WEBVTT
Kind: captions
Language: en
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today we're going to talk about Exponential's and so just to remind you
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how these work let's look at what the graph of say 4 to the X so of course
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it's always positive
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we're going to let x very so when we plug in four to the one we get the power
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four I can't my scale is going to be off so i'm going to put for about here then if
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we take four to the to this will give us 16 already i'm off board but widely a
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bit and then four the three will be four times 16 or 64 which is like up here
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somewhere about here so let's just call it there and 4 to the 0 will be at 1 which I'm going to
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exaggerate a little bit higher just to get some more detail here four to the
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negative 1 is 1 over 4 which is pretty small quarter four the negative 2 is one
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over 16 and so on so it already see when you get something like that
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now let's imagine four example with for the one-half will be at square root of
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4 at 2 my scales off so i have to put two there thats 1, 4 to the three-halves will be eight because
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four the three-halves is fuor one-half to the third power to the third which is eight and
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so on we can fill in all these points in between and in general for any
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fractional exponent four to the P over Q
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this is going to be the cubed root of four and then I raise that to the p
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power and so i can fill in all of the fractions or rational numbers in between
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and then to get the irrational numbers to do something like four pi i have to
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take a limit so this is going to be 4 to the 3.1415 blah blah will certainly be less
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than four to the 3.1 no 3.2 and bigger than 4 to the 31 so that will be 4 to the take
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the 10 through four and then raise it to the 31st power or raise it to the 32nd
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power anyway we can fill in all of these numbers so we get a graph looks
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something like that instead of using well let's stick with 4 for now now this is
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a calculus class so we want to do some calculus so if i take an arbitrary point
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here like let's call it a and I want to know what you slope our usual question
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what is the slope of the tangent line here what I want to know the slope of
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the tangent 4 to the X at X=a and this is something that we already know how to
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do well in theory we know this in practice that means that that would be
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the limit as X goes to a of 4 to the X minus 4 to the a /
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x minus a hmm maybe I should have written this in the other form so i can
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also write that find refer the limit as H goes to 0 of 4 to the a plus h minus
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4 to the a over a plus h1 and which is H this is the same thing here x is a plus h h
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is the distance between the point move a little bit and then we slide them
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together now I want to do this limit that's not a nine that's an a so I'll
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factor remember that four to the a plus h is four to the a times four H so i can
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factor four to the a out of both of these terms limit as H goes to 0 of 4 to the a
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over for 2h minus 1 h ok
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doesn't look like it helped much now a is a number it's like 2 or 7 or pie and so
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four the a is just some number so this is for to the a time limit as H goes to 0
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four the h minus 1 over H now I don't know what to do with this limit if i use
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a computer or take out a calculator and compute values of this closer and closer
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and closer to 0
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I'll get a number like 1.386 and change times four to the a
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so it looks like i didn't really get much of anywhere but in fact if you
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interpret this carefully this tells us something
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it tells us if we want to know the slope anywhere so if my question was like at a
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equals two then i would say so the slope of 4 to the x and x equals 2
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is going to be 4 squared times this number one 1. 386, so that's
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like 16 is like 20 means that at herw even at X square the slope is like
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20-something yeah maybe 22 some number like that and i wanted to know the slope
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at back here at negative 2
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well that's 16 of 1.386 so we actually did figure something out if we
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know the slow if we know this number then we know the slopes everywhere so
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really what we figured out with all of this
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let me just write it slightly differently
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let's say instead of x equals way so the slope of the tangent to the curve for
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the x is 4x times this limit limit as H goes to 0 of 4 to the h minus 1 over H
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and there's nothing magic about four in this calculation i could have done this
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with 2 to the x 10 to the X 3 to the x 2.7 to the X whatever is what always
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work this way now let's interpret this in a slightly different way
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what is this limit this limit is the slope of the tangent slope of the
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tangent to 4 to the x at x equals 0 because this is the limit as H goes to
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zero a 4, 0 plus h minus 4 to the 0 which is one divided by H so this is the slope here
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so if I know it is slope here at one at zero then I always hope everywhere and
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there was nothing magic about four in this calculation i could have done this
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with eight it doesn't look much like an 8 I could have done this with any younger so
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let's put a two here, with a two get the same kind of an answer
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I don't get the same answer so here I see that this slope two is a number like
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. 69 so depending on which 1i use to get a different slope so let's think about
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that question a little bit more so we've done two things
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one thing that we just got you think about it we learned that the slope i'm
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just going to write b here for base of e to the X at any point X is slope of
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the tangent well let's call it what it is the derivative which is the slope of
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the tangent so the derivative of B to the X is b to the x times the derivative
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derivative at x equals 0 so we've transformed the question of knowing it
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anywhere to knowing it at one point and one point tells us all for Exponential's and
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then some examples we see that if B is fuor so the slope let's write as e DX of
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four the X is something like 4 to the x times 1.38 change if b is 3
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something like 3x times 1.09 change the base is 2, two the x times 0.69 and change
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and so on
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that's very teadius to calculate all of these slopes but the thing to notice is
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as i make the but base go from 4 to 3 to 2 this factor that i have to multiply 2
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goes from something like 1.42 1.1 2.6 if I tried a number between here like two
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and a half i get two and a half lets write that as 5/2, I have five halves to the
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x times 0.9 16 so i can see that there's some continuity here too
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that's saying that if i look at the graphs of Exponential's just near zero
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if this is like 2.12 the x it's very flat here but if i take something like
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4 to x it's quite steep and ones in between are in between so and so they're
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there is one that is just right now what do i mean to just right well I
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don't like this factor this is annoying
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we don't like annoying if the slope was one
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everything would be beautiful so there's one special guy here who slope is 1 so
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there's a number erase all of this now there's a number so that the limit is as
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H goes to 0 B to the X minus one lets use h sorry b to the x minus 1 over H i
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would like this limit to be one and i'm going to call that number not b im going
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to call e so there is some number so thats true and that number is about 2.71
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8 blah blah so this number it satisfies this relationship is this number that we
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call e and it makes our life very easy and so that means if we look at the
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function e to the X then if we take its derivative is exactly itself because we
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chose e just to make sure that e to the X is its own derivative is saying
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that the slope
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it's the function so that the slope at x equals the height so the slope of e to the x
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is at point x is e to the X, e to the X is a special function whose derivative is itself and
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from this we can calculate the slopes of any exponential but in particular we
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have this one special one
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so that's probably enough for now