| Start | So now we're going to do do today and Wednesday, we're going to maximization to the perception how to use a maximums minimums and concavity to draw graphs Next week we're going to be doing word problems using the same tools And then it's not much to do after that there's a couple of small topics and then we get the exam |
| 0:31 | Remember when you have a curve you know a couple things. let's just draw a random curve So we know a couple things we did this a little bit on Wednesday. look at the the curb and you say well, I can see theres a maximumish here and Minimum there, i a said ish that's sort of a maximum because in this zone its the high value but its not as high as there Okay, there. That's called a relative maximum |
| 1:07 | Sometimes it's called a local maximum. It's the same thing okay, so it's a hill in the road This would be and absolute maximum And also sometimes called a global maximum |
| 1:31 | So in general is no real difference both places Your flat your tangent line becomes flat at the top so right there the derivative is zero and Before you get there your curve is going up and afterwards the curve is going down this is increasing 0 is decreasing Increasing zero decrease ok so that's how you can tell something's a maximum, you look at the derivative and you say this is zero right here Right there, the derivative |
| 2:04 | is 0 and same here to the left of the derivative is going up, the function will be positive and the right It's going down further to be negative same here the difference is just the y value, this just has a bigger y value than this so that's why just Absolute maximum, and this is a local max in this spot the derivative is also 0 Right you have a zero here as well because the tangent line is flat however that's a minimum |
| 2:36 | That's a local minimum, or a relative minimum
why is it not the absolute min, well you're going down for it's negative infinity on either side
Okay, so that's not the lowest value that the function ever gets puts the lowest value in this zone
Locality. That's why it's a local min, and how can you tell its a minimum notice the derivative is going down, negative.
|
| 3:03 | then zero then it goes up. so you thought about what's happening on the number line Okay, let's call that point a point B and point C. A B and then C The derivatives are at zero Just to the left of point a The derivative is positive to the right of point a is negative so that's how we know |
| 3:35 | With a maximum this process of the curve is increasing and that tells us the curve is decreasing so increasing zero decreasing maximum Now you look here this is a minimum Curves going down and then it's going to up because we go down into the minimum and then up at the minimum so the derivative is negative |
| 4:01 | So that's how we know its a minimum
now we go to c. you say well once again the curve is going u. Its positive, then is 0, its going down.
One thing were gonna do is were going to take the function and we're going to do a sign test. Sign S I G N test. That will help us figure out where things are maximum and minimums or maxima and minima |
| 4:34 | Then in a couple minutes we'll look at concavity, so let's just take a typical Nice simple equation |
| 5:04 | So I
Give you this graph were gonna learn how to sketch the graph, so you guys can sketch that anyways you went to high school
But just in case
What's the rule number one this is calculus class, so you don't know what to do. What do you do?
take the derivative take the derivative and you get 6x squared plus 24x Rule number two after that what do you do with the derivative? |
| 5:31 | set it equal to zero okay
So this is what youre gonna do to find maximums and minimums and what we call critical numbers you set the derivative
equal to zero
So you can pull six x out?
You get X minus 4 equals zero, so that's at x equals zero and x equals four |
| 6:13 | So now what do we do? we found, these are called critical numbers critical values different books have different names for it
Why are they critical well
theres a zero there, so we have to look at them in bits that are important, okay?
|
| 6:31 | let's make a number line and
See what does the derivative look like on either side of those values?
So you pick a number less than zero so what's a nice number less than zero? how bout negative 1 when you test negative 1 if you plug negative 1 into the derivative, you plug it in here is nice easy to get negative 6 times negative 5 is 30 that's positive |
| 7:02 | That says all of the values of derivative to the left of zero will come out positive. The curve is going up so it gets to zero Now I'm gonna try a number between 0 & 4 1 is a good. you could do 2 or 3 or 1 point something or pi, depending on how statistic you are You try a number like 1 |
| 7:30 | you plug in the derivative and you get 6 times negative 3 Is negative 18 thats negative. It's going down there. So you say AHA It's 0 here it goes from positive to negative so that must be a max Amazing, but wait there's more, now try a number bigger than 4 like 5 |
| 8:00 | Excellent choice
you plug 5 in here you get 30
5-4 is 1 thats a positive number
so the derivative is going up there. I mean the functions going up there
You say ah it's going down to the left 0 and then up to the right
So I must have a min
minimum x equals four
That's great, but what's the value? Now I know by my graph my graph?
