07 - What is an Exponential Function? (Exponential Growth, Decay & Graphing). - By Math and Science
Transcript
00:00 | Hello . Welcome back today . We're going to cover | |
00:02 | the concept of the exponential function in math . It | |
00:06 | has applications in algebra and trigonometry and pre calculus calculus | |
00:10 | , all branches of advanced math and so on . | |
00:13 | And it's not every day that I have the opportunity | |
00:16 | to explain to you one of the most important functions | |
00:19 | in all of math , all of science and all | |
00:21 | of engineering . So I'm very excited to teach this | |
00:24 | lesson here on the exponential function At the end of | |
00:27 | this lesson you should understand what an exponential function is | |
00:30 | . Have some idea of its importance . Although you | |
00:33 | will continue learning its importance as we go on throughout | |
00:36 | your your education in math and physics and chemistry and | |
00:39 | other areas . Uh And also be able to understand | |
00:42 | intuitive intuitively what the graph of the exponential function looks | |
00:46 | like . And I have a computer demo that we're | |
00:48 | gonna integrate into this lesson as well . So you | |
00:50 | can see interactively what's really going on with this function | |
00:53 | . I cannot overstate how important the exponential function is | |
00:57 | in math . It's really honestly one of the most | |
00:59 | important functions ever . Right , let me give you | |
01:02 | a little motivation why . It's a crucially important function | |
01:05 | for lots of reasons . But just off the top | |
01:08 | of my head , uh if you like to make | |
01:09 | money and you like to invest money in the stock | |
01:12 | market or you like to put money in a savings | |
01:14 | account and have it grow in the bank , the | |
01:16 | interest that you earn on money ends up following an | |
01:19 | exponential curve . So we're gonna learn a lot uh | |
01:23 | after we understand what an exponential function is , we're | |
01:25 | going to study the growth of money and that it | |
01:28 | follows an exponential curve . So if you like making | |
01:31 | money , exponential functions will do that for you . | |
01:33 | Also when you take a loan out and you owe | |
01:35 | money in form of a debt , the debt grows | |
01:38 | monthly as well . And the growth of that debt | |
01:40 | is also an exponential function . So if you want | |
01:42 | to stay uh and keep your money and not have | |
01:45 | as much debt . Exponential functions really are important for | |
01:48 | you to understand . And going back more to the | |
01:51 | pure science is when you have a bacteria that is | |
01:54 | multiplying in a human body or in a laboratory , | |
01:57 | it's following a growth uh what we call exponential growth | |
02:01 | . So the population of the bacteria is not growing | |
02:04 | linearly , is growing exponentially . Right ? Also cancer | |
02:07 | when cancer grows in your body , same kind of | |
02:09 | thing is growing exponentially . Also populations , population of | |
02:13 | Australia , Population of UK , population of the world | |
02:16 | , population of some city somewhere . When you have | |
02:19 | generation after generation uh producing offspring , the population follows | |
02:25 | an exponential curve like this . So it's it has | |
02:28 | practicality there . Getting more back into the sciences . | |
02:31 | We have radio activity . Right ? When you have | |
02:33 | uranium and it decays into some daughter product , you | |
02:36 | might have heard the concept of half life when half | |
02:39 | of the thing decays into something else . That also | |
02:41 | follows an exponential type of decay . And here's the | |
02:44 | granddaddy of them all . You will learn later in | |
02:47 | your math and physics classes down the road engineering classes | |
02:51 | that all waves , sine sine waves , which we | |
02:54 | haven't even talked about sine waves yet . But these | |
02:57 | up and down wave motions , right , They can | |
02:59 | always be written in terms of exponential functions . So | |
03:03 | that means literally every wave that propagates radio waves , | |
03:06 | microwaves , x rays , gamma rays , also , | |
03:09 | all of the waves that we use in quantum mechanics | |
03:12 | to predict how matters going to behave . Those are | |
03:14 | all waves too . And all of those waves can | |
03:17 | always be written as exponential . All waves can be | |
03:20 | written as exponential . So this exponential function that we're | |
03:23 | learning literally has applications , every branch of math and | |
03:27 | science and engineering . So that's why it is so | |
03:29 | incredibly important . Now , what I want to do | |
03:31 | is get on the board to show you what it | |
03:33 | is . We're going to do the computer demo to | |
03:35 | give you an idea what the graph looks like . | |
03:37 | But then the rest of this lesson , we are | |
03:38 | going to dissect this function so that you understand exactly | |
03:41 | why the curve of this thing looks the way that | |
03:44 | it does . And there's a couple little kind of | |
03:47 | curves along the way . We'll be taking left turns | |
03:49 | and right turns to make sure you understand every little | |
03:51 | part of it . But ultimately , when you get | |
03:52 | to the end , I want you to know what | |
03:54 | this function is , what it looks like and why | |
03:56 | it behaves the way it does . So what is | |
03:58 | an exponential function ? Uh Let me just give a | |
04:01 | couple of quick examples of exponential functions and I'm just | |
04:04 | gonna write these down in no particular order . If | |
04:06 | you have a function F of X equal to two | |
04:09 | to the power of X , notice that the X | |
04:12 | variable this is different than X squared . We've been | |
04:16 | doing polynomial F of X is x squared . That's | |
04:19 | a parabola . But in that case the X is | |
04:22 | down low and the exponent X squared . The X | |
04:25 | . The exponent is a number , just the number | |
04:27 | two . In this case the exponent is not just | |
04:29 | a single number . The exponent is the variable . | |
04:32 | So when you see the variable up in the exponent | |
04:34 | like that , that's an exponential function . Right . | |
04:37 | Another example F of X is equal to 10 to | |
04:41 | the power of X . This is a different exponential | |
04:43 | function than this one , but it is still an | |
04:46 | exponential function . Ffx Equals instead of a number like | |
04:51 | two or 10 . We can have fractions as well | |
04:53 | , we can have one half to the power of | |
04:55 | X . Right ? So I have three examples here | |
04:59 | and I could keep going on , I could say | |
05:00 | , you know , 16 to the power of X | |
05:02 | , 1/10 to the power of X , whatever . | |
05:04 | But notice that they're all positive numbers down here at | |
05:07 | the bottom and the exponent and every one of these | |
05:10 | cases is up in the variable the variable is the | |
05:12 | explosion . So what we can say about this exponential | |
05:16 | function is the number down in the bottom is what | |
05:19 | we call the base . So this base In this | |
05:23 | case is equal to to the base is equal to | |
05:25 | two . In this case , In this case you | |
05:27 | might guess the base is equal to 10 . And | |
