth derivative will come out 0. That's because this is to the fifth power. So each time you take the derivative you lose a power. So by the fifth derivative you reduce the power all the way to 1- or to 0. |
| 18:04 | So after that there are no more derivatives to take.
So if you had x^100 you can take 100 derivatives of that and the 101st derivative would be 0. So a favorite test question is find the 1000th derivative of x^999. And you know somebody cries and says I can't take 999 derivatives. And they usually just didn't come to class. Because the 1000th derivative would be 0, okay? |
| 18:31 | Now the 998th derivative would be a bit of a pain in the neck.
But there's also something you could observe here. Let's look at these derivatives again. See if you spot a pattern. So this one is x^5, this one is 5x^4. Another way to think of this is 5*4x^3. This is 5*4*3x^2. This is 5*4*3*2x^1. |
| 19:04 | And this is 5*4*3*2*1x^0.
So 5*4*3*2*, do we know what that function is called? Any class you've learned this? Exactly it's the ! thingy. Okay? That's called a factorial. |
| 19:32 | So a factorial function. I don't know why it has that name. n!
I'm going to run out of room there. n! 1*2*3 up to n. So 4! is 1*2*3*4. 5! is 1*2*3*4*5. 10! is 1*2*3... up to 10. Okay? |
| 20:00 | And it shows up a lot especially in things that have to do with combinations and probability and stuff.
So if you had f(x)= x^n then the nth derivative is n! There's a factorial button on your calculator. In Excel you write factorial parentheses and you put in the number. |
| 20:35 | Factorials grow very quickly. 70! is too big for your calculator.
It's bigger than googol so that's big. googol is an actual number. It's spelled differently. Okay? So um I don't know if it'll show up if we'll use this at all in the class but we might. So the n+1 derivative is 0. |
| 21:02 | Okay? So if I said to you f(x)= x^4 then the forth derivative is 4!
Or 1*2*3*4 which is 24. Okay? Fifth derivative would be 0, sixth derivative would be 0, etc. |
| 21:31 | Got it? So what do we care about these second and third and so on derivatives?
You usually only care about the first and the second derivative. So the first derivative remember it tells you the slope. It tells you how fast something is changing. Rate of change. So you care about the rate that things are changing in all sorts of aspects. Okay, that gives you the slope of something. In physics it's velocity. In the business world it'd be the change of whatever it is that you're doing. Your profit, your cost, your manufacturing ratios, all sorts of things. |
| 22:01 | The second derivative is how fast your rate is changing.
So when you're in your car and you press on the gas pedal that gets you velocity, the car starts to move. But your velocity isn't constant your velocity is either picking up or slowing down depending on how steadily you're pressing the pedal. Assuming you don't have cruise control. Okay? So that's your acceleration. Your acceleration is how fast your velocity is changing. |
| 22:30 | If you're going from 40 mph to 50 mph you've changed 10 mph so you accelerated.
So the second derivative tells you how fast something is accelerating. If we think about it graphically, Remember the first derivative gives you slope. |
| 23:05 | So the first derivative means you have a curve and it tells you sort of what the curve is doing.
Okay? That's the first derivative. This says you have a slope and the slope looks like this. Notice the slope is increasing so the velocity is increasing. So second derivative helps you figure out the curvature of your curve. It helps you figure out if it's curved this way versus say a graph that's curved this way. |
| 23:32 | Notice here the slopes are getting steeper. Here the slopes are getting flatter.
Okay? So this is something that's going up. This is also going up but it's going up more slowly it looks like it's either going to flatten out or start going down. So these slopes are increasing. This is a positive second derivative. We're going to do more of this on Monday so don't get too excited. And this is a negative second derivative. So the derivatives tell you curvature they tell you how the curve is changing. Remember I told you what's going to be important when you learn is how you find the maximum of a function or the minimum of a function. |
| 24:06 | So you're shooting a rocket and you want to hit something, you want to know when the rocket gets to its maximum arc.
Okay? You're kicking a field goal you know or any kind of projectile through the air. You want to know when it gets to the top because that will help you figure out how long it takes to get back to the bottom. It'll tell you how high it'll get. So if you were say trying to shoot down- suppose somebody's throwing a football right? |
| 24:32 | And then you want to throw a football from the side and hit the football the second best place to hit the football is right at the top of the arc.
Okay right when it hits the top because it stops for a second for an instant before it comes back down. The best place to hit it is before it leaves. So just as a person throws so when you're shooting missiles down the best place to hit it is before they take off. Obviously. Sounds dumb but in other words before they launch you want to take them out but assuming they've been launched |
| 25:00 | you'll have a moment there where they reach their maximum and then they come back down and you want to get them.
Leaving the weaponry out, you care about that say in your profitability. When you want the maximum profit. If it's cost where do I get to the minimum cost? So there's all sorts of things where you will care about maximums and minimums and we're going to need first and second derivatives. To figure that out so let's practice taking a couple first and second derivatives. |
| 25:38 | Okay how about why don't you guys take the derivative, take the first two derivatives of |
| 26:47 | Alright there's three derivatives to practice.
