How fast does a ball move?
Here we are at a lecture hall at The Ohio State University. I'm Bart Snap. Today, we're going to see how we can use Calculus to study the path of an object through space, in particular this orange ball.
When the ball leaves my hand, it's going pretty fast. But as it approaches the ground, it's going faster and faster, and faster until it hits the ground, bounces up at nearly the same speed. And then it's going quickly, but slower, slower, slower, until, it reaches its pinnacle and then it stopped. And then it starts falling back down again, slowly at first, but then quicker, quicker and quicker, and then I, grab the ball. So how fast was the ball moving well lets see. The total distance the ball traveled was about, 3.14 meters and the total time the ball traveled for, was about 1.1 seconds. Dividing this, we find this is equal to 2.85 meters per second.
This is the average speed of the ball. Now, of course, I took the ball and I threw it down with some force and then it came back up and hovered here and then came back down if you recall. So, what I'm trying to say is the speed of the ball was changing the entire time and this is the average speed and so sometimes the speed was faster and sometimes it was slower. Here. So, think about when I just threw the ball down. From leaving my hand to touching the ground, that was about 2. seconds. And the distance it traveled was about one meter. One meter divided by 2. seconds is 5 meters per second. So we found the average speed to be 2.85 meters per second. But that's not really the instantaneous speed of the ball. Because when I threw down the ball, the ball was traveling much faster. And then the ball bounced up. And then it started slowing down. And then it slowed down to almost no speed at all. And then it came back into my hand going a little faster. So how do we figure out how fast the balls going at say one of those other time intervals. We found the average speed over the course of the entire trajectory the full 1.1 seconds to be 2.85 meters per second. However if we only consider the first 2. seconds we find the average speed to be five meters per second, this makes sense because if you watch the video you see the ball is moving rather quickly in the first 2. seconds of its trajectory. You probably asking yourself, how does Bart know that the ball traveled one meter during the first 2. seconds of its journey and 3.14 meters during the whole of 1.1 second journey down up and down? Well, we know this because we got the video. Alright, here Bart is just about to release the ball from his hand. And 1, 2, 3, 4, 5, 6 frames later, the ball hits the ground. There's 30 frames being shot in every second so that means that it took 6 / 30 or 2. seconds for the ball to hit the ground. Making a bunch more measurements, I can combine all this information in a graph. This is a graph of a function the input to this function is time so along the x axis I'm plotting time and seconds. The output to the function is the height the height of the ball at that particular moment in time. Now, on this graph, I can go back now and try to figure out how fast the ball is moving at say 4. seconds after Bart releases it and at 8. seconds after Bart releases it. So let's try to figure out how fast the ball is moving at 4. seconds. Here I've marked the position of the ball, .4 seconds after Bart releases the ball. Now we don't really have any way of figuring how fast the ball is moving at that particular moment. What I can do is figure out the average speed of the ball during some time interval. So as sort of a first guess to how fast this thing is moving at 4. seconds, I'm going to figure out the average speed of the ball between 4. seconds and 6. seconds. Alright, so I've got this handy table here, of function values at 4. seconds. The the ball was 101.1 centimeters above the ground and 6. seconds after Bart released the ball, the ball was 16.8 centimeters above the ground. So I can put that information together to figure out the speed of the ball between 4. and 6. seconds. Alright. So
4. seconds to say 6. seconds at 4. seconds on my chart the ball was a 101.1 centimeters above the ground. At 6. seconds the ball was a 161.8 centimeters above the ground. Now this time interval has a length of 2. seconds. And how far did the ball move during that time interval? Well 161.8 - 101.1 is 60.7 centimeters. So during the 2. seconds that elapsed from 4. seconds to 6. seconds after Bart released the ball, the ball traveled a distance of 60.7 centimeters, which means the speed, which is the distance traveled over time is 60.7 over 2. centimeters per second, which is 303.5 centimeters per second or 3.035 meters per second. Which is about seven miles per hour. Now I could do a little bit better, alright? Here I'm calculating the average speed of the ball between 4. and 6. seconds, but I'm trying to figure out how fast the ball is moving at this particular moment. So instead of just calculating the average speed during this time interval to be about seven miles per hour, I could do it over a shorter time interval, all right? Instead of 6. to 4,. I could go from 4. to say 5.. Here's half a second after Bart released the ball. And I could figure out the average speed of the ball during this part of its trajectory. Let's see how we calculate that. Well, it's the same kind of game. All right?.4 seconds after Bart released the ball, the ball was 101.1 centimeters above the ground. 