05:31 | in this case you might guess that the base is | |
05:33 | equal to one half . So you can see that | |
05:35 | the base can be the base has to be positive | |
05:39 | . And I'm gonna write all of these rules down | |
05:40 | in a minute . It has to be positive and | |
05:43 | it has to be uh well , I'm just gonna | |
05:45 | leave it at that has to be positive . You | |
05:46 | want to stay away from zero being a base and | |
05:48 | you want to stay away from the number one being | |
05:50 | the base . And I'll tell you why as we | |
05:52 | get a little farther in why zero and one caused | |
05:55 | problems for this function . But any other number than | |
05:58 | that that's positive . We'll give you an exponential curve | |
06:01 | just with different bases . All right . So what | |
06:03 | I want to do now is go off to the | |
06:05 | computer before we get too far into this and just | |
06:07 | show you in general what this thing looks like . | |
06:10 | And then once we finish that will come back and | |
06:12 | we'll draw lots and lots of pictures . So you'll | |
06:13 | understand exactly what this function looks like and how it's | |
06:16 | calculated . Okay , welcome back here . We have | |
06:22 | our computer demo this is what we call an exponential | |
06:24 | function . This be right here is what we call | |
06:27 | the base . So let's first take a look at | |
06:29 | when the base is bigger than one , like two | |
06:31 | to the power of X or three to the power | |
06:34 | of X or 10 to the power of X . | |
06:35 | Or 100 to the power of X and so on | |
06:38 | . So what happens as I increase this guy ? | |
06:40 | You see what happens uh as the base here is | |
06:43 | the basis of 1.6 and then we have the power | |
06:45 | of X . As the base gets bigger and bigger | |
06:47 | and bigger . This curve starts out very very close | |
06:51 | to zero . It rapidly ramps up like a Kind | |
06:54 | of like a rocket launch and it kind of starts | |
06:55 | going up much more and more vertical . And as | |
06:58 | the base gets bigger and bigger and bigger all that | |
07:01 | happens is the steepness of this part on the right | |
07:04 | hand side gets more and more and more vertical . | |
07:06 | So if I crank this thing up all the way | |
07:09 | I think the maximum I have this thing set was | |
07:11 | to 20 . But if I go to 100 or | |
07:13 | 300 all that's gonna happen is it's going to ramp | |
07:15 | up more and more and more vertically . Right so | |
07:18 | you might think of this curve representing the population growth | |
07:21 | of bacteria starts off really really really small . But | |
07:24 | then as it builds momentum in each generation starts generating | |
07:28 | you know daughters and little daughter bacteria and little son | |
07:31 | bacteria . And they continue to reproduce the Whammo , | |
07:34 | the thing goes straight up vertical like this 20 to | |
07:36 | the power of X . Is a very steep exponential | |
07:38 | curve . As I back this thing down you can | |
07:41 | see we start to bend the thing down closer and | |
07:44 | closer and closer and closer notice what happens when we | |
07:47 | get down to around 1.5 to the power of X | |
07:49 | . With a very gentle curve that goes up . | |
07:51 | One thing I want to point out to you the | |
07:52 | most important thing is that this exponential function . Look | |
07:55 | at the scale of the Y axis , here's five | |
07:58 | and here's 10 . So the tick marks are 12345 | |
08:01 | So this tick mark right where the curve crosses right | |
08:04 | here is actually at the Y intercept of one . | |
08:07 | Notice what happens when I change this curve , no | |
08:10 | matter what it looks like , it always crosses the | |
08:13 | y axis at the same spot the curve to the | |
08:16 | left and the curve to the right changes shape but | |
08:19 | it always goes through the same location . Why is | |
08:22 | that ? Because if I take a value of X | |
08:24 | is equal to zero and stick it in this , | |
08:26 | to the equation 4.9 to the zero power is just | |
08:29 | one . So that's why it crosses at one . | |
08:32 | If I change this curve to 20 to the power | |
08:34 | of X , I put an X . Value of | |
08:36 | 0 , 20 to the power of zero is still | |
08:38 | one . So it still crosses here . If I | |
08:40 | go way way way down here to 3.33 point two | |
08:43 | to the power of X , sticking a value of | |
08:45 | X is equal to 3.2 to the zero power is | |
08:48 | still one . So no matter how I play with | |
08:50 | this curve , it always crosses there . Now notice | |
08:52 | when I get down all the way I told you | |
08:54 | get into problems when you have one to the power | |
08:57 | of X . C 1.1 you have a very gentle | |
09:00 | slope up . When you get down to one to | |
09:02 | the power of X . Um It just flattens out | |
09:05 | completely . And that's because one to the power of | |
09:08 | ex uh no matter what I put in for X | |
09:11 | , whether it's 21 to the power of two is | |
09:14 | is 11 to the power of four is 11 to | |
09:18 | the power of negative two is still one . No | |
09:19 | matter what I put in for the X value . | |
09:21 | When I say one to the power of anything , | |
09:24 | I always get one . So basically this exponential function | |
09:27 | turns into a flat line when you put one as | |
09:30 | the base . And that's why I told you need | |
09:32 | to stay away from one being the base . Uh | |
09:35 | And also the same thing , similar thing happens if | |
09:37 | you put zero as a base , if you put | |
09:39 | zero as a base zero to the anything power is | |
09:42 | still zero . So that's not an exponential function either | |
09:45 | . So you cannot have a base of zero and | |
09:47 | you cannot have a base of one . But any | |
09:49 | positive value notice everything bigger than one . Just yields | |
09:51 | a steeper and steeper curve . Now let's go down | |
09:54 | and take a look at what happens when the base | |
09:57 | is not zero uh over here and it's not one | |
10:01 | , but we just constrain it to be between 01 | |
10:03 | and one because the curve up here only looked at | |
10:06 | what happens when the base is bigger than one . | |
10:07 | Of course it gets steeper and steeper . But down | |
10:10 | here we're just saying what happens when the base is | |
10:12 | one half , what happens when the base is 3/4 | |
10:14 | ? What happens when the base is one ? 10 | |
10:15 | ? Things like that . So what I can do | |
10:17 | is crank this thing this way and you can see | |
10:19 | what starts to happen is the exponential function starts to | |
10:23 | look like the exact mirror image of what's above . | |
10:26 | Look at this guy here at .2 to the power | |
10:28 | of X . And then I can crank this guy | |
10:30 | up so that it looks something like a mirror image | |
10:33 | . I mean I can adjust everything to make it | |
10:35 | exactly the same . I'm not interested in making it | |
10:37 | look the same . But you can see that the | |
10:39 | shape of the curve looks the same as it does | |
10:41 | in the previous case , it's just that for the | |
10:43 | negative values of X , that's where the curve starts | |
10:46 | to go into the stratosphere and then it just goes | |
10:48 | down to zero as we go on to the positive | |
10:52 | X direction . We're gonna get into exactly why it | |
10:54 | does this in a minute . So here we have | |
10:56 | the positive values of X . It gets steeper and | |