|
| 8:30 | The function has a maximum at zero and a minimum at 4. I need to know the y value or i cant solve it
So I take my two critical numbers and plug it into the original
Equation that's how you find y coordinate that because this is y equals okay?
If you take the critical numbers you plug the derivative you're going to get zero Cause thats how you find them. youre trying to find y coordinates so when X is zero, let's see Y |
| 9:01 | zero
Zero zero eight so we have a point zero
comma 8
And y is 4
is 128
minus
192
Which is negative 48 plus 8 is negative 40?
Even if that's wrong, it sounds good Okay |
| 9:30 | That's not, right -52 and then I would take off a lot of points So we have a max here at 0 comma 8, a minimum here at 4 comma negative 52 |
| 10:01 | So our graph looks like that That wasn't so hard So you understand what we did take derivative you set it equal to 0 you put the value on the number line And you test to see whether the positive negative How do I know what the rest of the curve is doing well, I know it's increasing its zero, it's decreasing to zero and it's increasing Okay, lets try another one lets have you guys do one |
| 11:18 | So find critical numbers And figure out if there maximum or minimum, there's one of each Okay, so negative 4x cubed plus 6x squared plus |
| 11:31 | 24x minus 8 alright thats long enough So first take the derivative Dy/dx Negative 12x squared plus 12x plus 24 Hope you all got that. set it equal to zero and divide by negative 12 and get x squared minus X minus 2 equals 0 thank goodness it worked out |
| 12:08 | You know doing it in your heads a little little risky
but i pick easy numbers so I get x equals 2 then x equals negative 1 for those
My critical numbers so on webassign
Some of the problems say. What are the critical numbers, and there's actually more critical numbers than probable yet?
But those are critical numbers, so that's if you put it |
| 12:32 | Okay, now. Let's determine different maximums or minimums, so We've got negative 1 and 2 so you pick a number to the left and negative 1 like negative 2 And you plug it into the derivative, so notice the derivative is kind of complex here, its pretty easy when you have it in factor form So that's why I use that to plug in |
| 13:02 | if i take negative 2 and plug in. I get negative 4 times negative 1 is negative 1 is 4 the derivative is positive there so my curve is going up Now taken a number between negative 1 & 2 like zero Generally if you can use 0 as a test number sure, zeros very easy test you plug in 0 and get 24 so wait, sorry I |
| 13:32 | Can't hear you what?
Right so that should be negative, right? Yes I See you shouldn't have to use the factored form unless you have had a negative 12 in there. Yeah, now you can use the factored form Okay, you can edit that alright, so You put your negative 2 now it comes out negative now it's comes up positive |
| 14:01 | and itll come out negative. I got a clue for all of you when you have quadratics And theyre factorable these are going to open it it can either go negative positive negative or positive negative positive. I repeat when you have a quadratic With factors it will even be negative positive negative positive negative positive Okay, so the curve is going down and the curve is going up curve is going down so that's a minimum |
| 14:30 | And that's the maximum
All right now we're going of course the graph. we only have one more piece of important information. We need to find the y value
So when X is negative 1 Y is 4 plus 6 is 10
2 and negative 24 so negative 22
did i get that right?
Ok I heard one yes and no nos so im gonna go with that |
| 15:02 | And I got two yes, that's pretty good
now we'll try two
I get 32 yes?
