| Start | The practice problems. So we're practicing chapters zero and one.
I don't have a way to project the computer up there. 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. That I'm not sure about. But under study plan it'll give you lots of practice problems on every chapter. You should be able to access those. So I'll give you guys a minute to see if you can. |
| 0:31 | I mean it might come up on the phone. I don't know.
We'll find out. Is it coming up for you? Tablets are the same as phones. Yes it should be study plan which should be, there's assignments that's where your homework is. |
| 1:00 | And there should be something that says study plan. See my view is not necessarily the same as your view.
So if you can't see that then I have to unlock it somehow. Which I thought I had done but maybe not. You see homework but you don't see study plan? Alright so I'll have to find out how to unlock that. Although frankly what we're doing in class should be enough review. I'll be honest but- I'm sorry? |
| 1:30 | You have study plan? And you have study plan?
So look more carefully. It's on the left bar. Right so you should have course home, assignments, then study plan after that. And then you're going to want to click where it says all chapters. I don't know where to project this. And then you can do chapters zero and one. We are in chapter one up to sum and difference. The derivative using sum and difference which is... |
| 2:00 | So the exam will cover chapter zero or chapter R, excuse me, up through 1.5.
So anything in chapter R for review. And anything in chapter 1.1 through 1.5. Are you guys finding it? Yes? Yes? Did you find it? No? |
| 2:43 | Good okay so it's there.
That makes me happy. But in the mean time let's do some more review. So we did derivative stuff, the definition of derivative stuff, did we do remaining? Yes? |
| 3:04 | So everything in chapter R, and then up to 1.5.
So chapter 1 through 1.5. We don't get very far in the course. We get somewhere into chapter 5. We go nice and slowly just to make everybody happy. You all know the midterm is tomorrow night. Correct? And you all know where to go? You go to Harriman right? |
| 3:33 | Room 137.
The exam will end at approximately 10:15 at night. 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. Now who was not listening when I said that? Maybe you should've paid attention. We'll see you Wednesday then. Yes okay so who was not listening when I said that? |
| 4:02 | Good okay then we'll see you.
Alright so we did definition of derivative, we did some derivatives, we did um we did domains? We did not do domains. Let's practice a couple of domain problems. We will have class on Friday however. I know that'll excite you. |
| 4:30 | So the domain is what you're allowed to plug into a function.
I'll move this back up a little bit. Domain is what you're allowed to plug into a function for x. So it's not what you give out, that's range. It's what you put in. That's domain. So let's see. Closed f(x) = √ 6x-1. What is the domain? |
| 5:18 | This shouldn't take you very long.
Remember how to find the domain. What are you allowed to plug in that radical? What are you allowed to plug into the radical? |
| 6:19 | Okay that should be all the time you need.
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. So this cannot be negative. |
| 6:33 | So 6x-1 has to be ≥ 0.
So if you solve that you get 6x ≥ 1 and x ≥ 1/6. So that's the domain. You don't have to do any fancy notation anything like that. Okay the domain is what's under the radical must be ≥ 0. |
| 7:05 | Okay? What if instead I gave you...
What's the domain?
So you look at this and you say that's a fraction. You can put whatever you want in a fraction. You can have anything divided by anything. Except you can't have 0 in the denominator. |
| 7:31 | Because a number divided by 0 doesn't make any sense.
So the only thing you cannot have is x-2 ≠ 0 so x ≠ 2. So far so good? Alright let's put it together. Suppose I said f(x), what is I gave you that? |
| 8:34 | Alright you do not need to use interval notation for the exam.
Okay? You need interval notation to put the problems into mymathlab but you do not need it for the exam. You can write it just like that. Interval notation isn't better it's just compact. |
| 9:38 | Okay let's do the third one.
So what's changed? Well we still have a radical. So the radical, you still cannot take the square root of a negative number so 2x-1 still cannot be a negative number. And then you look and you say but wait. This told me I can't have 0 in the denominator of a fraction. |
| 10:01 | But can't have equal to either.