10:58 | steeper still goes through the value the y intercept of | |
11:01 | one . Here , if we have the base be | |
11:05 | a fraction like one half or one third or whatever | |
11:07 | to the power of X . We have the same | |
11:09 | shape but it's just a mirror image on the other | |
11:12 | side . Again , it goes through the y intercept | |
11:14 | of one . Anything to the one to the anything | |
11:17 | . Um I'm sorry anything zero any base to the | |
11:22 | power of zero is just going to give you a | |
11:24 | one . So it goes through the same spot here | |
11:26 | . Now what we're gonna do is combine those two | |
11:29 | things together , we're gonna say the base is gonna | |
11:30 | be bigger than zero , but um uh can also | |
11:35 | be a fraction as well . So when we go | |
11:37 | larger and larger values of the base , we have | |
11:40 | that nice skateboard type shape to the right , when | |
11:43 | we back it down , uh here what ? We | |
11:46 | have one to the power of X . We get | |
11:48 | to where it's flat . And then when I go | |
11:50 | just on the other side of one , then you | |
11:52 | can see this thing curving up the other direction . | |
11:54 | So you see the nice symmetry here , What's going | |
11:56 | on when the basis of fraction is , you can | |
11:58 | make it steeper and shallow over here . And then | |
12:01 | when you go through one it gets flat exactly flat | |
12:05 | . And when you go through one you get you | |
12:06 | have that nice shape happening on the other side . | |
12:08 | So this is what the exponential function does notice it | |
12:12 | always crosses through the y intercept of one . Okay | |
12:16 | , Like this and that is all I'm going to | |
12:20 | show you right now . I do want to show | |
12:21 | you what happens when the base gets less than zero | |
12:24 | , but I'm not gonna do that until the end | |
12:25 | of the lesson because I think it's gonna make a | |
12:26 | lot more sense then for now , what I want | |
12:29 | you to realize is when the base is bigger than | |
12:31 | one , you have this very steep curve on the | |
12:34 | positive part of the X values . When the base | |
12:37 | is a fraction like one half or one third or | |
12:39 | whatever , you have the steep part of the curve | |
12:41 | on the negative X . Side . And when you | |
12:44 | put those two things together , you can just see | |
12:45 | how it behaves . Going through a flat curve , | |
12:48 | a flat line and being exponential on both sides of | |
12:52 | this guy . So what I want to do now | |
12:53 | is stop the computer demo go back to the board | |
12:55 | and really understand why this function behaves the way that | |
12:59 | it does . All right , welcome back . I | |
13:02 | hope you've enjoyed the computer demo to really get an | |
13:05 | intuitive feel for what these exponential functions really look like | |
13:09 | . Now . What we learned from the computer plotting | |
13:12 | that we did was that the exact shape of the | |
13:14 | exponential function really is going to depend upon the base | |
13:17 | , the number that we have down in the bottom | |
13:19 | below the exponent is going to determine the exact shape | |
13:22 | of the curve . However , the general shape of | |
13:24 | the thing , as far as its sloping upward like | |
13:27 | that generally looks the same among all of those . | |
13:29 | But the exact steepness of it of course , depends | |
13:32 | on the base . So , just a couple of | |
13:34 | rules , I want to make sure and get down | |
13:36 | here on the board to make sure they're not lost | |
13:39 | is . So I'll just kind of put this under | |
13:41 | some miscellaneous note , right ? The base must be | |
13:48 | positive . Notice that when we did the computer demo | |
13:53 | , we did uh fractions like one half , one | |
13:56 | third . We did fractions all the way , just | |
13:57 | above zero all the way to 22 positive 20 . | |
14:00 | So any positive numbers . Okay . Uh except there's | |
14:04 | a couple of gotchas that you have to be careful | |
14:06 | about . Base must be positive . However , the | |
14:09 | base The base must not equal one . Why ? | |
14:16 | Because if I put a base of one here , | |
14:17 | one to the power of X No matter what I | |
14:20 | pick for X . 1 to the power of anything | |
14:22 | affects his one affects us to affect his negative 11 | |
14:25 | to the power anything is just one . So we | |
14:27 | saw that when the base was one in the computer | |
14:30 | demo , it was just a flat line that's not | |
14:32 | really considered exponential . In order to be an exponential | |
14:34 | curve , it needs to slope upward one direction or | |
14:37 | another . So the base cannot be won . If | |
14:39 | you put a one in there , it's just not | |
14:40 | gonna really be an exponential curve anymore . Also the | |
14:44 | base must not equal zero for similar reasons . If | |
14:49 | I put a zero in here , zero to the | |
14:51 | power of anything is just going to give me zero | |
14:53 | . So if you put a number one in for | |
14:55 | the base , you're not gonna get an exponential function | |
14:57 | . If you put a zero in for the base | |
14:58 | you're still not going to get an exponential function . | |
15:00 | Any other positive number is going to give you an | |
15:03 | exponential function with varying shapes . Now You might probably | |
15:07 | be wondering what happens if the base is negative ? | |
15:09 | What if I put a negative two for the base | |
15:11 | ? What if I put a negative 10 for the | |
15:12 | base ? I'm gonna save that for the end of | |
15:14 | this lesson . I would like you to know what | |
15:16 | happens , but I don't want to talk about it | |
15:17 | now because it doesn't give you an exponential function either | |
15:20 | . For now , just know that the base has | |
15:22 | to be positive , but it can't be one and | |
15:24 | it cannot be zero . All right now , some | |
15:27 | of you might be looking at this and saying , | |
15:28 | well , we've kind of studied functions similar to this | |
15:31 | in the past . We've studied things like for instance | |
15:33 | , ffx is equal to x squared . This kind | |
15:37 | of has an exponent there , we've studied ffx uh | |
15:40 | x cubed X to the fourth , but this is | |
15:43 | different . This is f of x is something like | |
15:46 | , you know , whatever , I can pick the | |
15:47 | base that I want . Six to the power of | |
15:48 | X . You see , I really want you to | |
15:50 | understand the difference here . These things are just polynomial | |
15:54 | , is they are not exponential . These are not | |
15:57 | exponential . Yeah , But this one is exponential . | |
16:05 | It's an exponential function . So the only thing you | |
16:07 | have to look for is if the variable is in | |
16:09 | the exponent , not a solid number here , but | |
16:12 | the variable itself is up top in the exponent . | |
16:14 | Then you have an exponential function . Then the exact | |
16:17 | shape of the curve is going to be governed by | |
16:20 | the base , right ? So we have to split | |
16:22 | it up into two pieces first . Like we did | |
16:24 | in the computer demo , I want to talk to | |
16:25 | talk to you in more detail what happens when the | |
16:27 | base Is bigger than the # 1 ? Right . | |
16:30 | And then we're gonna talk about what happens when the | |