5. seconds after Barb released the ball, looking back at my table I find that the ball was 136.5 centimeters above the ground. 136.5 centimeters. This time interval was 1. seconds long. And how far did the ball move during that time interval? Well, that's 35.4 centimeters. 136.5 minus 101.1 is 35.4. So, the ball moved 35.4 centimeters. During the point one seconds that elapsed point four seconds to 5. seconds after Bart released the ball. Speed is how far you've traveled over how long it took you so if I divide these this is the speed of the ball the average speed of the ball between point four and point five seconds and this works out to be 354 centimeters per second I mean I'm dividing by this very nice number point one. which is the same as 3.45 meters per second which is about eight miles per hour. And indeed, if you look back at this, this chart, between 4. and 6. seconds, yeah. Maybe the average speed was u, about seven miles per hour. Between 4. and 5. seconds the average speed was a little bit higher. You know, the average speed here worked out to be eight miles per hour instead of seven miles per hour. The average speed of the ball during this time interval is higher than during this whole time interval. We're still not there. We are trying to figure out how fast the ball is moving at this particular moment right not the average speed between 4. and 5. seconds. To get closer, right, we should take an even smaller time interval. Instead of 4. to 5. well, why not look back on our handy chart here and see well, here is where the ball is at 4. seconds. Here's where the ball is at 42. seconds. We could use this information to figure out the speed of the ball just during the very tiny time interval between 4. and 42. seconds after Bart releases the ball. Well, let's do that. All right. So again, .4 seconds after Bart releases the ball, the ball is 101.1 centimeters above the ground. .42 seconds after Bart releases the ball, the ball is 109 centimeters above the ground, that's what this chart is telling me .42 seconds after Bart releases the ball, 109 centimeters above the ground. So that means during the very tiny time interval 02. seconds that elapsed between 4. seconds and 42. seconds after Bart releases the ball, the ball has traveled how far well 109 - 101.1 centimeters is just 7.9 centimeters. So in two hundredths of a second the ball has traveled 7.9 centimeters. To figure out the speed I again divide 7.9 / 02. is 395 centimeters per second. Which is 3.95 meters per second. Which is about 9 miles per hour. This is a much better approximation to the instantaneous speed of the ball at 4. seconds. Look, here's the graph again. Between point four and point six seconds, the ball is travelling maybe seven miles per hour on average. Between point four and point five seconds, the ball is travelling maybe. Eight miles per hour between 4. and 42. seconds we just calculated that the speed of the ball is about nine miles per hour. And that makes a whole lot of sense. Right? The speed of the ball from here to here is slower than the average speed from here to here. Which is slower than average speed from 0.4 to 0.42 seconds. The ball's slowing down in its trajectory, so the average speed over these shorter time intervals is decreasing. So, let's figure out how fast the ball was moving at 8. seconds. That's when the ball was at the top of its trajectory. we can't really do that. All I can really do is figure out the average speed of the ball over some time interval but I've got a table of values of the function. And I know how high the ball was at 8. seconds after release. It was 182 centimeters, and I can compute its average speed over a very short time interval, like the time interval between 8. and 81. seconds after release, all right? And the ball didn't move very far during that time interval but of course that time interval also isn't very long, so it's not super clear how fast the ball might be moving on average during that time interval. We can do the calculation though. Let's do it now, so. 0.8 seconds after the ball was released, the ball was 182 centimeters above the ground. 0.81 seconds after the ball was released, the ball was 181.9 centimeters above the ground. Now this time interval between 8. and 81. seconds has the duration of just one hundredth of a second. 