okay so we're gonna continue with uh our discussion of exponential functions and I wanted to introduce in this uh this irrational number called e so we probably have seen Pi before in our life which is 3.14159 so on and so forth e is approximately 2.71 it's also an infinite decimal expansion and the way it's actually derived or one of the methods it's derived well before it's how we see it's derived why is this important well it turns out that the function y equals e to the X has a it's a special um it's a special base it's also sometimes called the natural base part of the reason why it's called the natural base you may see when you take or if you take a calculus course but it's also important because a lot of the growth rates that we see in nature things like bacterial growth or decay of certain naturally occurring compounds they Decay via some type of um Decay that can correspond with this base e so e turns out to be a really important number um and the part of the method it's derived e is basically um take a really and this is I'm going to put this in kind of quotes because there's a little bit more formal way to say it but essentially Take N to be a really large number and take one plus one over n to the nth power so keep taking and larger and larger and essentially you you approach you approach uh the value of E so what we're going to do also so we'll see some examples of this and we're going to do it in Desmos so you can see some examples of computations of particular n values and then what we're going to do is we're going to just see the graphs of e to the x squared and e to the negative x squared just to give you another example of some exponential functions again e being the same 2.71 and then we'll come back and look at an application okay so let's start with uh this one plus one over n to the N so what we did was Take N to be a thousand and notice we get 2.7169 if we take say ten thousand so one plus one over ten thousand to the 10 000 power you get 2.718 let's just do one more let's take like a million uh so we have a hundred thousand and then we have a million so we have a hundred thousand oops a million so now again we have 2.718 and if we just put e here this is the exact value 2.7182818 blah blah blah so you can see where we're getting there we're getting pretty close to the actual value of E okay so um that's kind of how e is constructed and now let's just look at these uh these graphs so e to the x squared is this um the red graph it's going to zoom in just a bit and then there's e to the negative x squared which has this notice with e to the negative x squared it it has its maximum value uh F1 occurring at zero and the reason is well essentially what you're you're looking at for the blue one you're looking at y equals one over e to the x squared so when you plug in any x value you're taking e to numbers that are getting bigger and bigger so you're getting 1 over numbers that are getting really small really fast so um doesn't have much hope of of growing up on the other hand e to the x squared grows exponentially fast on both sides of the axis because x squared is positive for any non-zero value you're looking at graphs or values that are going up really fast on either side okay so those are just a couple of other graphs and what we're going to do is go ahead and look at an application that's not you know it's not too too hard computationally but it's uh just a little example so it says the annual amount of wine in millions of gallons consumed in the US can be approximated by the function I'm just going to rewrite the function here 34 E to the 0.029 X um in millions of gallons where X say for example equal 50 corresponds to the year 1950 and this is the data from wineinstitute.org how much wine is was consumed in the year 2000 or year 2015 so year 2000 what would that correspond to well if x is 50 corresponds in 1950 we're looking at x equals 100 and then 115. so we can plug in so we're looking at 34 E to the 0.029 x so 34 E to the 0.029 x and let's get rid of some some of these guys so that we just have um this graph and we want to find F of 100 that'll tell us in millions so in in the year 2000 according to this model we're looking at 617 million gallons of wine consumed in the US F of 115 954 million gallon that's quite an increase from the year 2000. so the question comes according to this model when is it gonna when are we gonna uh the second part of this says find the first year where the consumption exceeds 1100 million gallons so um what we're going to do is we're going to look at the graph and kind of see where the Y value goes above 1100 so right now we're at 900. um at 115. so go to let's see this would be a thousand there's 1200 so right in between is Eleven Hundred and we're looking the first year we'll look across uh 1100 would be at 120. so 120 Is 2020 and we can check here 2020 would be F of 120 and indeed if we take uh 120 that's the year 2020 we have one um basically 1.1 billion gallons of wine consumed just to check 2019 that would be under 1.1 billion so 2020 um at least according to this model we crossed a 1.1 billion gallon mark and that is wow so compared to 20 years earlier that's almost 100 increase in the amount of wine consumed interesting all right you can make of that what you will but um that is what our our function shows and um um that's that's pretty much that application okay so we're going to stop here for now and we will continue with more topics in the next video