So here I could plug in 1/6 because I'm going to get √ 0. Here this is x is going to have to be > 1/2. I cannot plug in 1/2 here because then the denominator would be 0. So far so good? Okay what if I gave you |
| 10:37 | that?
|
| 12:21 | Alright that's long enough.
So now we're combining. So here, still can't take the square root of a negative number. |
| 12:30 | So 3x-4 ≥ 0. Is it allowed to be 0?
Yes because you could have 0 in the numerator of a fraction. And here x ≠ 5 right? Because x-5 can't be 0. So here you have two parts to the domain. x has to be ≥ 4/3 and x ≠ 5. Again, you don't need to use interval notation. |
| 13:06 | Now what if I flipped that the other way. What if I said f(x) is x-5/√ 3x-4?
Well it's the same numbers. But now notice, can I have 0 in the numerator of a fraction? Yes, so x can be 5 now. And check by the way we plug in 5 that's the square root of a positive number. |
| 13:32 | So I only have to worry about the radical in the denominator.
So 3x-4 has to be greater than 0. It can't be equal to 0 because you can't have 0 in the denominator. So the domain would just be x > 4/3. Everybody got the domain stuff down? We good? |
| 14:00 | Alright let's do some linear stuff.
|
| 16:36 | Okay.
A plane is flying from New York to London. One hour after takeoff it is 300 miles from New York. 4 hours later it is 15 miles from New York. Assuming constant velocity find an equation of the plane's distance from New York t hours after takeoff. And then if London is 3,000 miles from New York, how long is the flight? |
| 17:01 | This is as good as my handwriting gets folks.
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| 20:16 | You want to think about a pair of coordinates.
?? |
| 20:35 | I'll give you guys one more minute.
See when we edit the videos we can cut all this stuff out. |
| 22:53 | Alright that's long enough.
I put the chalk somewhere. |
| 23:01 | Walk with the chalk. It's not over there.
Alright how about this chalk. Alright so we want to find the equation of a line. 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. Okay? If you used x we wouldn't take off points. So one hour after takeoff the plane is 300 miles from New York. |
| 23:31 | So at time 1 it's location is 300 miles, 4 hours later, so at time 5 it's location is 1,500 miles.
Okay? Because it's 4 hours later not time 4. Okay? So let's find the slope. So m, that's 1,500-300/5-1 which is 1,200/4 which is 300. |
| 24:09 | Okay?
And I think you could say that you could use either point so it doesn't really matter. So y-300 is the slope times x-1 except instead of x we should use t. |
| 24:31 | And if you want to simplify that
the 300's cancel and you get y= 300t.
Now some of you wanted to just use rate * time = distance. On this problem that would come out the same but I can give you a different type of problem where it wouldn't. So be careful about just assuming that formula. Okay this wasn't very hard. |
| 25:00 | So if London is 3,000 miles from New York, that's about right, how long is the flight?
That's a. This is b. 3,000 = 300t. That's a terrible t. So t = 10. I don't know how long it actually takes the fly to London. Yes. |
| 25:32 | Oh sure you could say time equals x.
I mean I don't care if you use t or x honestly. 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. Okay? So we could do a slightly different one. |
| 26:07 | Okay you're given a choice of telephone plans.
Plan number one you pay a flat rate of $40 plus $0.10 per minute of using the phone. |
| 26:39 | Because it's some sort of burner phone I guess.
Plan number two you pay a flat rate of $25 but it's $0.12 per minute. |
| 27:07 | Okay so what's the question I'm going to ask?
Give it a shot what's the question I'm going to ask? What's the better plan? And you say well it depends on how many minutes you use right? So what's the sport where we don't care which plan we're at? Okay which would be called the break even point or here it would be called the indifference point. |
| 27:43 | Indifference. I don't care which plan I'm on.
Okay? If I'm only going to use one minute I clearly want plan 2. If I'm going to use a billion minutes I clearly want plan 1. That ends up cheaper. So somewhere I don't care. That's indifferent. So what's the indifference point? |
| 30:30 | Alright let's set this up as a pair of equations.
