| Start | Okay so test is over. The TAs are grading so I told them to have the grading done today.
I'm sorry I don't have the microphone on. Whether they actually have the grading done today remains to be seen but they're pretty reliable. So we should have everybody's scores by the end of the week at worst. Remember how long you had to wait for your SAT results? Am I on now? No? Alright so I guess I'll just talk louder so the microphone can get me. |
| 0:31 | Anyway as I said we should have the grades done by no later than today or tomorrow.
There's only about 100 of you. It doesn't take as long as my other class, etc. Anybody have any questions? The exam wasn't so bad I hope. Maybe you were worried beforehand and then you weren't worried afterwards. That's a good sign. I'm sure not everybody did perfectly. |
| 1:03 | But we'll see. I have no idea what or if the curve will be. There may not be a curve.
Generally we don't do anything about establishing what the grades are till we get to the end. Alright do at the end of the course that's when I determine was it an A, was it a B, and so on. Not now. Because if we do it now then you guys tend to misinterpret some of the information and you do your version of what the grade should be |
| 1:31 | ...
because then you disagree with me and you're fighting with me and you feel bad when I say too bad that's your grade
and you go "wow I didn't know that guy could be a jerk just because I do the calculating my way."
It's not a power thing it's just that's how the grades are calculated.
So I'd rather not have that happen and not tell you what your letter grade is just yet. Because as I said if I tell you this is a B and something else is a B and when it's all done and you get a B- and don't understand why, |
| 2:01 | that's the way the class came out. Sometimes there's no curve.
So take a few deep breaths. Many of you are doing well. I walked around the room I saw how people were doing. Okay? It's nice to get so many people here. I figured we'd have about 12. It's Columbus Day on Monday. We have class. We used to get those days off but they stopped that a few years ago. They decided that we were enjoying ourselves too much. That's not the real reason. Alright I'm goin to go over this. |
| 2:31 | Just so you can see how you did. Did you guys all go to recitation?
Good. Many none of you will speak. Okay so we'll start it off with one of these domain things which we'll certainly practice. So this was the question on the exam. Find the domain. Okay? We did I don't know ten of these? You shouldn't be surprised. Well how do you find the domain? |
| 3:01 | There's two factors you have to take into account.
They're assuming I'm not going to be doing anything else like this. Each time I turn around I'll look back and they will be fewer of you. Reminds me of my dates. Okay do domain you have two things to think about. You have the square root and then you have the denominator. The denominator cannot be 0. So x ≠ 2. |
| 3:31 | And the radical, you can't take the square root of a negative number so that has to be ≥ 0.
So x has to be ≥ -1/3. As we've emphasized you don't need to do interval notation so that's all that was involved. Easy? Not too much stress? I think this was 3 points and this was 2 points so a total of 5. Then we did another one where we just turned it upside down. |
| 4:03 | And if you're clever you say well if he turned it upside down it can't have the same answer.
Because if it comes out the same answer then what's the point? Unless we're trying to trick you. Always something fun to do. So now once again we have the radical so 3x+1 but now its not ≥ 0 it's just > 0. And the reason is you can't have 0 in the denominator. |
| 4:32 | So x has to be > -1/3.
As far as the numerator is concerned you can have 0 in the numerator so you stop. So if you put x ≠ 2- it's okay, that's why I leave the door open. Honestly it makes less noise when they go in and out. So if you put x ≠ 2 you lose a couple points. |
| 5:00 | Okay? Because you have to know the difference. That's really what we're doing.
W'ere not doing more complicated domain stuff because the only other functions we're going to deal with in this course are where there's domain restrictions and ? Those exponentials are unrestricted. We're getting into good stuff on Monday so brace yourselves. So far it's been pretty easy pre-calc review. Now we're going to get you into calculus and intro calculus. Which means some of you will not be doing stuff you learned in high school. |
| 5:35 | Alright the second question we said f(x) = 2x^2 -4x + 5.
Find f(x+h) - f(x) / h. You've seen- again we've done several of these. Shouldn't be a shock. We know if we do the derivative it's going to come out 4x-4. |
| 6:04 | So this has gotta come out something like 4x-4.
So let's do f(x+h). That would be [2(x+h)^2 + 4(x+h) +5]. - (2x^2 - 4x + 5). |
| 6:37 | Okay now we just have to simplify that. Notice I didn't ask for the limit.
Don't you wonder why I didn't ask for the limit? Any of you wonder about that? Did some of you take the limit anyway? Because it's reflexive and you just did? We decided not toe punish you. We said we'll take it either way. |
| 7:00 | But this is 2(x^2 +2xh +h^2) -4x -4h +5 -2x^2 +4x -5 all over h.
So far so good? Okay distribute the 2. |
| 7:34 | 2x^2 +4xh +2h^2 -4x -4h +5 -2x^2 +4x -5 all over h.
Somebody was writing pressing really hard on the board. |
| 8:03 | Okay, plan B.
