| Start | So Monday's class will be a review and maybe Friday we'll also do a review.
It depends on how fast we go today and Wednesday. Okay? I haven't written the exam completely yet so that's why I don't have practice problems for you guys. As soon as I've written the exam then I'll write up a set of practice problems for everybody. So just be patient. You all know where to go? It's in Harriman. If you don't know where the building is Harriman is uh that way. |
| 0:32 | After the big metal bagel sculpture it's next it's the brick building on your left.
I mean it's not actually a bagel. It's an umbilic torus. Which means nothing to almost anybody here. Um including me. Um and you go in Harriman, you go in the doors and it's just immediately to your left. It's the one big room in the Harriman building. Okay? Terrible room to take tests in but they didn't give me a choice um and there you go. |
| 1:00 | The exam will be 500 questions. Each of which will have several sub-parts.
Statistically people will ask questions like "How many questions will be on the exam?" and as I just said, I haven't written it yet. I have no idea. Alright Shannon did you start? Yup. Okay. Alright so we were doing all the derivative stuff with the f(x+h)-f(x)/h stuff. Now, for those of you who don't know this, we're going to learn how to do it the fast way. Because it's sort of annoying to keep doing these derivatives a long, messy, complicated way. |
| 1:33 | And some of these expressions can be quite long so you really don't want to do a derivative that way.
So first thing is let's make sure we get some notation down so another way to write derivative is you write d/dx of whatever the function is. Or you write, it's usually y, so it's usually dy/dx. |
| 2:02 | And that stands for derivative and it says that the variable that you care about is x.
So the variable that we do the derivative with respect to is x. That means a function defined in terms of x. This isn't really important because in this class we only do these kinds of derivatives. But if you have a function that was defined in terms of x and y, this would say we're only looking at the changes in x. We're not going to do more than one of these because then you're in what's called multi-variable calculus. |
| 2:32 | Which means that you're in the middle of that nightmare where you wake up and you're in the wrong class.
So this isn't multi-variable calculus. We're not gonna really worry about that. But that's one way to write a derivative. Another way is with the prime notation. So you can write- eh I didn't like that piece of chalk anyway. So you can write f prime of x. |
| 3:03 | You write f'(x).
Or sometimes y', that's sort of the lazier notation. It's easier because you don't have to write all of this work, but these are all ways to say the derivative and if you want to evaluate a derivative at a number let's say you're doing it at 3 |
| 3:34 | you could use this notation, you could use this notation, or this notation.
Or you could do this. All of which are perfectly valid ways to write derivative. The one we will generally use will be dy/dx. Okay. Time to learn how to do some derivative stuff. |
| 4:00 | So the first rule that we learn is what is called the power rule.
The power rule is very simple. It says that if we have an equation y=x^n then the derivative of y is n * x^n-1. |
| 4:30 | We could prove this. We could derive this. But you guys don't care you're willing to take my word for it.
So this is a very handy and useful formula. For example we did some derivatives last week. We started with let's say y=x^2. This formula now says that the derivative is the following. You take the power and you put it in front so you're going to multiply. |
| 5:04 | 2 * x^1.
And since x^1 is just x, this is 2x. So the people who have taken calculus before they already know this so they look so smart when you're sitting there working away they're just telling you what the answer is. That's because you just use this. If we did y = x^5 see this formula is useful because if you had to do x^5 by hand you'd have to multiply out (x+h)^5. |
| 5:39 | It's pretty annoying.
But here it just says well look if you did all that and did all the simplifying etc. you'll end up with 5 * x^4. In fact if we had x^100 |
| 6:03 | we'd have 100x^99.
I mean you certainly wouldn't want to expand out x^100. That would take you all day. This is the shortcut for it so what Newman and ?? figured out was they said look, there's a formula for how to do x to any power. And then they took that formula and they realized, remember I was telling you the other day about terms that have h in them. |
| 6:30 | That when you do a little bit of the algebra a lot of these terms are going to vanish.
Because when you do the limit as h goes to 0 they're all going to be 0 and all you're going to be left with is this term. Okay so I think you've got the basic concept of this. Now there's some special cases. So what if we have a negative power? Well, same rule. |
| 7:02 | Bring the negative power in front and you get to the -6. Because remember -5 -1= -6.
If I had say 1/3 bring the 1/3 down and we get to the -2/3. |
| 7:42 | So far so good?
So let's like I said do some special cases. So if I had y= x^1/2 let's figure that out for a second. This is what a couple of you might want to memorize. |
| 8:01 | So if y is x^1/2 the derivative is 1/2x, 1/2-1 = -1/2 so do the -1/2.
But x^1/2 is square root. So this is really 1/x^1/2 or 1/√ x. |
| 8:32 | So when you see y= x^1/2 or you see y= √ x the derivative is 1/2√ x.