Am I doing okay? I'm not that old yet alright so that means the other point is 2,32 All right a minimum at negative 1 comma negative 22 |
| 15:35 | Negative 1 negative 22, and I have a maximum here at 2 comma 32 Thats a maximum ish and miimum ish by the way I know the y-intercept is negative 8 okay that's the y-intercept So that could also help you graph it a bit and there you go |
| 16:00 | not drawn to scale as they say how'd you do on this one Okay, of course there's more stuff to learn So this by the way is what call the first derivative test That's used to determine if things are maxima or minima you run the first derivative test because we havent done anything the second derivative yet. Now im going to So the second derivative is |
| 16:32 | The derivative it it tells you whether the derivative is increasing with derivative decreasing So you have something it's going up It's going up Faster and faster which means the slopes are getting steeper. Okay, you have something that looks like this notice going up |
| 17:01 | Getting steeper but if something's going up and going like this then notice The derivative is getting less steep. It's flattening out so here the derivative of the derivative would be positive but here the derivative would be Negative so that's second derivative thats tells us about something called concavity Okay, so to write these down. A few rules to remember |
| 17:32 | if the first derivative greater than 0 you're increasing Less than zero your function is decreasing if your second derivative Is greater than zero or concave up and if your second derivative |
| 18:02 | Less than zero concave down or what we like to say second derivative is positive and You get this fun little faces like that I I didn't invent that But anyway, that's how you can tell the cocavity of course the second derivative can be 0, which means its neither concave up or concave down. thats called a point inflection |
| 18:37 | thats i dont know if we consider those critical numbers or not.
but when the second derivative is 0 would help you figure out to concavity, but Let's go back in this equation for second and take the second derivative you get negative 24x plus twelve Set that equal to zero and you get X is a half |
| 19:09 | So if we put that on the number line and we detected the values into the second derivative We get positive and then negative Just try a couple values so that tells us that at a half Remember the y coordinate, but somewhere where X is a half before that |
| 19:32 | It's concave up and after that It's concave down thats where the point of inflection Okay And you can find a y coordinate by plugging a half here and you get negative half 1, 11 you get three so, this is a half comma 3 a 1/2,3 that's your point of inflection |
| 20:01 | So concavity can also help you graph curves
Concavity is also pretty helpful cause you can also figure out whats a maxima or minima
Because what's a concavity of a maximum?
its concave down whats the concavity of the minimum is concave up So far so good a lot of people rather be doing anything else, right? Okay another handy thing |
| 20:30 | another little handy chart Okay you did first derivative the second derivative positive negative |
| 21:01 | So if you had the first derivative is positive The second derivative is positive, so what do you know? curve is going up and it's concave up is going to have this shape If the first derivative is positive but the second derivative negative, then the curve is going up, but the concavity is down That shape so your graph would look like that |
| 21:35 | Okay now if the first derivative is negative , the graph is going down. It's concave up, it would look like that And if your first derivative is negative and the second derivative is negative Then the curve is going down and the concavity is down it would look that, and this is four Basic shapes and you can see that. Youre going down, concave up |
| 22:05 | up and concave up up and concave down down and concave down got the idea Very useful, alright lets look at another curve |
| 22:55 | So suppose we have the equation y equals x to the fourth minus four x squared |
| 23:01 | plus one So let's find the max maximums and minimums and any points of inflection So let's see What we do> |
| 23:30 | well take the derivative dy/dx is four X cubed minus 8 X And then take the second derivative By the way all the numbers come out very messy when you start to do these |
| 24:01 | Takes a little work to make the numbers come out nicely and im lazy I take the first derivative and set it equal to 0 pull out the four X. get x squared minus 2 4X plus the square root of 2, x minus the square root of 2 |
| 24:33 | square root of 2 is not so bad lots of times we'll just ask for the x values Sometimes we wont, if we ask for the Y value with no calculator have to be relatively easy to find All right now, let's take a second derivative and set it equal to 0 |
| 25:00 | You get x squared is 8 over 12, which is 2/3 so X is plus, or minus?
equals the square root of 2/3, which is whatever And now we're gonna do some sign testing So Let's just look at the first derivative so the first derivative has three 0's |
| 25:32 | negative radical 2
0
and radical 2
Now let's check what happens to them?
