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Graphing Functions Using Derivatives · Acedra
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Graphing Functions Using Derivatives
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0:00
let's see if we can use everything we know about differentiation and contivity and maximum and minimum points and inflection points to actually graph a function without using a graphing calculator so let's say our function let's say that f ofx is equal to 3 x to 4th minus - 4 x to the 3 + 2 and of course you could always graph a function just by trying out a bunch of points but we want to really focus on the the points that are interesting to us and then just to get
0:35
the general shape of the function especially we want to focus on the things that we can uh uh take out from this function using our our calculus toolkit or our derivative toolkit so the first thing we probably want to do is figure out the critical points we want to figure out I'll write it here critical critical points and just as a refresher of what critical points means it's the points where F where the the derivative of f ofx is zero so critical points are fime of X is either equal to
1:06
0er or it's undefined or if it's undefined this function looks differentiable everywhere so the critical points that we worried about are probably well I can tell you they're definitely just the points where fime of X are going to be equal to zero this this this derivative frime of x is going to actually be defined over the entire domain so let's actually write down the derivative right now so the derivative of this F Prime frime of X this is pretty straightforward the derivative of 3x to
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4th 4 * 3 is 12 12 x to the we'll just decrement the four by one three right you just multiply times the exponent and then decrease the new exponent by one minus 3 * 4 is 12 * x to the 1 Less Than 3 is 2 and then the derivative of a constant the slope of a constant you can almost imagine is zero it's not changing a constant by definition isn't changing so that's fime of X so let's figure out the critical points the critical points are where this thing is either going to
2:13
be equal to zero or it's undefined now I could look over the entire domain of real numbers and this thing is defined pretty much anywhere I could put any number here and it's not going to blow up it's it's going to give me an answer to what the function is so it's defined everywhere so let's just figure out where it's equal 0 so fime of X is equal to 0 so let's solve which X's so let's solve well I don't have to rewrite that I just wrote that let's solve for when
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this is equal to zero and I'll do it in the same color so 12 x 3r - 12 x^2 is equal is equal to zero and so let's see what we can do to what can we do to solve this well we could factor out of 12x so if we factor out of 12x then this term becomes just X and then actually let's factor out of 12 x^2 if we factor out of 12 x^2 if we divide both of these by 12 x^2 this term just becomes an X and then minus 12 x^2 ID 12 x^2 is just 1 is equal to zero I just rewrote this top thing like this
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you could go the other way if I multi if I distributed this 12 x^2 times this entire quantity you would get my derivative right there so the reason why I did that is because to solve for zero or if I want all of the X's that that make this equation equal to zero we now have I now have written it in a form where I'm multiplying One Thing by another thing and in order for this to be zero one or both of these things must be equal to zero so 12x^2 or equal to0 which means that X is equal to0 will
3:56
make this quantity equal Z and then the other thing that would make this quantity Zer is if x -1 is equal to Z so X - 1 is equal to 0 when X is equal to 1 so these are our two critical points our two critical points are X is equal to 0 and x is equal to 1 and remember those are just the points where our first derivative is equal to zero where the where the slope is zero they might be maximum points they might be minimum points they might be inflection points we don't know they
4:27
might be uh you know if this was a constant function they could just be anything so we we really can't say a lot about them just yet but they are points of interest I guess that's all we can say that they are definitely points of interest but let's keep going and let's try to understand the contivity and maybe we can get a better sense of this graph so let's let's figure out the second derivative so the second I'll do that in uh let me do it in this orange color so the second derivative of my
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function f let's see 3 * 12 is 36 x^ 2ar - 24 x so let's see well there's a couple of things we can do now that we know the second derivative we can answer the question is my graph concave upwards or downwards at either of these points so let's figure out what at either of these critical points and it'll all fit together remember if it's concave upwards then we're kind of in a u- shape if it's concave downwards then we're in a in a kind of upside down U shape so
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fime Prime our second derivative at X is equal to 0 is equal to what it's equal to 36 0^ 2 - 24 * 0 so that's just zero so FR prime prime so it's just equal to zero so we're in neither concave upwards nor concave downwards here it might be a transition point it may not if it is a transition point then we're dealing with an inflection point we're not sure yet now let's see what FP prime prime or our second derivative evaluated at one is so that's 36 * 1 let me write it down
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that's equal to 36 * 1^ 2ar which is just 36 - 24 - 24 * 1 so it's 36 - 24 so it's equal to 12 so this is positive our second derivative is positive here it's equal to 12 which means our first derivative our slope is increasing the rate of change of our slope is positive here so this at this point right here we we are concave upwards upwards which tells me that this is probably a minimum point right the slope is zero here but we are concave upwards at that point so that's