Whoever was writing on that board had anger issues. Alright so this now. Cancel the two x^2's, cancel the 4x's, cancel the 5's. And you get 4xh +2h^2 -4h all over h. |
| 8:33 | Factor out an h.
And divide it by h. That's an h. Cancel and you get 4x +2h -4. If you took the limit you get 4x-4 which is the derivative. So that's one way you can know you were doing it correctly. |
| 9:02 | So some of you will get lost in the algebra on this problem. So this was a 10 point problem.
Okay. You got a lot of points just for the first line. I think you got 4 points just for filling it in at the top. Okay any questions so far? Good you guys are silent. So I assume silent means good. See what you really do is afterwards you come up to me and you talk to me which is |
| 9:35 | you could've just raised you hand in class. But that's okay.
A car is traveling along a straight road and it's distance is changing linearly with respect to time. Mathematical model. A linear model. Semi-realistic. It's hard to keep your car's speed actually constant. I mean, cruise control, but other than that it's hard to do. |
| 10:05 | The time t= 10 minutes to travel four miles.
So you're not going very fast. And time 34 minutes it's traveled 16 miles. So this car is not in a rush. And you want to find the equation for a line. So we've got slope is 16-4/34-10. |
| 10:32 | Which is 12/24 which is 1/2. Some of you got it upside down and you got 2.
And if you want an equation you say y-4 = 1/2(x-10). And you can leave it like that. Or we actually asked to leave it in terms of t. But we said we would give you credit either way. |
| 11:11 | Alright? And occasionally it was- some of you did it backwards. You did t in terms of b.
|
| 11:38 | Yes you should try to get the letters right but it really doesn't matter.
Okay? There are teachers who take off for that. Remember when you were in the 7th grade and you had the teacher who took off for that. That's when you decided you hated math. And you came home and your mom said you didn't study hard enough and you said yes I did. Then how come you didn't get an A and you say my teacher sucks and she says stop always blaming your teacher. So now you gave up, you've got a fight with your mom. And then a few years later people ask about math and you say I hate math. |
| 12:04 | And you're justified. That's on video so we'll see how the math department feels about that.
That's a lot of you, your math teacher punishing you for stuff that's stupid and not rewarding you for the fact that you work hard and you know what you're doing. I'm going to do the same...OK? How many of you had one of those types of teachers? Good so you know what I'm talking about. Right, isn't that great? I think it's usually about 7th or 8th grade. |
| 12:30 | That's usually the decisive moment.
Alright so where's the car at the time t=48? So plug in 48. d-4 = 1/2(38). So d-4=19, d=23 miles. |
| 13:03 | How are people feeling so far about this?
Good? Sinking feeling in your stomach right there? Not sure because it was so long ago that you've forgotten what you wrote down. |
| 13:30 | Alright let's do the next one. Which was a function problem.
These are fun. |
| 14:05 | Okay f(x)= x^3 +12 /4 and g(x)= 8/x-1.
We said what's f(g(5))? I don't know why we do this stuff either but what is f(g(5))? Let's see so first we need g(5). So g(5)= 8/5-1 which is 8/4 which is 2. |
| 14:39 | And then f(2)
is 2^3 +12 /4 which is 8+12/4 which brings you back to 5. Does that mean those two functions are inverses?
Not a chance. That's just an accident. |
| 15:02 | If that's always true then yes they're inverses.
Now we go the other way. We say what's g(f(0))? So f(0)= 0^3 +12 /4 which equals 3. |
| 15:30 | And then g(3)
is 8/3-1 which is 8/2 which is 4.
I think we've lost another one. |
| 16:08 | Alright on to the next one, limits.
f(x)= 8x^3 -5x^2 +6 / 2x^3 -7. There isn't a lot of work for this. |
| 16:40 | Oh I'm sorry you had to do the lim x-> ∞ of f(x).
So we're doing the limit to infinity when you have these ratios you look at the degree of the top which is x^3, you look at the degree of the bottom which is x^3. Since they're the same the limit's going to be 8/2 which is 4. |
| 17:05 | There's not really much involved in this so there wasn't really any partial credit you either got it right or you didn't.
This was worth half the exam. And then we had- I think it was worth 5 points. But one of the professors had one of these exams, they had 5 limits all or nothing no partial credit. And they were 5 points each so if you didn't do one of the limits that's it you were down to like a C. |
| 17:34 | Which is kind of rude but he didn't ask me.
Okay now find the limit as x-> 4 plus. 4 from the right side. Okay so you plug in 4 on the bottom you're going to get 0. So this is going to go towards infinity. You plug 4 in on top you just get a number so this is going to be positive infinity. |
| 18:05 | Okay?
Because 0, again there wasn't really any partial credit on this one either. So you kind of needed to get these right. I forget how many points the exam is worth. It wasn't a 100 point exam. 85. |
| 18:30 | Okay then we had f(x)= x^2 -2x -15 / x^2 -25.