So the plan 1 equation, your cost, is going to be 40+.10t. Because it's 10 cents. Or you could make it 4,000 cents instead of $40. It's really up to you. Plan 2. |
| 31:01 | Is 25+.12t.
And then you want to set those equal to each other. So 40+ .1t = 25+ .12t. And you get 15= .02t. So you should get 750 minutes. |
| 31:33 | So if you wrote 10 instead of .1 I think you got 7.5 minutes or something.
That's because you weren't paying attention to units. Okay how do we feel about this one? So we could do linear stuff. Or we could do domains. |
| 32:05 | Alyssa is asking if we could to the vertical line test stuff.
No. Did we do finding the inverse of a function? Did I show you guys how to find the inverse of a function? I think I should do those. No we don't want quadratics. But we didn't really do anything with quadratics so. We did polynomials. Did we do any limits? Let's review some limits. |
| 32:46 | There's not a lot on this exam.
So very few chances to trick you up. Okay let's do some limits. |
| 34:15 | There's 4. See how you do on those.
|
| 37:08 | You got this?
You need another minute? There's got to be a difference between these two limits right? One's squared one's cubed. |
| 37:35 | Alright let's do these as a team.
So the top one. The lim x-> ∞. You look at the power of the top. The degree of the top, the highest power. It's cubed. And you look at the bottom and it's also cubed. 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. |
| 38:01 | Which is 4.
You don't have to reduce 8/2 but you should. So again you look at the powers. Since the powers match it's just the coefficient over the coefficient. Now when you do the second one when you plug in 3- oh yes, yes? Question: Is it only for infinity you can do that? It's only for infinity. Now we look at the second one. When you plug in 3 on top you get 0. |
| 38:34 | You plug in 3 on the bottom you also get 0.
So does that mean the limit does not exist? You have to factor it. So let's factor this. We get (x-3)(x-4) on top. (x-3)(x+3) on the bottom. So our problem was the (x-3) term. You plug in 3 here you get 0, you plug in 3 here you get 0. |
| 39:05 | Now you cancel and now if you plug in 3, and that's -1/6.
So far so good? Okay now for these 2 on the bottom. |
| 39:35 | So remember when you plug in a number and you get 0 in the bottom it's going to be some kind of ∞.
The problem is you always have to check the left and the right side for these. If you check the left side you do the lim x->2 from the minus side you get 8/x-2. Okay well, (x-2)^2 sorry. You'll get 8/0 this will be positive so you get positive ∞. |
| 40:04 | 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 ∞.
Therefore this limit is ∞. You check the left side and the right side. They both get you positive ∞ so the limit is ∞. And it's always positive because 8 is always positive and (x-2)^2 is always positive. |
| 40:34 | If you look at the bottom one, however.
When you cube a negative number you get a negative. So this will approach negative infinity. Okay? Because if x is just a little bit less than 2 at 1.99, -2 is a negative number. When you cube it it's still a negative so this comes out negative infinity. |
| 41:03 | On the other hand,
if you plug in a number greater than 2, x-2 is positive so this will come out positive infinity.
Because they don't match the limit does not exist. |
| 41:30 | How do we feel about those?
Let's see what other limits I could give you. Oh yes one other kind. So far so good? Well you sort of are plugging in 2, when you plug in 2 you get 0. Okay so remember 8/0 is nonsense. So 8 over a very tiny number is a very big number. Because you have a fraction when you plug the fraction in you get a very large number. |
| 42:03 | So if you have 8/1/1,000,000,000,000 it becomes 8 trillion.
Alright one other type of limit. |
| 43:00 | Okay how about this one.
|
| 44:47 | Okay so if you want to do the lim x->2 for this one.
The lim x->2 from the minus side, that's a number less than 2. You use the top branch. |
| 45:01 | That would be 16-3=13.
And if you go the lim x->2 from the other side you use the bottom piece which is 7*2 is 14-1 which is also 13. So therefore the limit is 13. Okay? |
| 45:32 | What if I took this away?
Would that effect the problem? Why does it not matter? Because you don't care if it's equal to 2. 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. It would affect continuity but I didn't ask about continuity. It's not on the test. 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. |