Stony Brook MAT 125 Spring 2015
The derivative of exponentials, pre-lecture

Start   today we're going to talk about Exponential's and so just to remind you how these work let's look at what the graph of say 4 to the X so of course it's always positive we're going to let x very so when we plug in four to the one we get the power four I can't my scale is going to be off so i'm going to put for about here then if
0:32we take four to the to this will give us 16 already i'm off board but widely a bit and then four the three will be four times 16 or 64 which is like up here somewhere about here so let's just call it there and 4 to the 0 will be at 1 which I'm going to exaggerate a little bit higher just to get some more detail here four to the
1:04negative 1 is 1 over 4 which is pretty small quarter four the negative 2 is one over 16 and so on so it already see when you get something like that now let's imagine four example with for the one-half will be at square root of
1:334 at 2 my scales off so i have to put two there thats 1, 4 to the three-halves will be eight because four the three-halves is fuor one-half to the third power to the third which is eight and so on we can fill in all these points in between and in general for any
2:04fractional exponent four to the P over Q this is going to be the cubed root of four and then I raise that to the p power and so i can fill in all of the fractions or rational numbers in between and then to get the irrational numbers to do something like four pi i have to
2:31take a limit so this is going to be 4 to the 3.1415 blah blah will certainly be less than four to the 3.1 no 3.2 and bigger than 4 to the 31 so that will be 4 to the take the 10 through four and then raise it to the 31st power or raise it to the 32nd
3:03power anyway we can fill in all of these numbers so we get a graph looks something like that instead of using well let's stick with 4 for now now this is a calculus class so we want to do some calculus so if i take an arbitrary point here like let's call it a and I want to know what you slope our usual question
3:33what is the slope of the tangent line here what I want to know the slope of the tangent 4 to the X at X=a and this is something that we already know how to do well in theory we know this in practice that means that that would be
4:04the limit as X goes to a of 4 to the X minus 4 to the a / x minus a hmm maybe I should have written this in the other form so i can also write that find refer the limit as H goes to 0 of 4 to the a plus h minus
4:374 to the a over a plus h1 and which is H this is the same thing here x is a plus h h is the distance between the point move a little bit and then we slide them together now I want to do this limit that's not a nine that's an a so I'll factor remember that four to the a plus h is four to the a times four H so i can
5:08factor four to the a out of both of these terms limit as H goes to 0 of 4 to the a over for 2h minus 1 h ok doesn't look like it helped much now a is a number it's like 2 or 7 or pie and so
5:37four the a is just some number so this is for to the a time limit as H goes to 0 four the h minus 1 over H now I don't know what to do with this limit if i use a computer or take out a calculator and compute values of this closer and closer
6:01and closer to 0 I'll get a number like 1.386 and change times four to the a so it looks like i didn't really get much of anywhere but in fact if you interpret this carefully this tells us something
6:31it tells us if we want to know the slope anywhere so if my question was like at a equals two then i would say so the slope of 4 to the x and x equals 2 is going to be 4 squared times this number one 1. 386, so that's
7:04like 16 is like 20 means that at herw even at X square the slope is like 20-something yeah maybe 22 some number like that and i wanted to know the slope at back here at negative 2 well that's 16 of 1.386 so we actually did figure something out if we
7:34know the slow if we know this number then we know the slopes everywhere so really what we figured out with all of this let me just write it slightly differently let's say instead of x equals way so the slope of the tangent to the curve for
8:01the x is 4x times this limit limit as H goes to 0 of 4 to the h minus 1 over H and there's nothing magic about four in this calculation i could have done this with 2 to the x 10 to the X 3 to the x 2.7 to the X whatever is what always work this way now let's interpret this in a slightly different way
8:33what is this limit this limit is the slope of the tangent slope of the tangent to 4 to the x at x equals 0 because this is the limit as H goes to zero a 4, 0 plus h minus 4 to the 0 which is one divided by H so this is the slope here
9:06so if I know it is slope here at one at zero then I always hope everywhere and there was nothing magic about four in this calculation i could have done this with eight it doesn't look much like an 8 I could have done this with any younger so
9:30let's put a two here, with a two get the same kind of an answer I don't get the same answer so here I see that this slope two is a number like . 69 so depending on which 1i use to get a different slope so let's think about that question a little bit more so we've done two things
10:04one thing that we just got you think about it we learned that the slope i'm just going to write b here for base of e to the X at any point X is slope of
10:37the tangent well let's call it what it is the derivative which is the slope of the tangent so the derivative of B to the X is b to the x times the derivative derivative at x equals 0 so we've transformed the question of knowing it
11:20anywhere to knowing it at one point and one point tells us all for Exponential's and then some examples we see that if B is fuor so the slope let's write as e DX of
11:41four the X is something like 4 to the x times 1.38 change if b is 3 something like 3x times 1.09 change the base is 2, two the x times 0.69 and change
12:24and so on that's very teadius to calculate all of these slopes but the thing to notice is
12:34as i make the but base go from 4 to 3 to 2 this factor that i have to multiply 2 goes from something like 1.42 1.1 2.6 if I tried a number between here like two and a half i get two and a half lets write that as 5/2, I have five halves to the
13:11x times 0.9 16 so i can see that there's some continuity here too that's saying that if i look at the graphs of Exponential's just near zero
13:33if this is like 2.12 the x it's very flat here but if i take something like 4 to x it's quite steep and ones in between are in between so and so they're there is one that is just right now what do i mean to just right well I
14:03don't like this factor this is annoying we don't like annoying if the slope was one everything would be beautiful so there's one special guy here who slope is 1 so there's a number erase all of this now there's a number so that the limit is as
14:37H goes to 0 B to the X minus one lets use h sorry b to the x minus 1 over H i would like this limit to be one and i'm going to call that number not b im going to call e so there is some number so thats true and that number is about 2.71
15:188 blah blah so this number it satisfies this relationship is this number that we call e and it makes our life very easy and so that means if we look at the
15:40function e to the X then if we take its derivative is exactly itself because we chose e just to make sure that e to the X is its own derivative is saying that the slope
16:02it's the function so that the slope at x equals the height so the slope of e to the x is at point x is e to the X, e to the X is a special function whose derivative is itself and
16:41from this we can calculate the slopes of any exponential but in particular we have this one special one so that's probably enough for now