the derivative So then pick a number less than the square root of 2, square root of 2 is somewhere between one and two right? Square 1 is 1 and square root of 4 is 2. negative 1.4. So how about negative 2 as a test number? You take negative 2 when you put it in you get negative 8 |
| 26:02 | times 4 minus 2 is 2 that's a negative number the graph is going down If you trying a number between negative radical 2 and 0 thats negative 1 You plug in and you get negative 4 times negative 3 is positive, so the graph is going up now you try a number between 0 & 2 like 1 you get 4 times negative 1. It's a negative, grph is going down again |
| 26:37 | And we try a number bigger than the square root of 2 like 2 you plug it in and You get 8 times 2 is positive, and the graph is going up. So that's how this we have a minimum maximum Min Down up down. In continuous curve you have to alternate you can't get to Max's in a row |
| 27:05 | Now we'll do the second derivative We've got negative square root of two thirds, positive square root of two thirds. You dont have to know the square root of two thirds, we dont care You try a number less than the negative square roots then negative square root of 2/3 like 1 Plugging into the second derivative you get a negative value the curve is concave down |
| 27:32 | Right then and then at zero Sorry no backwards, its conave up there And it's zero, it's concave down and at 1 it's concave up Okay So far so good |
| 28:01 | Now you should be able to graph this, except were gonna need y-coordinates So were gonna need y coordinates for negative square root of 2, so 0 square root of 2 So we take negative square root of 2, square root of 2 is 2 2 squared is 4 that's 4 minus |
| 28:32 | 8 minus 4 plus 1 is minus 3 and at 0 we get 1 so we had a minimum here minimum here Maximum there and We wouldnt ask you to do the square root stuff on the test, thats two hard. 4/9 |
| 29:03 | minus 4/9 is negative 29 plus 1 So we have a maximum at zero comma one we have a minimum at Negative square root 2 comma negative 3 you can check arithmetic later square root of 2 comma negative 3 |
| 29:37 | And that point is Is negative square root of 2/3 comma negative 11/9 at that point is positive square root of two thirds comma negative 11/9 this graph is symmetric Okay, even function |
| 30:04 | So this was more annoying most because the numbers are more annoying, but the concept should be pretty straightforward So lets have you guys try one that's problem. I'll try to make up one that doesnt have such bad numbers Sure could i go through what i did again? okay We got this We take the function, and need to figure out where has maximum and minimum And where the point of inflection. So first thing we do is take the derivative |
| 30:32 | then take the second derivative so to find out what function is going up and down, you take the first derivative you set it equal to 0 So we factored it we got x equals 0 plus or minus the square root of 2 i took the second derivative and set it equal to 0 and I got an x equals plus or minus radical 2/3 Now I sign test them to help me figure out what the curves going up and down and what the concavity looks like |
| 31:05 | And then I need Y coordinates, I'm actually gonna graph, so notice concave up concave down notice also at the critical number Maybe square root of two you could tell something's a minimum by by plugging that in to the second derivative If i take negative square root of 2 and plug it in we get a positive number. Which tells me It's concave up so in a minute I'm going to show you that its called the second derivative test |
| 31:35 | Okay Ready to try one sure Let's do a cubic again their not too painful |
| 32:39 | Find maximum minimum point inflection Let's take the derivative, but prime cause were lazy 6x squared minus 30x plus 24 and the second derivative is 12x Minus 30 |
| 33:01 | So far so good?
How we dod? take the derivative get some partial credit Okay, set those equal to zero So 6x squared lets do it over here get x squared minus 5x plus 4 equals 0 X minus 1 X minus 4 equals 0 x equals 1 or 4 |
| 33:32 | Okay, so we know if we have any maxima or minima there going to be a 1 in 4, right?