None of them are really that hard so let's see how you do. |
| 32:12 | Alright long enough.
Alright first derivative. 3x^2 +16x. Second derivative. 6x+16. |
| 32:34 | Notice that after the first derivative we got rid of the 5.
And the second derivative, if we did the third derivative we'd get rid of the 16. And by the forth derivative we'd just be down to 6 which would go away. We'd be down to 0 I'm sorry. So each time you take the derivative with polynomials you start to lose the lower powered terms. Okay? They become less important. Third derivative doesn't really show up very much. Third derivative is the jerk. It's the change in acceleration. |
| 33:04 | So that whiplike motion.
Alright here you have an option. So if you do the product rule you're going to have to do two product rules a second time. So what I would advise is multiply this out first. And get 4x^3 +18x^2 -10. |
| 33:34 | Because now the two derivatives is easy.
So one thing and one reason I put that is to show you that there are times when you want to do work before you take the derivative. And there are times when you want to do work after you take the derivative. And you want to sort of do whatever will make your life easier. Now this isn't too complicated to do two product rules. So let's see what happens. We would get the first function |
| 34:01 | times the derivative of the second plus the second function times the derivative of the first.
And now you should want to clean that up. Because otherwise you have to do the product rule again here. Here you don't because of the 4. But here you do. So I would before I took the second derivative do this and multiply that out. |
| 34:34 | Which simplifies to 12x+ 36x- 10.
For the second derivative now it becomes 24x+ 36. So it's not a tremendous mount of work but you can imagine how much tougher it would be if I just made the problem a little messier. |
| 35:01 | Here if we took the second derivative, let's see the first derivative is 12x^2 + 36x.
And the second derivative is 24x+ 36. Good thing they're the same. Okay? So as I said it's up to you whether you would rather you know foil it out first and simplify and then take the derivatives, or if you'd like to take the derivatives and then simplify. Part of it depends on what you're doing. |
| 35:31 | If you're plugging in numbers I would probably deal with the first derivative and then plug in numbers right here.
If I'm trying to find 0's I certainly would want to do it this way. It'd be much easier. Question: What happens to the -10 in the first derivative? I take the derivative here. The derivative of 10 is 0. Right so the derivative here. I foil this out I get 10 so the derivative of 10 is 0. So either way you're going to lose the constant. Okay? Everybody sure about that? |
| 36:01 | You have to practice some of these. That's why I made the mymathlab for you guys.
And feel free to practice others as I said they have the study plan. I only gave you like 4 for homework. Alright so here again you say gosh I wish I could simplify that first. I'll show you a trick for it in a second, but let's take the first derivative. It's 4x+ 3 times the derivative of the top which is 4. |
| 36:32 | Minus 4x-3 times the derivative of the bottom which is also 4.
Or (4x+3)^2. If you multiply that out you get 16x+12 -16x+12 |
| 37:01 | over (4x+3)^2.
You would certainly want to clean this up. This would be a mess to take the second derivative of right there. You've got product rule, quotient rule, it's a lot. And here the 16x's cancel. So you get 24/(4x+3)^2. That's still first derivative. |
| 37:35 | And notice, by the way, that's the same as saying 24 (4x+3)^-2.
So if you want to take the second derivative bring the power in front you get -48(4x+3)^-3. |
| 38:05 | Times the derivative of the inside.
Which you could then leave alone or you could make -192/(4x+3)^3. |
| 38:30 | Okay? Now I said there was a trick for simplifying this.
Whenever you see one of these kinds of things when the top and the bottom have the same variable here and they just differ by a constant say to yourself what would i have to do to make this plus 3? Because I have plus 3 on the bottom. But by adding 6 to that, right, that would be plus 3. But of course you can't just add 6 to something just because you feel like it. So you have to subtract that 6. |
| 39:00 | Because that's 0 right?
This is now 4x+3-6/4x+3. Why is that useful? Because you can then make this 4x+3/4x+3 - 6/4x+3. Now this is 1. |
| 39:33 | That's easier to take the two derivatives.
Okay? So if you had something of the form ax+b/ax+c if you just tweak the constant so this also comes out c by adding and subtacting the same number. Then you can turn it into 1 plus or minus whatever's left over. You agree that's easier to take the derivative of right because the first derivative would be 6(4x+3)^-2 |
| 40:04 | and then minus 12 times 4x- whoops.
Times 4 which is 24. And then you'd have -48(4x+3)^3 (4) so you get the answer. Much faster. Okay? That's a little trick I learned in I don't know 7th grade algebra which we don't seem to teach anymore. It's sort of sad. |
| 40:34 | Anyway alright that's enough for today. Monday we're doing all new stuff. Get excited.
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