| Start | 20 minutes of Maximum minimum problems and we're going to review for what's going to be on the exams starting next week Okay, I'm just gonna give you guys a bunch of problems if you can do those then in youre fine So after consultation with professor Sutherland we reduce the exam from 15 questions to 12 Okay, you need to get nine to pass. You all can do that Okay, nine out of 12 you get two tries if its a problem after the second try, then your last try is the final exam |
| 0:33 | I expect all of you who were paying attention to do just fine at this it's okay Even some of you who were not paying attention. maybe you can text your mother and she can keep the answers to the test You won't be able to bring youre cellphone over to the testing center, I know Something might happen while you're out you will know about it Okay We're supposed to get a few storms. I guess since were underground will be okay They're suppose to be lightning hail and all that stuff starting at about the next half hour thanks |
| 1:31 | Let's see other stuff, I've scheduled my final exam review
the final exam is on May 13th. That's a Wednesday is it 8:00 in the morning. Did I say 8 oclock at night?
No, one of you will show up at 8 oclock at night. Then I get an email it says. Oh wow I thought the final was at 8 oclock at night you fail You fail That's not a good move. You get to take the final the next time math 125 is offered |
| 2:00 | And it wont be my final, it will be someone else final. So I strongly suggest that you not confuse 8 a.m.
And 8 p.m.. I Plan for 8 a.m. Thats what they gave us But anyway my review session will be that Monday from 12 to 2 in javitis, i forgot what room I'll be posting it ok so I have a nice review session before the final plus week before the final is review We'll do practice problems all week Yeah, good. You should go through your webassigns |
| 2:33 | You know practice problems All the stuff that you hate one more round of it As I said the other day because I know how many of you were focused if you go to our page And then where is it. that wasn't a page it was an assess |
| 3:13 | Yeah, asset.math.sunysbu.edu
this is the page i showed you guys a couple days ago. and then you go to practice problems
Differential calculus, that's what we're doing differential derivatives, okay?
|
| 3:30 | And then you can do practice problems Okay, so that'll be another chance to practice this stuff so again assess.math.sunysbu.edu Click on practice problems you click on differential calculus all good Alright, I'll leave that up there for a couple mins |
| 4:06 | you ready to practice? so much fun that whenever you ready, or have you started already you did. What did you start?
a couple mins ago. Is it up okay. You did get any of the bad stuff in there no why would there be bad stuff This is Stony brook, this is math. What could it possibly be? fun exactly |
| 5:30 | All right a
rectangle is to be inscribed note use of passive place
Rectangle is to be inscribed
Under the parabola y equals 16 minus x squared with its base on the x axis
Maximize the area of the rectangle, so what does that mean?
I have a parabula that is a non parabolish parabola |
| 6:02 | Okay, you're gonna stick a rectangle under the parabola through the base on the x axis
Like that
Okay
I'm gonna figure out. What's the biggest rectangle area? What's the biggest rectangle area you can make?
I'll give you one other clue this Is x2 trying to maximize that area |
| 6:32 | Let's see if you do it for those you realize That's a silly number because there's all those people who don't check it every hour That's so we dont have people checking it every min Except for those people whoe never get off your phone so how can we figure out x So this is also x so the base of this rectangle is 2 X the height of this rectangle |
| 7:01 | is y this is the y-coordinate to find the x height right you find
the height of the rectangle by going out X and down Y so the area will be 2x
times y
does That make sense
2x times y okay and why is 16 minus x squared?