81. - 8. is 01.. That's a very short amount of time and during that short amount of time, how far do the ball move? Well, 181.9 - 182 centimeters, that's just.1 centimeters. And if we're being pedantic, it's negative 1. centimeters. All right? The ball fell between 8. and 81. seconds, so this number is recording not only how far it moved but also the direction that it moved in. It's really displacement instead of a distance. Anyhow, .01 centimeters divided by 01. seconds, that will give me the velocity, right? Displacement over time. So if I divide these, this ratio here is 10 centimeters per second or -10 if I am keeping track of the direction its moving in. It's falling down at a speed on average of ten centimeters per second during this time interval. That's -.1 meters per second. Which is about.2 miles per hour or -.2 miles hour if I'm keeping track of the direction it's going. Anyway, .2 miles per hour is a really slow speed, right? The ball is not moving very much on average between 8. and 81. seconds. In light of this, it might make sense to say that the instantaneous speed of the ball at.4 seconds is nine miles per hour. Now, why? Well, the average speed between 4. and 6. seconds is maybe 7 miles per hour. The average speed from 4. and 5. is about eight miles per hour. The average speed between 4. and 42. seconds is about 9 miles per hour. You know and based on this, it seems like if we took a really short time interval, just after 4. seconds and tried to calculate how fast the ball was going on average during that very small time interval, you might conclude that the average speed during a very small time interval is about 9 miles per hour. It's in that sense that we're going to say that the instantaneous speed of the ball at point four seconds is 9 miles per hour. When you play the same game 8. seconds into the balls journey, alright. When it's just to the top of its trajectory. So the average speed of the ball, between 8. and 81. seconds is exceedingly slow, and you can see that in the video, alright. The ball is barely moving, at the top of its trajectory. What's the instantaneous speed of the ball at the top of its trajectory? It's zero, right? I mean yes. The average speed over a time interval between 8. and 8000001. seconds isn't zero. But if you look at an average speed over an exceedingly small time interval, those average speeds over shorter and shorter time intervals are as close to zero as you like. That's the sense in which the instantaneous velocity at the top of the trajectory, the limit of the average velocities over small time intervals, is zero. Isn't calculus amazing? We're using the idea of limits to compute instantaneous speed. Using a little bit of math, we can understand the world around us. That's the power of calculus.
What else is there to study about functions and limits?
We've already have been thinking about functions and about limits. And now I want to think deeper into those two ideas.
In terms of functions, I want to think, what does it even mean for a function to be continues. How does that relate to limits. And what are some consequences of a function's being continuous?
And in terms of limits, what would I think about different kinds of limits now? I want to think about one side of limits. I want to know what can I say about a function's output if I'm looking at input points that are just a little bit below some point. Or if I'm just looking at input points that are a little bit above some point. It'd be a one sided limit. I want to know what happens when I evaluate a function at very large input points. That'd be a limit as x approaches infinity. I want to be able to say, maybe, that my function's output value is as large as I'd like as long as my input point is close enough to some chosen point. That'll be saying that the limit equals infinity.
What is a one-sided limit?
Up to now, we've been thinking a lot about limits. We've been considering f(x) when x is close to but not equal to a. Let's refine that a bit now. Instead of thinking about x close to a. I'm going to think about x close to the right side of a or x close to the left side of a. Super-sized definition. So we say the limit of f(x) as x approaches a from the right, I'm going to use this little + sign to mean from the right. So if this is equal to l, it means that f(x) is as close as you want to l, this is just like before, provided x is near enough a on the right-hand side. So we're going to use a little + sign to denote approaching from the right-hand side. Now we're going to play a same game for approaching from the left-hand side. So let's say the limit of f(x) as x approaches a from the left-hand side is equal to l means that f(x) is as close as you want to l just like usual, provided x is near enough a on the left-hand side. Let's go see some picture of this at the blackboard.