Alright what about the second derivative well said that equals 0 And you get x equals 5 halves or 2.5 So thats the second derivative of 0 and that's where we're going to look for a point of inflection |
| 34:00 | Okay, let's do some sign testing The first derivative is 0 at 1 and at 4 and the second derivative 0 at 5/2 All right Let's try a number less than 1 like zero we take zero and we plug it in the first derivative we get 4 |
| 34:34 | So it's positive The graph is increasing Going up to the left of one Now you try a number between one and four like two You plug into derivative you get a negative number so the graph is decreasing going down Then you take a number bigger than 4 like 5 You put it in the derivative you get a positive numbers. so the graph is going up |
| 35:01 | down, up So we have a maximum here, and minimum here And now we do the second derivitive Take a number less than 5/2 like 0 you plug into the second derivative And you get a negative number so graph it down there which is what you'd expect And then you take a number bigger than 5/2 and plug it into the second derivative. Its positive so its concave up there |
| 35:37 | All right now we just need to find y values, lets see.
When X is 1 we get a 7 right when X is 4 |
| 36:04 | lets see 128 minus 40 is -112 plus 96 is Negative 20 5 1/2 125 over 4 minus 75 over 4 is negative 50 over 4 plus 60 which is 240/4 , 190/4 minus |
| 36:35 | So I get it 176 over 4 Which is 444 did i get that wrong? I think i got that wrong. thats definitely wrong, what did you get? I dont want to do it twice Negative 13 over 2 that's about right. I had problem with subtracting |
| 37:01 | Maybe this is the first sign im getting old So negative 13/2, so now we just gonna graph this And we get 1 comma 7 We've got 4 comma negative 20 |
| 37:37 | By the way the y-intercept is at negative 4 and Let's see what else we've got 5 1/2 comma negative 13 halves So Something like that Okay, what would we be looking for on the test we would be looking for whether you found |
| 38:02 | These points these X values and this x value
Whether you got the shaped right
I'll be looking for the Y coordinates, but they won't be an important part again unless yours are way off
So one of the things you know is it kind of has to do something like this
So you know for example at that point of inflection has to be in the middle here somewhere?
Okay, that's a couple of clues on what should be going on |
| 38:30 | Im gonna do one more I know It's only 5:05 will do one is team suppose I have y equals, let me make sure this is right |
| 39:29 | Thats to hard, let's just do this |
| 39:37 | X to the 2/3 All right, this is annoying let's see if it could figure this out. Lets take the derivative derivative is 2x to the negative 3 Second derivative is negative 2/3 X To the negative 4/3 |
| 40:02 | And remember what X to the negative third means 1 over the cubed root of x And This would be negative 2 over 3 times cube root X to the fourth All right, so some interesting things happen here There's no place where this is zero |
| 40:32 | Okay, because for a fraction to be zero the numerator has to be zero and the numerator is 2 But the denominator is zero at zero so you also get critical numbers where the derivative is undefined Okay, something funny can happen when the Nominator is zero Okay, what we maximum and minimum under the regular sentence But something funny can happen and get the second derivative. notice the function exist at 0 |
| 41:00 | But the derivative is undefined and the second derivative is undefined so what's going on well let's see It's zero Well take another less than zero like negative cube root of negative one is negative one so this is negative The curve is going down. I'm sorry. the curve is going down here and to the right of zero like one the curve is going up |
| 41:37 | Now let's do the second derivative Well the cube root of x to the fourth anything to the fourth is positive or 0 Times three, is still positive so this whole thing is going to come out negatives. So no matter what's going on concavity is negative Got that shape |
| 42:02 | And when X is zero Y is zero, so what does this graph look like Well the graph is going down and it's concave down till you get to here its going down, concave down, and then it's going up and it's concave down so you get a tusk |
| 42:34 | how did i know it was gonna be a tusk? x to a fractional power, between 0 and one youre gonna get something funny happening Alright, because a function will exist at zero, a derivative will not so how do i know it looks like this?Once again the curves going down and it's concave down that has had that shape and Then curve is going down is concave up, so it has to have that shape |
| 43:00 | And know that there's no derivative here because the derivative is undefined, and youll get something funny Okay, remember looking at these two months ago All right, I think that's enough, well do more of this on wednesday |