|
| 7:31 | So therefore the area Is 32 X minus 2 X cubed That's not very hard nope now you take the derivative eight prime 32 minus 6 x squared set it equal to zero Welcome to calculus class, you solve this and you get 32 six x squared |
| 8:04 | 32 over six which is 16 over three is x squared so X is plus minus 4 over the square root of 3 So either you go for over the square three. or you go minus 4 over the square root of 3. that doesnt make sense so this has to be the minimum But you can double check so the easiest way to double check when you have a simple first derivative is take the second derivative |
| 8:33 | And run the second derivative test To remind you the second derivative test You take those critical numbers And you plug it in further into the second derivative if you get a negative that means your curve is concave down so you're looking at a maximum, so if you find a double Prime at 4 over the square root of 3 that would be negative 12 times 4 over the squared root of 3 we don't care what that number is This is negative so that's the maximum |
| 9:02 | and if you do it at the critical value You get a positive number That's greater than 0 that's the minimum Remember it's less than 0 that means your curve is concave down so you're looking at the maximum Here it's positive so its concave up. okay. That's the second derivative test you can certainly do the number line and nice thing about doing the second derivative test is it's such an easy second derivative |
| 9:34 | Sometimes your do problems and the second derivative will always be negative or always be positive then its even easier, okay got the idea, yeah, let's do another one |
| 11:05 | All righta soda can, a closed soda can, a closed Can and it's cylindrical Is supposed to ahold 144 cubic centimeters of soda the top cost four cents per square centimeter the sides and bottom cost two cents per square centimeter What radius |
| 11:31 | minimizes the cost of the can so to help you set up you draw The cylindrical soda can That can be red bull which ever you prefer okay, and as a radius the can has a height You know from the volume of the can |
| 12:02 | I know you guys love geometry, so just in case you forgotten surface area would be 2 pi r h + 2 pi r squared, the 2 pi RH is the 2 PI R squared the top Pi r squared of the bottom. the volume Is PI R squared H |
| 12:31 | With that information you should be able to solve this problem
So let's see how would you figure out what the cost is, well the cost is going to be?
2 pi. RH. that's how much material we need for these sides and 2 cents The bottom Also cost me two cents and the top cost me four cents |
| 13:01 | or to simplify that 4 cents per square centimeter So that's the surface area of the side times two cents, the surface area of the bottom times two cents Surface area of the top times four cents. why would the top of soda can cost more than one That's the thing at the top. the pening, yea it cost money this is 4 pi r h plus six pi r squred Okay |
| 13:30 | That's the cost. Okay, great but we have two variables, but we can now use this information to get rid of the other variables
sence volume is PI R squared H. And that has to equal
144 or how we say in brooklyn
144 right
We know that H
is
144 over pi r squared
Okay, so you can plug that in for H, do a little simplifying, take the derivative set it equal to zero and off to med school we go.
|
| 14:04 | So let's see four PI R times 144 over PI R squared plus 6 PI R squared four times 144 is 576 the Pi cancels and the r cancels. so you get 576 over R plus 6 PI R squared. that's actually a very simple derivative |
| 14:41 | So C Prime, the derivative of 576 over R is the - 576 over R squared because remember I told you guys the derivative of 1 over R or 1 over anything. 1/x is Negative 1 over x squared. you wrote that down several times |
| 15:04 | And of course you do the hard way. and the derivative of 6 PI R squared is 12 PI R and
You set that equal to 0
did i lose you?
Whered i lose you guys? So lets go through this again Okay, we know that the volume is PI R squared H. Which equals 144 |
| 15:31 | H is 144 over PI R squared We also know the surface area so the surface area is involved in the cost to The tub, the cylinder part Is 2 pi r h and it cost two cents per square Centimeter. the bottom is pi R Squared, 2 cents a square centimeter and the top is pi r squared times 4 cents a square centimeter So you can simplify that the 4 pi RH plus 6 PI R squared |
| 16:03 | Now I know that H is 144 over pi r squared. i put that in here for H And I get this Okay 4 pi r times 144 over PI R squared plus 6 PI R squared. now I could do some canceling the PI's cancel One of the R cancels and you get 4 times 144, which is 576. you get R in the denominator Plus 6 PI R squared it my beautiful handwriting. 6 PI R squared |
| 16:37 | Ok now to take the derivative of 576 over R That's 576 times R to the negative 1, so the derivative of that is minus 576 r to the negative 2 plus the derivative of 6 pi r squared is 6 pi times 2r |
| 17:01 | So one way to solve it is put the 576 over R squared to the side And cross multiply. you get 12 PI R cubed is 576 Okay divided by 12 and you get R PI R cubed |
| 17:34 | pi r cubed
equal 48 so R cubed is
48 over
pi
So R is the cubed root of 48 over pi.
I Want that to be a minimum, Im trying to minimize the cost So I could |
| 18:00 | Sign test it, make a little number line and plug in values. or i can take the second derivative Either way you'll find, thats a minimum Okay Yes I i have pi R squared H is 144 Okay So much fun not so much fun. should i give you another one |
| 18:32 | I'll give you the one I gave up my final last semester. Well. It's very similar to this |
| 20:07 | Okay a coffin is in the shape of a rectangular box The cost of the top, the part that opens is six dollars per square foot The cost of the bottom is $2 square foot because who looks at the bottom partner okay. the sides are four dollars a square foot iif the volume is 96 cubic feet, minimize |
| 20:31 | The cost. very similar to this problem, except now you do it with a box instead of the cylinder
Okay, so you should be able to do it. Let's say
Okay now we have to figure out what this cost for the top and the dimensions x time y?