16:32 | basis of fraction somewhere between zero and one . Because | |
16:34 | we saw in the computer demo it flipped everything around | |
16:37 | and made the curve kind of mirror image and that | |
16:40 | we need to explore a lot more in a lot | |
16:42 | more detail . But for right now let's go and | |
16:45 | talk about something very very important . Let's take a | |
16:47 | look at an example . One of these ffx is | |
16:51 | equal to two to the power of X . That's | |
16:53 | an exponential function with the base of two . And | |
16:56 | what I want to do is just spend a minute | |
16:58 | trying to draw a pretty good graph of this . | |
17:02 | Now is this graph gonna be as good as a | |
17:03 | computer ? No that's actually why I showed you the | |
17:06 | computer first . Um It's not gonna be as good | |
17:08 | but we can still learn a lot . Why does | |
17:11 | it start out low and why does it get steep | |
17:13 | like this ? And why does the steepness change depending | |
17:16 | on what the base is ? I want to get | |
17:18 | very very very comfortable with that . Now we saw | |
17:22 | in the computer demo that when X . is equal | |
17:24 | to zero all of these curves cross at an intercept | |
17:29 | point of one . I'm gonna put the number one | |
17:32 | up above here . Okay , so we saw that | |
17:35 | when the base was a positive number bigger than one | |
17:39 | like this , it starts way over here , close | |
17:42 | to zero . I'm probably gonna mess this curve up | |
17:44 | but it goes really really gently slowly increasing right through | |
17:47 | this point and then kind of going up something like | |
17:51 | this . Is that perfect ? No it's a little | |
17:52 | bit lumpy . It should be nice and smooth now | |
17:55 | the exact shape of it of course , depends on | |
17:57 | the base . As the base gets bigger , bigger | |
17:59 | , bigger . This thing gets steeper , steeper steeper | |
18:02 | as the base gets smaller smaller smaller it lays more | |
18:04 | flat till eventually it's just a flat line and then | |
18:07 | it flips over and goes the other direction . We | |
18:09 | saw that in the computer . But why is the | |
18:11 | thing curve this way ? Because when you take and | |
18:15 | put an X . Value of zero into this thing | |
18:17 | , what do you get ? I'm going to do | |
18:18 | a lot of little substitution is down here below you | |
18:21 | get two to the power of zero but two to | |
18:24 | the power of zero is just one . So you | |
18:26 | see the number one it crosses at the uh the | |
18:29 | uh Why Value of one . Now let's go and | |
18:32 | put a few values in there . 12345 negative one | |
18:37 | negative two negative three negative four negative five . So | |
18:40 | this is negative five negative four , negative three negative | |
18:43 | two negative 11234 and five . And what I'd like | |
18:50 | to do is just spend a second to understand how | |
18:55 | this thing uh increases . If we put the value | |
18:58 | of one in here then what we're going to get | |
19:00 | is two to the power of one . That just | |
19:03 | means to because that's all it is . Two to | |
19:05 | the power of one . So we get a value | |
19:06 | of two . So I'm not going to plot it | |
19:07 | . This graph this why access I have is not | |
19:10 | exactly right . My curve is not exactly right . | |
19:12 | But you can still see that as I go up | |
19:14 | here . When I put a two in there , | |
19:16 | it'll be two to the power of to this goes | |
19:18 | into the exponent location which gives you a four . | |
19:22 | Right ? And then you can see that as I | |
19:24 | put three in there , it's two to the power | |
19:25 | of 32 times two times two . This gives you | |
19:28 | an eight . So you see what's happening is it's | |
19:30 | increasing faster and faster . When I put a value | |
19:33 | of foreign to to the power of four , I'm | |
19:36 | going to get 16 two times two times two times | |
19:39 | 2 16 . And then when I put a value | |
19:42 | of five and I'll get to to the five uh | |
19:45 | and I'm gonna get 32 . So you can see | |
19:47 | these numbers 248 16 32 . It's doubling every time | |
19:53 | . And that's kind of the genesis of the exponential | |
19:56 | growth part . When when you put money in the | |
19:58 | bank and it earns interest , we're going to study | |
20:00 | it more later . But you have an exponential effect | |
20:03 | because when you earn money one year that money is | |
20:06 | still in the bank for the following year to help | |
20:09 | earn more money . So the more money you earn | |
20:11 | , the more interest you earn the following year . | |
20:13 | Because the money you earned in year one starts to | |
20:17 | earn more money in year two . But then in | |
20:19 | year two you have even more money which then generates | |
20:22 | more money . And so it goes up and up | |
20:23 | and up in an exponential fashion . So that's why | |
20:26 | the curved gets so steep . So quickly look at | |
20:28 | how fast these numbers are increasing . Now when we | |
20:30 | go to the other side here switch colours for this | |
20:33 | , I want to go to the negative one . | |
20:35 | A lot of students don't really quite understand why this | |
20:37 | works the way it does . If you put a | |
20:39 | negative one in for the experiment it'll be two to | |
20:41 | the power of negative one . But this is just | |
20:44 | the same thing as 1/2 to the first power , | |
20:48 | Which is the same thing as 1/2 . Okay , | |
20:52 | so we have one half here . Um And that's | |
20:56 | why it gets smaller than one , notice it gets | |
20:57 | smaller than one . Now we have a similar thing | |
20:59 | right here . If we put two and this exponent | |
21:01 | of negative to here , then this becomes 1/2 squared | |
21:06 | which becomes 1/4 . So it gets even smaller and | |
21:10 | you can see the pattern here . This I'm not | |
21:12 | gonna substitute every little thing , it's just gonna be | |
21:14 | 1/8 . This guy is gonna be 1/16 and this | |
21:18 | guy is gonna be 1/32 . So you see the | |
21:22 | numbers that you have on the positive side , Just | |
21:25 | become one over those numbers on the negative side . | |
21:27 | So as fast as this thing is increasing this direction | |
21:30 | , it is chopping it down just that fast going | |
21:33 | the other direction . This is why the shape of | |
21:35 | the curve for the exponential function in this case we | |
21:38 | just pick two to the power of X starts out | |
21:40 | very small because 1/32 is a really small number . | |
21:44 | But as we get It builds very very , very | |
21:46 | slowly . It always goes through the .1 , but | |
21:49 | then it just rockets up because when you get into | |
21:52 | positive territory , everything starts stacking as we just described | |
21:56 | here and that's called exponential growth . Okay , if | |
22:00 | we change the base , we're still going to be | |
22:03 | going through this crossing point of why is equal to | |
22:06 | one . That's really , really important for you to | |
22:08 | know . Um In fact , um I do want | |
22:11 | to take just a second before we talk about the | |
22:13 | negative side just to kind of draw some comparisons . | |
22:15 | Right ? So we might , you might say , | |
22:17 | well we've studied functions like this before and we just | |
22:19 | talked about the fact that these are not exponential functions | |