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interesting so let's see if there are any other potential inflection points here we already know that this is this is a potential inflection point let me Circle it in red it's a potential inflection point we don't know whether our function actually transitions at that point we'll have to uh experiment a little bit to see if that's really the case but let's see if are any other inflection points so to figure that or potential inflection points so let's see if this equals zero anywhere else so 36
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x^2 - 24x is equal to 0 Let's solve for x let us factor out a well we can factor out of 12x 12x * 3x right 3x * 12x is 36 x^2 - 2 is equal to 0 so this this these two are equivalent expressions if you multiply this out you'll get this thing up here so this thing is going to be equal to zero either if 12x is equal to Zer so 12x is equal to 0 that gives us X is equal to 0 so at x equals 0 this thing equals 0 so the second the second derivative is zero there and we already
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knew that because we tested that number out or this thing if this expression was Zero then the entire second derivative would also be zero so let's write that so 3 x - 2 is equal to 0 3x is equal to 2 just adding two to both sides 3x is equal to 2/3 so this is another interesting point that we haven't uh really hit upon before that might be an inflection point and the reason why I say it might be is because the second derivative is definitely zero here you put 2/3 here you're going to get zero
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what we have to do is see whether the second derivative is positive or negative on either sides of of 2/3 and we already have a sense of that I mean could try out a couple of numbers we know that you know if we take X's if we say that X is greater than 2/3 let me scroll down a little bit just so we have some space so let's see what happens when X is greater than 2/3 x is greater than 2/3 what is fime of X so let's see or sorry what is frime prime what is the second derivative so let's try out a
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value that's pretty close just to get a sense of things just to get a sense of things so let me rewrite it frime Prime of X is equal to let me write it like this I mean I could write it like this but this might be easier to deal with it's equal to 12x * 3x - 2 so if x is greater than 2/3 this term right here is going to be positive that's definitely you know any it's any positive number times 12 is going to be positive but what about this term right here 3 * 2/3
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- 2 is exactly zero right 2 - 2 but anything larger than that 3 times you know if I had 2.13 right so if here this is going to be a positive quantity any value of x greater than 2/3 will make this thing right here positive right this thing is also going to be positive so that means that when X is greater than 2/3 that tells us that the second derivative is positive it is greater than zero so in our domain as long as X is larger than 2/3 we are concave upwards and we saw that here at F at at
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X is equal to 1 we were concave upwards but what about uh X being less than 2/3 so when X is less than 2/3 let me write it let me scroll down a little bit when X is less than 2/3 what's going on I'll rewrite it frime Prime of X second derivative 12x * 3x - 2 well if we goes you know really far or less we're going to get a negative number here and this might be NE let's see if we just go really like you know right below 2/3 when we're still in the positive domain so you know if this was like
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1.93 which is you know a mixture of a decimal and a fraction or even 1/3 this thing is still going to be positive right below 2/3 this thing is still going to be positive we're going to be multiplying 12 by a positive number but what's going to be going on what's going on right here what's going on right here at 2/3 we're exactly zero but as you go anything less than 2/3 3 * 1/3 is only 1 1 - 2 you're going to get negative numbers so when X is less than 2/3 this
11:06
thing right here is going to be negative so the second derivative if x is less than 2/3 the second derivative right right to the left right when you go less than 2/3 the second derivative of x is less than zero now the fact that we have this transition from when we're less than 2/3 we have a a negative second derivative and when we're greater than 2/3 we have a positive second derivative that tells us that this indeed is an inflection point that X is equal to 2/3 is
11:39
definitely an inflection point for our original function up here now we have one more candidate inflection point and then we're ready to graph then you know once you know all the inflection points and the maximum minimum you are ready to graph the function so let's see if x is equal to 0 is an inflection point we know that the second derivative is 0o at 0 but what happens above and below the second derivative so let's let me do our little test here so when X is let me
12:06
draw a line so I don't so we don't get confused with all of the stuff that I wrote here so when X is greater than zero what's happening to the second derivative remember the second derivative was equal to 12x * 3x - 2 I like writing it this way because you've kind of decomposed it into two linear expr and you could see whether each of them are positive or negative so if x is greater than zero this thing right here is definitely going to be positive and then this thing right here right when
12:40
you go right above X is greater than zero so we have to be make sure to be very close to this number right so if this number is like let's say it's 0.1 right so you know you're right above zero so this isn't going to be true for all of X greater than Z we just want to test exactly what happens right when we go right above zero so if this was 0.1 you would have 3 3 minus 2 that would be a negative number right so right as X goes right above zero this thing right
13:06
here is negative so as X is greater than zero you will have your second derivative is going to be less than zero you're concave downwards which makes sense because we're at some point we're going to be hitting a a transition remember where we were concave downwards before we got to 2/3 right so this is consistent from 0 to 2/3 we are concave down downwards and then at 2/3 we become concave upwards now let's see what happens when X when X is Right less than when X is just barely just barely less