Find the limit as x approaches 5. Don't confuse this with the type where you go to infinity. The limit for infinity of this is 1. But this isn't to infinity this is to x=5. Correct? Goes to 5. |
| 19:00 | If you plug 5 in the top you get 0 if you plug 5 in the bottom you get 0.
If that's confusing you say what if I factor this? Okay now we see the problem. When I plug in 5 I'm getting 0 in the top and 0 in the bottom because of these terms. |
| 19:32 | Is there a save because I could reduce this equation to this because x is not actually 5.
If x is 5 you're not allowed to do that. Now when I take the limit as x goes to 5 thats going to equal 5+3/5+5 which equals 8/10. Or 4/5. To anticipate, no you did not need to reduce the fraction from 8/10 to 4/5. So this you got partial credit. You got credit for factoring and you got credit for the right answer. |
| 20:07 | Okay everybody feeling good right now?
Some of you feel better than others. Okay then we had another fun one. A piece-wise function. |
| 20:44 | f(x)= 3x^2 -5x +2 when x < 2.
And x^2 +6x -10 when x > 2. |
| 21:09 | And we want to find the limit as x approaches 2.
Naturally. What else would we ask for? So if I plug 2 into the- wait it doesn't exist at 2. |
| 21:31 | It's only got greater than or less than.
Does that matter? Nope. It doesn't matter because we don't care about 2. We're doing the limit. So the limit when x approaches 2 from the minus side of f(x)= 3(4) -5(2) +2. Which is 4. |
| 22:12 | Much better.
And then the limit as x approaches 2 from the plus side is 2^2 +6(2) -10. |
| 22:31 | Which is 6.
These two do not agree. Since they don't agree the limit as x approaches 2 does not exist. I figured in honor of October 3rd I should have at least one limit that does not exist. Can expect at least one of those. Good. So far so good? Still happy? Still think you're going to get through college? |
| 23:04 | You'll get in somewhere. Maybe even Stony Brook.
Alright find some derivatives. So our first one is nice and easy and then I throw in that annoying radical. I had to throw a little plot twist. |
| 23:44 | So dy/dx is, bring the 3 down you get 24x^2, bring the 2 in front you get 8x, minus 7.
I didn't put in a pi or anything annoying like that. |
| 24:03 | Okay and then we had a y= 2√ x - 4 cube√ x.
This tends to send people into a panic. Okay well √ x, square root of x is just x^1/2. And cube root of x is just x^1/3. The way you do these is just convert to a fractional power. |
| 24:46 | Okay so dy/dx you bring the 2 times 1/2 you just get 1 so you get x^1-/2 -4/3x^ -2/3.
|
| 25:01 | You could leave it like that or you could make this 1/√ x -4/3 cube√ x ^2.
You don't get extra points for converting it. You're just showing you can. Okay? Then we had more. |
| 25:35 | Did any of you use that study plan from mymathlab?
One? Oh two, three, four, good. Did you find it useful? It was kind of confusing okay not that helpful. Because we're evaluating the program so if you guys don't think thats useful that's good to know. |
| 26:04 | Student: It was hard to know which parts of the material were going to be the main focus for the exam.
Okay good to know. Thank you. y= 9/x - x/9. Well these were, I had homework problems that were practically identical to these. So dy/dx- so again rewrite this. |
| 26:30 | This is 9x^-1 -1/9x.
This tends to bother people. I don't know why. You have to know that if you take x and divide it by a number it's like multiplying it by a fraction. But okay. So dy/dx= -9x^-2 -1/9. |
| 27:01 | Because 1/9 times x well 1/9 is just a constant.
And again you could leave it like that. Or you could make it -9/x^2 -1/9 because some of you are going to go directly from here to here. And then my favorite, the equation of a tangent line. |
| 27:47 | Find the equation of a tangent line to y= x^3 -2x^2 +1 at x=3.
|
| 28:04 | Okay so dy/dx, going to need that.
And we're gonna need y. Well, find dy/dx first. That's 3x^2 -4x. So dy/dx at x=3 becomes 3(3^2 ) -4(3) which is 15. |
| 28:37 | And at x=3 y= 3^3 -2(3^2) +1.
Which is 27- 12- 1 which is 16. No it's not. That's not 12. |
| 29:04 | That's 18.
It's 10. So your equation is y-10= 15(x-3). Which you leave like that. You don't have to make it into mx+b form. How do we feel about that one? Not so bad? You guys like these exam questions? |
| 29:32 | Well if you changed it to y=mx+b form I hope you got it right.
But we'll look for this. Okay? If you had y= 15x-35 that's close to fine. You sure it's not 35? Some people were yelling 45 but they forgot I moved the 10 to the other side. Did that all in one step. Master. |
| 30:03 | I can't believe he can do two things at once. Neither can I.
Anyway How'd we feel about this? So this wasn't that hard of an exam was it? Did we all feel relatively well prepared? On top of things? Alright so we have some new stuff to do but it's Friday afternoon, well almost afternoon so we'll see you all on Monday. |