22:22 | , but a curve that looks kind of like this | |
22:24 | is the X square curve . I'm not going to | |
22:26 | clutter up this curve , but you remember the X | |
22:27 | square parabola goes down and then up . So it | |
22:30 | kind of has an upward slope like this . But | |
22:32 | one thing that's not , I think made clear enough | |
22:35 | in most books , is that an exponential function like | |
22:39 | this ? If you give it enough time , if | |
22:41 | you go out far enough will always go up steeper | |
22:45 | than a polynomial of the of a similar degree . | |
22:47 | Right ? So in other words , if I were | |
22:49 | to plot X squared , what's going to happen is | |
22:52 | if you square this guy , you're gonna get one | |
22:55 | . If you square this guy , you're going to | |
22:57 | get four . If you swear this guy you're gonna | |
22:59 | get three times three is nine . So you see | |
23:01 | the X Squared functions actually beating this function here Because | |
23:04 | nine for X squared if I did three square being | |
23:08 | nine is bigger than this guy . Four square to | |
23:10 | 16 . So right at this point the X squared | |
23:12 | function is going to kind of intersect here and it's | |
23:15 | going to start to not win anymore because if you | |
23:17 | square this guy five square you're gonna get 25 that's | |
23:20 | not as big as 32 . So what happens is | |
23:23 | the the the parabola function X squared , which is | |
23:26 | kind of similar because the basis to an X . | |
23:28 | Squared has a two in the exponents . It's not | |
23:30 | gonna it's gonna actually beat this function in the beginning | |
23:33 | . But then eventually if you give it enough time | |
23:35 | , the exponential function will always overtake a polynomial and | |
23:39 | rocket higher . So that's why if you ever want | |
23:41 | to you know , you know , earn money or | |
23:44 | interest or study bacteria or anything that has an exponential | |
23:46 | kind of growth , that's why it gets out of | |
23:48 | control so fast . Especially in terms of debt when | |
23:51 | you can owe so much money so fast because it | |
23:54 | just builds on each other on itself year after year | |
23:57 | after year and it kind of just rockets up and | |
23:59 | starts to go vertical like that . Okay . Um | |
24:02 | One more thing I want to bring very uh importantly | |
24:06 | home is that uh zero comma . One is uh | |
24:13 | or I should say lies on all exponential functions . | |
24:22 | I've said this several times but it's important enough to | |
24:24 | make sure you understand what I mean by this is | |
24:27 | the Y intercept is one . Right ? If I | |
24:29 | take for instance ffx let's take a crazy exponential function | |
24:33 | . Let's say it's 100 to the power of X | |
24:35 | . That's an enormously steep exponential function . Right ? | |
24:39 | But if I put the value of X is equal | |
24:41 | to zero in here , I'm gonna get 100 to | |
24:44 | the power of zero , which is just equal to | |
24:45 | one . So this means that the 10.0 comma one | |
24:48 | is on this curve . So just like we saw | |
24:51 | in the computer demo even if I crank the bass | |
24:53 | up to 100 all that's gonna happen is this is | |
24:55 | going to get very , very steep , but it's | |
24:56 | still going to come down through this point and then | |
24:58 | it's gonna get really , really small on this side | |
25:01 | . Okay , now so far , we've talked about | |
25:05 | positive values of the base . I want to draw | |
25:07 | a few more pictures about the positive values of the | |
25:09 | base and then we're gonna move on and talk about | |
25:11 | what happens . I should say we've talked about positive | |
25:13 | basis that are bigger than the number one base is | |
25:16 | equal to two . Base is equal to three . | |
25:18 | Base is equal to four in just a second . | |
25:20 | We're gonna talk about fractional basis where we see the | |
25:23 | exponential function flip over to the other side . Okay | |
25:26 | . But before I do that right underneath this , | |
25:29 | I want to draw a few different cases which we | |
25:32 | saw in the in the demo , but I want | |
25:35 | to draw a couple of cases to make sure you | |
25:36 | understand why this thing gets steeper and steeper and steeper | |
25:40 | as you increase the base . So let's take the | |
25:42 | case that we just have right here . Ffx is | |
25:45 | equal to two to the power of X . And | |
25:48 | we're gonna graph this guy right next door to a | |
25:51 | couple of alternatives just so we can kind of see | |
25:53 | what happens . So it's gonna go through the value | |
25:56 | why is equal to one . We know that it's | |
25:58 | going to start something like this and something like this | |
26:03 | . And we know we can just pick whatever number | |
26:05 | we want . Let's pick the number three just as | |
26:08 | a point of comparison X . Is equal to three | |
26:10 | when we put a value . Uh And and of | |
26:13 | course my scale is not right at all here , | |
26:15 | but if I go up here and I go right | |
26:17 | here , what's going to happen whenever I do this | |
26:20 | ? Is that um this value right here , it's | |
26:24 | gonna be two to the power of three , which | |
26:26 | is eight . So my scale is not right okay | |
26:28 | because there's one here and there's an eight here , | |
26:30 | but you get the idea we're gonna cross up and | |
26:32 | we're gonna come up to a value of eight . | |
26:34 | Okay , now let's take another example and let's see | |
26:38 | what happens . If we increase the base to F | |
26:41 | . Of X . Uh is equal to 3 to | |
26:44 | the power of X . What will something like this | |
26:47 | look like ? Well we're gonna have a curve that's | |
26:50 | gonna go like this is gonna be X . And | |
26:52 | ffx like this . We know it's going to go | |
26:54 | through the number one . Yes , I know that | |
26:56 | my scale is not perfect . So don't don't you | |
26:58 | know , don't get too mad at me . But | |
27:00 | basically what I'm telling you is just what we saw | |
27:02 | in the computer program . It's gonna start out very | |
27:04 | very slow and then it's gonna go a little bit | |
27:07 | steeper . Why is it going to go a little | |
27:09 | bit steeper ? Because again , if I take The | |
27:12 | same number three here and put it in here , | |
27:15 | It's going to go , it's going to be much | |
27:17 | much higher . Why ? Because this point is going | |
27:19 | to be three to the power of three , and | |
27:22 | three to the power of three is 27 . So | |
27:25 | the crossing point here is actually at 27 . So | |
27:28 | that's why increasing the base makes the thing gets steeper | |
27:32 | because here we're just as we go along and X | |
27:34 | . We're just going to times two times two . | |
27:36 | However many times you're doing it here , we're going | |
27:38 | three times three times three . However many times we | |
27:40 | do it . So if we pick the same value | |
27:42 | of X , of course the curve is going to | |
27:43 | be steeper here and we can make a similar argument | |
27:46 | on the other side is going to get closer to | |
27:48 | the X axis on the other side as well . | |
27:50 | Okay , so as the base gets bigger and bigger | |
27:53 | , larger than one , the exponential function gets steeper | |
27:56 | and steeper , but it always goes through the y | |
27:58 | intercept of one . Okay , now , what we | |