13:41
than zero so once again FR Prime the second derivative of x is equal to 12x * 3x - 2 well right if you know if x was -.1 or -01 no matter what this thing is going to be negative this expression right here is going to be negative the 12x right you have some negative value here time 12 is going to be negative and then what's this going to be well 3 * Min -.1 is going to be Min -.3 minus 2 is min - 2.3 you're definitely going to have a negative this value right here is going
14:14
to be negative and then when you subtract from a negative it's definitely going to be negative so that is also going to be negative but then if you multiply negative times a negative you're going to get a positive so actually right below X is less than zero the second derivative is positive the second derivative is positive now this all of might have been a little bit confusing but we've we should now have the payoff we now have the payoff we have all of the interesting things going
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on we know that at X is equal to one we know that at X is equal to one let me write it over here we we figured out at X is equal to one the slope is zero so FR prime prime is sorry let me write this way I should have said we know that the slope is zero slope is equal to zero and we figured that out because the first derivative was Z this was a critical point and we know that we're dealing with the the the function is concave upwards at this point and that tells us that this is
15:16
going to be a minimum point so what is f of and we should actually get the coordinates so we can actually graph it that was the whole point of this video so and F of one is equal to what F of one let's go back to our original function is 3 * 1 right 1 to the 4th is just 1 3 * 1 - 4 + 2 right so four * so it's it's it's well 3 * 1 - 4 * 1 which is min - one + 2 well that's just a positive 1 so F of one is 1 and then we know at X is equal to 0 we also figured
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out that the slope is equal to Z but we figured out that this was an inflection point right the the contivity switches before and after so this is an inflection point inflection point and we are concave below zero below zero so when X is less than zero we are upwards we are upwards our second derivative is positive and when X is greater than zero we are downwards we are concave downwards right above not for all of the domain X and0 just right above zero downwards and then what is f
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of0 just so we know we want to graph that point F of0 let's see F of 0 this is easy 3 * 0 - 4 * 0 + 2 well that's just two F of 0 is 2 and then finally we got the point x is equal to 2/3 x is equal to let me do that in another color we had the point x is equal to 2/3 we figured out that this was a inflection point the slope isn't necessarily or the slope definitely isn't zero there because it wasn't one of the critical points and we know that we are downwards
17:15
we know that when X is less than 2/3 or right less than 2/3 we are concave downwards and when X is greater than 2/3 we saw it up here when X was greater than 2/3 right up here we were concave upwards the second derivative was positive we were upwards and we could actually figure out what's F of 2/3 that's actually a little bit complicated we don't even have to figure that out I don't think to graph I think we could do a pretty good job of graphing it just with what we know right
17:49
now so that's our takeaways let me do a rough graph let's see so let me do my axes just like that so let's see we're going to want to graph the point 02 so let's say that the point 02 so this is X is equal to 0 and we go up one 2 so this is the point 02 maybe I'll do it in that color the color I was using so that's this color so that's that point right there then we have the point x we have F of one which is the point 1 one right so this point right here is we
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go up one so that's the point one one this was the point 02 and then we have the x is equal to 2/3 which is our inflection point so when X is 2/3 we don't know exactly what number F of 2/3 is might be here someplace let's say F of 2/3 is right there so that's the point 2/3 and then whatever F of 2/3 is it looks like it's going to be 1 point something F of 2/3 you could calculate it if you like you just have to substitute back in the function but we're ready to graph this
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we're ready to graph this thing so we know that at X is equal to 1 the slope is zero we know that the slope is zero it's flat here and we know it's concave upwards so we're dealing it looks like this it looks like that over that interval we're concave upwards and we know we're concave upwards from X is equal to 2/3 and on right for let me do in that color we knew x 2/3 and on we're concave upwards and so that's why I was able to draw this u-shape now we know
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that when X is less than 2/3 and greater than zero we're concave downwards so the graph will look something like this over this interval will be concave downwards let me draw it nicely over this interval the slope is decreasing and you can see it if you keep drawing tangent lines it's flattish there it gets negative more negative more negative more until the inflection point and then it starts increasing again because we go back to concave upwards and then finally the last interval is
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below zero and we know below zero when X is less than zero we're concave upwards so the graph looks like this well graph looks like that and we also know that X is equal to Z was a critical point the slope was zero so the graph is actually flat right there too so this is an inflection point where the slope was also zero so this is our final graph we're done after all that work we were able to use our calculus skills and our knowledge of inflection points and
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contivity and transitions in contivity to actually graph this fairly fairly hairy looking graph but this this should be kind of what it looks like if you graph it on your calculator
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