And you'll find it's faster because lots of problems we have will have a square root in them so you'll find it's faster if you just jump to that step. So we could say find, if y= √ x find dy/dx at x=25. |
| 9:17 | And then you would say okay we could do that.
|
| 9:33 | So dy/dx = 1/2√ x.
So dy/dx at x=25 is 1/2√ 25 which is 1/10. Questions? |
| 10:10 | There's another very handy one to memorize.
How are we doing on this? Good. Okay because I'm going to cover the board up in a second. The other very handy one to memorize is x^-1. |
| 10:32 | Because again that shows up a lot.
So if you have y=x^-1 we would use 1/x. For the derivative bring the -1 in the front and make that x^-2. That's -1 * 1/x^2 which is -1/x^1. |
| 11:09 | So if y= 1/x then the derivative is -1/x^2.
So these are all very handy to memorize. |
| 11:30 | Now of course you don't have to. You can work them out all the time.
But as I said you'll see a lot of problems that have square roots in them and 1/x's in them. And if you could do it a little faster that'd be more useful. Also when you're plugging in numbers or you're solving you'll find it's more useful. Alright well, these are just powers of x. Clearly there's more stuff that we're going to have to know how to take derivatives of. |
| 12:00 | Particularly in the beginning, polynomials. So let's do some polynomial stuff.
But what if I had y=5x^3 and you wanted to do the derivative? Well the good news is when you're multiplying by a constant you just leave the constant in. You don't do anything. The derivative is 5 * 3x^2. |
| 12:37 | Which is 15x^2.
So another rule is if y is a constant times x^n then the derivative is the constant times nx^n-1. |
| 13:19 | The constant can be positive, it could be negative, doesn't have to be an integer it could be a fraction.
It could be pi, it could be log, it doesn't matter. |
| 13:35 | So if you had y= 8x^5
the derivative is 40x^4.
Okay? So far so good? If the function you had was πR^2, you all know that one. |
| 14:09 | Okay so the derivative, this would be dy/dR would be π * 2R also known as 2πR.
Both of those look very familiar I hope. Right? That's area, that's circumference. That's not a coincidence that the circumference is the derivative of the area. |
| 14:31 | It's not important for this class but. So in other words it doesn't matter what the numbers are.
Okay? You just continue to use this rule. What if I had y= 8x, well first of all that's a line. Right? Remember y= mx+b. That's the line y= 8x. The slope of a line is just that number. |
| 15:00 | So if you do the derivative, well, this is to the one.
So 8 * 1 is 8 and then you have x to the 0 but since x to the 0 is 1 it's just 8. So in general special case if you have y= kx then the derivative is just k. |
| 15:35 | So as I said if y= 8x it's 8.
You don't want to confuse this, however, with what's coming next. Which is if y is just a constant, so if y is a constant that's just a horizontal line. What's the slope of a horizontal line? What's the slope of a horizontal line? It's 0 so the derivative is going to be 0. |
| 16:04 | You could also think of that as y=k would be the same as y= k*x^0.
So you just bring the 0 in front and then the whole thing just becomes 0. And then the other fun thing is you can now put these together. Say you had |
| 16:40 | and you wanted to get the derivative.
Well when they're added or subtracted you just add or subtract the derivatives. So 6x^3 bring the power in front you get 18x^2. |
| 17:02 | 5 times 2 is 10, derivative of 4x is 4, and the derivative of 2 is 0.
Easy? You're waiting for a trick. There's no trick. Ready for a trap? There's no trap. Because this is the very easy part of calculus. Taking the derivative of a polynomial is actually quite easy. It's what we need to do with the polynomial that can be tricky. Okay? But the actual taking the derivatives is not hard. |
| 17:33 | So let's give you a couple to do.
And very important. Don't confuse the constant times x with just the constant. They're two different things right? So the derivative of 4x is 4 because it's 4 times x. The derivative of 2 is 0 because there's no x there. Okay? The derivative of a constant is just 0. I'll give you a few to practice. |
| 18:12 | How about- I walked away with the eraser.
|
| 18:53 | ? |
| 19:16 | That one's a fun one.
|
| 19:49 | Okay there's a couple that'll keep you busy for a minute.
|
| 21:37 | That third one might be a little tricky.
|
| 23:21 | Alright I'll give you one more minute.
|
| 24:01 | Alright that's probably long enough.
Mic power back up I am back on. Alright. First one should be nice and straight forward. Okay we have y= 6x^2 - 5x + 9. |
| 24:36 | Look at that. It's magic.
So derivative. Okay the derivative of 6x^2 is bring the 2 in front you get 12x^1. 12x. Derivative of 5x that's -5 and the derivative of 9 is 0. Okay the second one. |
| 25:00 | y= 3x- √ x.
Well the derivative of 3x is just 3. And the derivative of √ x is? 1/2√ x. Nothing tricky. The third one now some of you might mess up. So you have to rewrite that. So first when you see y= 2/5x^6 you might want the think of this as 2/5 x^-6. |
| 25:41 | Okay?