28:02 | want to do is we want to take a look | |
28:05 | at what happens when this base here , in this | |
28:08 | case it's the number two . When this base is | |
28:09 | allowed to be below the value of one , because | |
28:13 | remember when the base is one , it's exactly equal | |
28:15 | to one . We don't have an exponential function at | |
28:17 | all . But when it's less than one but bigger | |
28:19 | than zero , we have something really interesting happened . | |
28:22 | We saw in the demo that this curve flips backwards | |
28:25 | and starts high on this side and gets very low | |
28:27 | on this side . So we want to describe um | |
28:31 | exactly why that happens . So , let's take a | |
28:34 | look at that . Let's take another exponential function . | |
28:38 | Let's say f of X is going to be equal | |
28:41 | to uh instead of two to the power of X | |
28:44 | . Let's take a look at one half to the | |
28:46 | power of X . Same kind of thing . We're | |
28:49 | just gonna draw crude graph . It's not going to | |
28:51 | be exact at all , but we're gonna do our | |
28:54 | best . All right ? So let's do the same | |
28:58 | kind of numbers 12345 So 543210 negative one , negative | |
29:06 | two negative three negative four negative five -4 -3 -2 | |
29:11 | -1 . Alright , What I'm claiming is that this | |
29:14 | function is going to go through the exact same point | |
29:17 | . It's still going to go through one . It's | |
29:19 | still going to go through one . But instead of | |
29:21 | being high on this side , it's going to be | |
29:22 | high on this side . So I'm probably not gonna | |
29:25 | be able to draw this right . Apologize let me | |
29:26 | start this way and see if I can get it | |
29:29 | go like this right through this point like this . | |
29:33 | That's not actually too bad . You see it's a | |
29:35 | mirror image of what we have over here . Why | |
29:38 | is it a mirror image ? Let's take a look | |
29:40 | at it . If we take the number zero and | |
29:43 | put it in there , we're gonna have one half | |
29:45 | to the power of zero , that equals one . | |
29:47 | So we have to go through the 10.1 . Just | |
29:49 | like always this guy is going to be one half | |
29:53 | to the power of one . So you can see | |
29:55 | right away . This is going to give you a | |
29:56 | value of one half . So it has to be | |
29:58 | lower . And then this guy , I'm running out | |
30:00 | of space here , it's gonna be one half to | |
30:03 | the power of to which is gonna be 1/4 . | |
30:07 | So it's gonna be 14 so it's going to get | |
30:09 | smaller and smaller . And then this guy you can | |
30:11 | see it's gonna be uh one half to the third | |
30:16 | power . So two times two times two is 1/8 | |
30:19 | whoops , 1/8 . So it's gonna get smaller . | |
30:22 | And then over here you're gonna have 1/16 . And | |
30:26 | then over here you're gonna have 1/32 . Okay , | |
30:30 | now let's go the other direction , we'll pick another | |
30:32 | color for this and go this way . Now take | |
30:34 | a look at what happens right here . Um Right | |
30:36 | here we have 1/2 To the -1 . Power to | |
30:42 | the negative one . Power means it's the same thing | |
30:44 | as 1/1 half to the one power . But I | |
30:48 | can flip this thing over and then that means this | |
30:50 | value is going to give me a two . So | |
30:52 | this side actually pops up higher than the one . | |
30:54 | Okay , here it's gonna be one half to the | |
30:58 | power of negative two , and it's going to be | |
30:59 | the same kind of thing , 1/1 half squared . | |
31:05 | So this is gonna be 1/4 . And when you | |
31:07 | flip this guy over , when you do the division | |
31:09 | one divided by means you flip it over , you're | |
31:10 | gonna get a four . Same thing is going to | |
31:13 | happen here except you're gonna get an eight and right | |
31:16 | here you're gonna get a 16 and right here you're | |
31:18 | going to get that 32 . So you see the | |
31:21 | numbers the 32 16 , 842 is are the same | |
31:24 | numbers on the bottom here , which are the exact | |
31:27 | same numbers we got here . So it literally is | |
31:30 | the mirror image here . The positive values of X | |
31:32 | gave me large values and here they are over here | |
31:35 | when the base is one over the base we had | |
31:37 | there the numbers are the same . It's just that | |
31:40 | the larger side is on the left and the fractional | |
31:42 | side is on the right . So it literally is | |
31:44 | like taking that function and reflecting it to the other | |
31:47 | side . Like this . It's an exact mirror image | |
31:50 | because this base is just one over this base . | |
31:53 | But of course , if I change this base so | |
31:56 | that it's different than one half , then I'm going | |
31:59 | to of course have a slightly different curve there . | |
32:01 | And as I play around with that base , uh | |
32:04 | making it closer to zero and closer to the number | |
32:07 | one . Then I'm going to change and make it | |
32:09 | steeper steeper or less steep . And I do want | |
32:11 | to take just a second to show you what that | |
32:14 | would look like . All right . So what I | |
32:16 | want to do is just do a little bit of | |
32:18 | a comparison to see how this curve changes as we | |
32:21 | change the base . We saw that in the computer | |
32:23 | demo also . So , let's start out by coming | |
32:26 | over here and taking a look at what happens if | |
32:28 | we change . Well , let's go in and draw | |
32:30 | what we have actually , let's say we have one | |
32:32 | half to the power of X . This is what | |
32:34 | we just drew . I'm just going to kind of | |
32:35 | re sketch it over here as a point of comparison | |
32:38 | like this . We know this function is going to | |
32:40 | go through the number one comma zero , uh sorry | |
32:43 | , zero comma one . And we know it's gonna | |
32:45 | be pretty gradual in general . It's gonna go start | |
32:48 | out something like this . It's gonna go through this | |
32:51 | point as close as I can get it . Something | |
32:53 | like this . Now , let's pick a point of | |
32:55 | comparison . Let's pick the point negative three out here | |
32:58 | . Which is going to intersect something like this . | |
33:01 | What is this point gonna look like It's going to | |
33:03 | be when we do uh one half to the negative | |
33:07 | three power , Right ? It's one half to the | |
33:09 | negative three power . We go over here and we | |
33:11 | see we're going to get an eight . How do | |
33:13 | we know that ? One half to the negative three | |
33:15 | power ? It's gonna be 1/1 half cube . So | |
33:18 | the thing is really big and what we're gonna get | |
33:21 | is eight . We just saw that here . So | |
33:23 | we see that it's higher . Now let's do the | |
33:25 | exact same curve . But let's change it so that | |
33:29 | we say F of X Is equal to change the | |
33:33 | base to make it even smaller . 1/10 to the | |
33:36 | power of X . Kind of draw a little dividing | |
33:38 | line right here . So I'm comparing as to what | |
33:40 | happens over here . What's going to happen whenever we | |
33:43 | make the base really , really tiny . Close to | |
33:44 | zero ? Well , we're gonna show in just a | |
33:48 | second what's going to happen is this function is going | |
33:51 | to be really , really steep , but still going | |