and it cost six bucks per square foot. because it is made out of good stuff |
| 21:01 | then the bottom also has the area XY It only cost two dollars, because the bottom doesn't have to open That's very amusing when it does but you try not to have that open Okay, then anything else is four dollars so two of the sides are x squared And to the sides are x,y So to simplify that |
| 21:31 | You get twelve XY plus 4x squared Okay That makes sense so again the top is XY six bucks the bottom is XY 2 bucks the side, well two of the sides are x squared, the ends. the two sides are xy They cost four dollars each Just wait wait wait its two sides |
| 22:05 | plus 4 times 2xy
So that's eight and eight so
That's 24, that's why okay?
2x squared sides at four bucks a piece and two XY sides at 4 bucks a piece, so you get 6 XY 8 XY 8 is 16 xy and |
| 22:31 | and 8x squared You know Thursday, get a little mellow So y 96 over x squared 96 over x squared Now we plug in and get rid of y, and you get the cost is 16x |
| 23:03 | times 96 over x squared plus 8x squared 16 times 96 is 1536 get that Someday, I'll show you guys a trick on how I do that plus 8x squared |
| 23:30 | should we take the derivative of that So C prime is minus 1536 over x squared plus 16X set it Equal to zero okay solve. so there's two different ways to solve this but I'll do the same as last time put the 1536 over x squared on the other side |
| 24:01 | Cross multiply you get 16x cubed is 1536 we divide 15 36 by 16 and you get 96 So x is the cubed root of 96, cube root of 96 is 2 cubed root of 12, who cares |
| 24:32 | x is
cube root of 96
So far so good. How did you do on that one?
Ok and to actually find the cost you Have to take the cube root of 96 and plug it Into the cost equation, but we just has dimension that minimize the cost ok But that's the general idea. Did you set that one up? How'd do we do on this one better? We just waiting for me again |
| 25:04 | Ok that's not gonna be effective strategy for the final. Just so you know works well up until that point All right, let's do some exam practice so one more time starting Monday go during your recitation to fray hall 109 don't be late, okay You have 53 minutes to take the exam it is form of a WebAssign. It is twelve questions |
| 25:33 | You must get nine rights to pass, if you get something wrong you get a half a point
If you get a second try at it, okay?
Try to take your time you can use scrap paper try not to screw it up. Okay If you cannot for some reason go during your recitation. I strongly suggest to avoid that problem, but if you have a legitimate Conflict send an email to professor Sutherland and he will help you figure out a different time to take it, okay? |
| 26:01 | Remember there are many recitation times. You'll just end up taking with a different recitation. Dont do that without checking with us Okay, we dont want to accuse you with dishonesty. That would be bad, and then no med school Okay, so lets practice some problems stuff you should be able to do |
| 27:55 | Should we go through our limits the first one you plug in four in the top, and you get 0, you plug it four in the bottom and you get 0 |
| 28:05 | the limit does not exist but wait it does what do you do you factor the top and get X minus 2 times X minus 4 and The bottom is X minus 4 and you cancel now you plug in 4 and you get 2 One down a hundred to go. the second one |
| 28:32 | this is the limit as x goes to infinity
remember what you do with these? you take the power of the top
and you look at the power of the bottom
rule is
Power at the top, power the bottom, the highest power the top
highest power the bottom are the same
then the limit is just the coefficient of the highest term on top
Divided by the coefficient of the highest term on the bottom
its jut 3/7
What would you do if the bottom at a higher power?
|
| 29:00 | say you had 3x to the fifth plus x squared plus one over 7x to the sixth minus 8x now able to pick up power in the bottom you get zero And if you have a bigger power on top you get infinity Okay, so the rule is bigger power in the bottom you get zero Bigger power on the top you get infinity The highest power in the top and bottom is the same It's just the coefficient of the 1 over the coefficient of the other. so far so good get a little of that rust off |
| 29:35 | Okay, remember the rules for the signs Don't use the squeeze theroem. That's what this is infinity This just equals five So what's the rule, rule is if the limit as X goes to 0 sine x over X That equals 1 Okay, now if you have the limit as X goes to zero of |
| 30:03 | Sine
b x over
X
What you would do is you would multiply top and bottom by b?