Now to do the derivative 2/5 * -6 = -12/5. |
| 26:02 | And that's to the -7.
How many of you managed to mess that one up? That's what I thought. Okay. Those fractions they've been annoying you guys for years now. Yeah. Not going to get better. The good news is this is your last math class. The bad news is then you take statistics but it's not the same. Alright next one we had is y= 4/x - 3√ x +7. |
| 26:38 | So derivative of 4/x is going to be -4/x^2.
Since the derivative of 1/x was -1/x^2. You're just multiplying it by 4. It becomes -4/x^2. The derivative of √ x is 1/2√ x so this is going to be -3/2√ x. |
| 27:01 | And the derivative of 7 is 0.
So notice you just leave the constants in. You don't do anything with it. The last one that's just annoying because they're fractional powers. So you've got -3/4x. What's -3/4 -1? |
| 27:32 | -7/4. If you're rusty at that stuff whenever you want to subtract 1 from a fraction you just take the denominator, which is 4, and subtract it from the numerator.
Okay? So if you had any fraction. If you had -5/8 and you want to take away 1 it's just going to be that. Okay? So you just subtract whatever's in the denominator. |
| 28:01 | That's the fast way to do it in your head.
Okay so for 2/3 2/3 -1 is -1/3 because 2-3 is -1. And this is -5/4x^1/4. So far so good? |
| 28:31 | How do we feel about these?
Good let's give you some more. |
| 29:33 | How about this?
You have to do 2 steps. You have to find the derivative then plug in 1. |
| 31:24 | Okay so first we need to find the derivative.
f'(x) is first bring the 3 in front. 15x^2. |
| 31:33 | The derivative of 8x is 8.
And the derivative of √ x is 1/2√ x so that's 2/2√ x. And then f'(1). Be happy I didn't set it as a nasty number like 18. It's going to be 15(1)^2 -8 +1/√ 1. |
| 32:10 | That comes out to 8.
We do okay on that? No? Where did we get lost? The definition of the derivative formula? Oh no we're done with that. |
| 32:31 | The only time you'll need that is if on the exam, there will be one, I'll say use the definition of the derivative formula to find the derivative.
Okay? Once you learn the shortcut you never go back. Okay. So far so good? Alright another thing to be able to do now is to find equations of tangent lines. So remember that the derivative gives you the slope of the tangent line. When we have a curve it gives you the slope of a line tangent to the curve. |
| 33:02 | Now we can use it to actually find the equation of what that line is.
Remember you have some function and some curve. And the derivative enabled us to find the slope right at that point. So if I want to find the equation of a line well because of the derivative I now have the slope and I have the point. |
| 33:34 | So I can just use my point slope formula and find the equation of a line.
For example. I could say |
| 34:22 | Suppose I said find the equation of the line tangent to y= 3x^2 + 4x-1 at x=2.
|
| 34:30 | Why do we want these equations of these tangent lines?
Well it'll show up later. Okay? Remember I told you when you know, when you have the curve and you know the slope you can find places where the slope of that line is 0 and that'll be very useful. To find the equation of a line alright so we're going to need 2 things to find the equation of a line. Remember the equation of a line is y-y1= m(x-x1). So we're going to need the slope and we're going to need x1, y1. |
| 35:09 | Well we've got x=2 so that's our x1.
So if we want to find y1 we're going to plug in 2. And we're going to get y=3(2^2)+4(2)-1. |
| 35:32 | Or 19.
So our point is (2, 19). So we're going to have x1, y1 is (2, 19). Now we're going to find the slope. Well this is calculus class. You only know one tool. You're only going to have one tool for a while. |
| 36:01 | It's the derivative. So let's take the derivative of this.
You get dy/dx= 6x+4. So dy/dx at x=2 is 6(2)+4= 16. |
| 36:30 | So that's our slope.
Okay? So again to get the y coordinate you plug 2 into the function. To get the slope you plug 2 into the derivative of the function. Because remember the original function, the original function gives you y. The derivative gives you the slope. So our equation would be y-19= 16(x-2). |
| 37:02 | And you can stop there. You don't have to simplify that.
Alright let's have you guys practice one. Find the equation of the tangent line. |
| 37:57 | That looks like a good one.
|
| 39:18 | Okay so you need 2 things. You need the slope and you need the point.
|
| 40:24 | Alright so let's find the point.
So y= 9+√ 9 which is 12. |
| 40:34 | So we're going to go through the point (9,12).
And the derivative. Okay the derivative of x is 1. And the derivative of √ x is 1/2√ x. |
| 41:06 | They're going to keep calling me back because I can't pick up the phone and say stop calling.
Alright so now we plug in 9 and you get 1+ 1/2√ 9 which is 1+ 1/6 which is 7/6. Okay? So then the equation is y-12= 7/6(x-9). |
| 41:38 | How'd we do?
Feeling good about this? Alright so we'll pick up more of this on Wednesday. One more thing to learn. |