33:54 | to go through this point and it's going to get | |
33:55 | like this . You see how this one's a lot | |
33:57 | steeper . I may not have drawn it perfectly , | |
33:59 | but this one's supposed to be a lot steeper . | |
34:01 | Why is it gonna be steeper ? Because if I | |
34:03 | go to the same place , negative three , what's | |
34:07 | this point gonna look like ? It's gonna be 1/10 | |
34:10 | to the negative three , Which is one over 1/10 | |
34:15 | to the power of three . Which this is going | |
34:17 | to be won over 10 to the power of three | |
34:20 | is gonna be 1000 and then one over . Uh | |
34:24 | Or I should say , let's just do it like | |
34:26 | this 1/1 over 1000 And that's going to equal 1000 | |
34:32 | . You see it's one over the fraction one over | |
34:35 | 1000 . You flip this guy over , you're gonna | |
34:36 | get 1000 . So this value right here is actually | |
34:39 | way up here at 1000 . Now of course I | |
34:42 | didn't draw it right . This thing is actually much | |
34:45 | , much steeper . It should be going really , | |
34:46 | really low and getting really really really steep when you | |
34:49 | make the thing uh so small like this . So | |
34:52 | the bottom line is that as you start off with | |
34:56 | a base very very close to zero , you have | |
34:58 | a really steep curve . Really steep as you make | |
35:02 | the base bigger and bigger and bigger closer to one | |
35:04 | . This bends down so that it's getting closer and | |
35:07 | closer . When you make the base equal to one | |
35:10 | . Exactly , you have a flat curve that's not | |
35:13 | even really an exponential function . But then as you | |
35:16 | increase the base past one , so you get to | |
35:18 | to it gets a little steeper . And then when | |
35:20 | you get even higher gets a little steeper . This | |
35:22 | behavior of starting high getting flat and then going up | |
35:26 | again is what we saw in the computer demo . | |
35:28 | And that's why it's because when you have these negative | |
35:31 | values here for X and you're putting them in here | |
35:34 | , then you have to do the fraction arithmetic to | |
35:36 | flip it upside down . And that's why it gets | |
35:37 | getting bigger on this one side here . It's extremely | |
35:41 | confusing when you first learned an exponential function , why | |
35:44 | is it getting steeper when the base is a fraction | |
35:47 | on the on the opposite side of when it's getting | |
35:50 | steep when the base is bigger than one ? Okay | |
35:53 | , so now that we have all of that background | |
35:55 | in place , Mhm . We can finally write down | |
35:59 | what you'll probably see in your textbook this is an | |
36:03 | exponential function , exponential function . And in general , | |
36:10 | you know , we've already talked about this but I | |
36:13 | want to put it all in one place . The | |
36:14 | exponential function is the following . It is a function | |
36:18 | F of X is equal to some base to the | |
36:22 | power of X , right ? With be greater than | |
36:29 | zero . The base has to be greater than zero | |
36:31 | and B is not equal to one . This is | |
36:35 | exactly what I told you in the beginning , I | |
36:36 | told you the base can't be zero and the base | |
36:39 | can't be one , but the base must be positive | |
36:42 | . Okay , so that's all I'm saying is how | |
36:45 | you would typically see it in a book . It'll | |
36:47 | be exponential function with B to the power of X | |
36:49 | . B has to be greater than zero . That | |
36:51 | means be can't be zero , but it cannot be | |
36:53 | equal to one . Any other number is fair game | |
36:55 | . And then you have different cases , right ? | |
36:58 | And we've already written all this down , but I'm | |
36:59 | putting it all in one place if B is greater | |
37:04 | than zero , but less than one . This means | |
37:06 | it's a fraction somewhere between zero and one than in | |
37:09 | general . The exponential function is going to look something | |
37:12 | like this , It's going to go through the .001 | |
37:17 | on the left hand side , right through this point | |
37:19 | , something like this . Okay . If uh B | |
37:26 | is exactly equal to one , which you can't really | |
37:29 | have , but I'm just putting it here for completeness | |
37:31 | . So you can see what it actually in your | |
37:33 | mind , you can see what would happen . Is | |
37:35 | the line is not an exponential function at all , | |
37:37 | it's just flat . So you can't really have this | |
37:39 | is not an exponential function . I'm just putting it | |
37:41 | here to show you what would happen if it is | |
37:42 | equal to one . And then finally , if B | |
37:47 | is greater than one , what you get is that | |
37:50 | standard exponential function that we have already talked about , | |
37:53 | where it goes through zero comma , one starts off | |
37:56 | really , really small , goes through this point and | |
37:59 | rapidly goes up high like this bigger than one . | |
38:03 | If you have a base of one , it's not | |
38:07 | really an exponential function . If you have a base | |
38:08 | of zero , it's not an exponential function . So | |
38:11 | that's why those constraints are there . Now , The | |
38:13 | one point of curiosity that most people have is what | |
38:16 | happens if the base is negative ? We said the | |
38:19 | base has to be bigger than zero . We understand | |
38:21 | zero to the power of whatever is not gonna be | |
38:23 | an exponential function . But we said it has to | |
38:24 | be bigger than zero and it has to not be | |
38:26 | one that all makes total sense . What happens if | |
38:29 | we put a value a negative value for the base | |
38:32 | into this equation ? The answer is it's not an | |
38:35 | exponential function and I want to tell you just a | |
38:38 | second why it's not an exponential function . Let me | |
38:41 | go find some room . I think over here let's | |
38:45 | say the base is negative . Let's just pretend you | |
38:49 | cannot have . This is not going to be an | |
38:51 | exponential function , but what's going to happen ? Let's | |
38:53 | say F of X is equal to let's just say | |
38:57 | negative two to the power of X . I mean | |
39:00 | , at first glance it looks like it would be | |
39:01 | an exponential function . But think about it for a | |
39:04 | second . What's going to happen if you put a | |
39:07 | value of one here , Let's just pretend let's say | |
39:11 | f of one . Okay , that's gonna be negative | |
39:14 | two to the first power . So F of one | |
39:18 | is going to be equal to negative two . All | |
39:20 | right , so so far so good . Let's say | |
39:22 | we go up to the next guy and say let's | |
39:24 | put a value of X . Is equal to two | |
39:26 | in here . Two . Okay . And that means | |
39:30 | that we have negative two to the second power . | |
39:34 | Now , you see because negative times negative is positive | |
39:37 | . This is actually going to be a positive for | |
39:41 | All right . Now let's do one more . What | |
39:42 | if we go up one more and say F . | |
39:44 | Of three ? It's gonna be negative two to the | |
39:47 | three power . So F of three is gonna be | |
39:50 | negative two times negative two times negative two . So | |
39:53 | negative two times negative is positive for positive . Four | |
39:56 | times negative two is actually negative . Eight . You | |