This would be become limit X goes to zero b. Sine b x bX this is now b x and bx is the same as saying X and X so this would be one so this comes out b. so lets erase that |
| 30:30 | Don't want to run into the other problem This equals B Okay, and if you have the limit as X goes to zero sine bX over sine ax is B over A Now all the rules for sine are also true for tangent, so this one is 3/4 |
| 31:04 | Really we did some of these it was a Long time ago we get her taken away there All right, okay now lets do some derivatives |
| 31:43 | find the derivative of |
| 32:29 | there will be problems where we say find f |
| 32:32 | Prime at five for example
You do not have to simplify that you could write in WebAssign five squared plus six times 5 plus ten
Parenthesis to the fifth in WebAssign do that for you, but if you do that don't screw it up, okay?
But you do not need to supply and you plug in numbers You can just leave them and plug in. you do not need to reduce from there but be careful, you don't mess it up |
| 33:00 | that's what will cost you a half point if you do. okay so you have plenty of time on these problems I alright lets start going through them Okay the derivative of this f prime of x We use the chain rule five Times this thing to the 4x, x squared plus 6x plus 10 raised to the fourth times the derivative of what's inside |
| 33:31 | Which is 2x plus 6y Second one you could multiply that out. It's not that bad to multiply out, but we could use the product rule Yes, oh forgot to write the fourth sorry. okay f prime of X. that would cost me half a point on Webassign You do the first times the derivative of the second plus the second times the derivative of the first. so the first |
| 34:07 | Times the derivative of the second, three x squared minus five Plus the second times the derivative of the first X cubed minus 5x Times 6x plus two and you're done No simplifying, leave it alone Okay, now we could say what's f prime of 1 well, then you would plug in one |
| 34:31 | And you get 3 plus 2 plus 1 is 6 Three minus five. but you can leave that three plus two plus one. Okay? Webassign is smart enough to be able to do that What's the third one well f prime of x, this is sine 2x plus five the whole thing cubed so this is three times the sine 2x plus 5 Squared |
| 35:00 | Which you can also write as sine squared
Cosine of two x plus five
Times two
So far so good
Passing? so far all right. How do you do the derivative of the log?
So easy you write a fraction bar, put the function of the bottom put the derivative on top |
| 35:31 | We hope they're all like this
Okay, this is what the test is going to be like stuff like this. Okay? except much much much harder. Yeah
Okay
I can't give you the exact questions
This is what the test is going to look like because if you can't do this you're not ready for 126
You should not pass. if you can do this you demonstrated minimum confidence. That's what you're trying to happen to do okay?
This isn't the beat you up part |
| 36:01 | the derivative of f prime of x is x to the tanx times Secant squared X Okay, let's do more practice Well some of you were feeling better |
| 39:23 | We want to find what interval is y increasing? so you take the derivative you Get 6x squared minus 30x minus 36 and set it equal to zero |
| 39:38 | Divided by six, and you get x squared minus 5x minus 6 equals zero Fortunately that factors you get X minus 6x, x plus 1 equals zero |
| 40:01 | So we have minus one and we have 6
and you do a little sine testing
Guess I should say intervals
So this is increasing
Minus infinity to minus 1 and it 6 to infinity
you get that one?
Yes, what are the coordinates of the maximum kind of same question, kind of |
| 40:31 | Yes
I'm sorry
in what intervals is why increasing? That's what that means right the function is increasin, okay?
You don't have to find the y values. Okay, you just use the x value What is the X coordinate of the maximum, take the derivative and you get it 3x squared minus 6x minus 9 and set it equal to 0 divided by 3 |
| 41:06 | And factor Then you can either use the second derivative test or you can sign test it so if you sign that And you'll find that the maximum at negative 1 x equals negative 1. ok x equals negative 1 |
| 41:34 | howd we do on that one Passing so far so good whats the x coordinate of the point inflection, take the derivative 3x squared minus 48 X Plus 8. That's the first derivative take the derivative again 6x minus 48 set equal to 0 you get x equals 8 |
| 42:04 | That is another right answer
Okay last one find dy/dx at x equals 1, y equals 1. We're gonna have to take
the derivative implicitly and plug in
This will be online don't worry so 3x squared minus 2y, dy/dx
Plus 3y squared is dy DX?
|
| 42:33 | Plus 4 equals 0. plug in 1 and you get three minus 2 dy/dx plus 3 dy/dx Plus 4 equal 0 that simplifies to dy/dx Plus 7 equals 0 dy/DX Equals negative 7 |
| 43:01 | Okay, that was a lot of work. That's the kind of stuff you should be able to do. see on Monday |