39:59 | see what's going on here as we as we put | |
40:02 | different values of action for our regular exponential function , | |
40:05 | it's a very smooth increasing value that just goes up | |
40:08 | and up and up . But here when the base | |
40:10 | is negative you have this oscillation happening here . We | |
40:14 | just had when we put a value of of one | |
40:17 | in we got a negative value out but then we | |
40:20 | put a value of two in and we got a | |
40:21 | positive value of out . But then we put a | |
40:23 | value of three in and we got another negative value | |
40:25 | out . So it's like it's going positive negative positive | |
40:28 | negative . So what's gonna happen is it's gonna get | |
40:30 | bigger . 248 16 32 but positive negative positive negative | |
40:36 | positive negative , it's going to oscillate and that is | |
40:38 | not an exponential function . So that's why when we | |
40:41 | say an exponential function is when the base is greater | |
40:44 | than zero but it can't be equal to one . | |
40:47 | The reason we're saying that is because you put negative | |
40:49 | bases in here , you can calculate the answers but | |
40:51 | it's going to have this oscillation effect up and down | |
40:54 | bouncing and it's not gonna be a smooth exponential function | |
40:57 | . So let's take just a second to go . | |
40:59 | Take a look at that in a computer just to | |
41:00 | satisfy your curiosity to see what that actually looks like | |
41:04 | . Hello , Welcome back . So here's what we're | |
41:06 | going to do is see what happens when the base | |
41:08 | is less than zero . Now , first , let's | |
41:10 | remind ourselves what happens when we increase the base . | |
41:13 | This is a positive base getting bigger , bigger , | |
41:15 | bigger , bigger , bigger . We have this nice | |
41:17 | exponential function going really , really , really steep as | |
41:20 | we get down here to where the base is now | |
41:22 | getting close to the number one , it gets very | |
41:24 | flat . The scale is a little different here , | |
41:26 | so it gets very flat . And then when we | |
41:28 | get to fractions , we see that the exponential function | |
41:31 | gets larger on the other side , mirror image like | |
41:34 | we just described . Now , this is .2 to | |
41:36 | the power of X . Let's still upside gave the | |
41:39 | punchline away , let's go and let it get to | |
41:42 | where we have a negative value in here for this | |
41:44 | base . It gets very , very high . And | |
41:47 | then if I can tweak it and get a negative | |
41:49 | value here , we have this oscillation kind of effect | |
41:52 | . You see how these oscillations are happening , We're | |
41:54 | bouncing positive negative , it's still getting bigger . It's | |
41:57 | just the absolute value is getting positive , negative , | |
41:59 | positive negative . And if we crank it up even | |
42:02 | more than that , we're going to see this thing | |
42:03 | go out of control and they're gonna see an oscillation | |
42:05 | effect on the other side . So you see how | |
42:08 | we still have a mirror image kind of thing going | |
42:09 | on , where it gets bigger on one side , | |
42:11 | then bigger on the other side when the base is | |
42:13 | negative . But this oscillation effect that I just described | |
42:16 | on the board is what prevents this thing from becoming | |
42:18 | an exponential function . This is a perfectly valid function | |
42:21 | . It's just not an exponential function . So let's | |
42:24 | go ahead and wrap up the computer demo and go | |
42:26 | off and wrap up the lesson . All right , | |
42:29 | welcome back . I hope you understand now what an | |
42:31 | exponential function is and why it is so important . | |
42:34 | The bottom line is an exponential function is when we | |
42:36 | have a number raised to a power raised to an | |
42:39 | X variable like this , the number on the bottom | |
42:42 | is what we call the base . The shape of | |
42:44 | the exponential curve depends on the base and went through | |
42:46 | all of this kind of background information . Ultimately arriving | |
42:50 | at something like this . We can see that we | |
42:52 | put bigger and bigger values in for a positive base | |
42:54 | bigger than one , it gets steeper and steeper . | |
42:57 | If we change the base , the curve still goes | |
42:59 | through one , but just gets steeper . If we | |
43:02 | increase the base , we can actually see it getting | |
43:04 | steeper right here . Then if we flip it around | |
43:07 | and say , well , what happens when the basis | |
43:09 | of fraction between zero and one ? We have the | |
43:11 | exact mirror image we have of getting steep on the | |
43:13 | left and getting very small on the right . The | |
43:16 | reason it does that is because of the way fractions | |
43:18 | work with powers . When we have negative power is | |
43:21 | applied to a fraction and ends up becoming one over | |
43:23 | that fraction , which flips it over and makes it | |
43:25 | very steep . Similar reasons is why this becomes very | |
43:29 | shallow right here . But still , when we adjust | |
43:31 | this base , even if it's in between zero and | |
43:33 | one , we can change the shape of this exponential | |
43:35 | curve as we make the base closer to zero . | |
43:38 | This is much smaller than that one . It gets | |
43:40 | much much much very very very steep . As we | |
43:43 | get it closer to uh closer and closer to the | |
43:46 | number one . Then of course it bends down as | |
43:48 | we have seen it do in the the computer demo | |
43:53 | and then finally we have the exponential function , it's | |
43:55 | B to the power of X . The base has | |
43:57 | to be bigger than zero , but it cannot be | |
43:59 | equal to one . If the basis of fraction between | |
44:01 | zero and one , you're gonna in general have a | |
44:03 | sloping downward type of function that's big on the left | |
44:06 | and small on the right . If you have a | |
44:09 | base larger than one , it's the exact opposite , | |
44:11 | small on the left , big on the right . | |
44:13 | Of course the exact shape depends on the value of | |
44:15 | the base . If the base is exactly equal to | |
44:18 | one , the function is not an exponential function anymore | |
44:20 | , it's completely flat . And we talked about why | |
44:23 | that's the case and that's why you cannot have a | |
44:25 | base uh equal to one . And we just saw | |
44:27 | on the computer demo if you actually got crazy and | |
44:30 | put a negative value in for the base , then | |
44:32 | you have this kind of like oscillating beast . That's | |
44:35 | not an exponential function at all . That's why the | |
44:36 | base can be negative . So I hope you understand | |
44:39 | exponential functions . Burn these graphs into your mind please | |
44:44 | because really and truly exponential functions is one of the | |
44:47 | most important functions in all of math . We are | |
44:49 | going to come back to it again and again and | |
44:51 | again . And when you get into calculus and other | |
44:54 | advanced subjects it's going to become even more critical . | |
44:56 | So please make sure you understand this . Follow me | |
44:58 | on to the next lesson . We're going to learn | |
45:00 | how to solve equations that contain exponential functions . |
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