Really helps to see a graph. Here's a graph of a made up function that I'm called f(x). You'll note that f(x) has some issues at the input 3. There's an empty circle here and a filled-in circle here so if I plug in 3, my output valley is 2. And indeed if I plug in numbers that are just a little bit above 3, I get out numbers that are close to 2. On the other hand, if I plug numbers that are a little bit less than 3, I get out numbers that are close to 1. We can summarize our observations with these two statements. The first is that the limit of f(x) as x approaches three from the right-hand side is equal to 2. And that makes sense because I can get the output of the function to be as close to 2 as I'd like if I'm willing to evaluate the function of inputs that are close to but just a little bit bigger than 3. Similarly, the limit of f(x) as x approaches two from the left-hand side is equal to 1. We think back in the graph, I was getting outputs that are close to one if I evaluated the function inputs which are close to, but just a little bit less, than 3. So in this case, the two sided limit doesn't exist. I can't say that f(x) is getting close to anything if all I know is that x is close to 3 but the left and the right hand limits do exist. I can say that f(x) is getting close to 1. If x approaches 3 from the left-hand side, and I can say that f(x) is getting close to 2 if x approaches 3 from the right-hand side. This situation comes up quite a bit where you compute and you find that the right and the left hand limits exist but they disagree and consequently, the two side of limit doesn't exist. I can summarize it like this. If the limit of f(x) as x approaches a from the right, is different than the limit of f(x) as x approaches a from the left, then the two sided limit the limit of f(x) as x approaches a does not exist. It works the other way as well. If the limit from the right-hand side is equal to the limit from the left-hand side, let's call our common value l, then the two sided limit, the limit of f(x) as x approaches a, no + or -. So this is just the usual old limit. Then this limit exists and it's equal to that same common value, l. The homework will challenge you with many more situations of one sided limits. If you get stuck, contact us. We are here to help you succeed.
What does "continuous" mean?
Let's think back to our friend the square root function. What's the square root of two? Well, the square root of two is about 1.414. What's the square root of 2.01? It's awfully close. It's 1.417 and a bit more, which is really close to the square root of two. What's the square root of 1.99? Well, it's also really close to the square root of two. It's 1.410 and a bit more, which is really close to 1.414. The point here is that nearby inputs are producing nearby outputs. Let's try to see these numbers. What does the graph of the square root function look like? Is this the graph of the square root function? Take a look at what happens right here. Nearby inputs are not being sent to nearby outputs. Inputs very close to two. Say something a little bit less than two and something a little bit bigger than two, are being sent to outputs that are quite far apart. The numbers show that couldn't have been the graph of the square root function. But what does the graph for the square root function look like? This is what the graph for the square root function looks like. It looks continuous, it's one nice curve. In particular, nearby inputs give rise to nearby outputs. Let's try to capture the concept of continuity. A bit more precisely than just a picture on the graph. Here's sort of a moral definition of what I mean when I say f of x is continuous at a. So morally I mean that inputs near a are being sent to outputs near f of a. From this perspective, it looks like the function, F of X equals the square root of X is continuous at two because inputs near two are being sent to outputs near the square root of two. How do we make this intuition a little bit more precise? Here is a precise definition, to say that F of X is continuous at A is to say that the limit of F of X as X approaches A is equal to F of A. Now think back to what we mean by limit, to say that the limit f of x equals f of a is to say that I can make f of x as close to f of a as you like as long as x is close enough to a. But that's really the spirit of continuity. Continuity is trying to say that nearby inputs are sent to nearby outputs, and this limit statement is capturing that sense. It's saying that I can make the output close to the output at A as long as the input is close to A. This definition's pretty involved. We've got to try to unpack this a bit. What does it really mean to say that the limit of F of X equals F of A as X approaches A? To make this statement I really need to know that f of x is defined at the point a. I can't talk about f of a unless I know that a is the domain of f. Talk about the limit of f of x I also need to know that the limit of f of x as x approaches a exists. I need this to be some number so I can talk about it being equal to some other number. Well once I've got these two statements, then it makes sense to claim that the limit of F of X as X approaches A is equal to F of A. But before I can make this third and final statement, I'm really assuming that these two preceding things hold, alright. So the definition of continuity is a little bit more subtle than it seems. It's really these three parts. The function has to be defined at the point. The limit has to exist and be equal to some number, and then I can say that number, the limit is equal to the functions value. So it makes sense to talk about the function at that point. Nothing we've done so far really captures the idea that the graph is a single curve, a single continuous curve. We've always just been working at a single point. This is the definition of continuity at the single point A. But often we want to talk about continuity on a whole interval at once. So we'll get rid of this and we'll make this solely a fancier definition. To say that the function is continuous on a whole interval from A to B is to say that for all points C in between A and B so C is bigger than A and less than B, so C is in the interval A to B. Then F of X is continuous at that point C. So this is what we mean when we talk about continuity on a whole interval at once. So that's the definition for open intervals. What about closed intervals? We have to be even a bit more careful when we talk about continuity on a closed interval, A B as opposed to the open interval A B. So if we say that the function is continuous on the closed interval from A to B. We mean that F of X is continuous on the open interval from A to B so in between A and B. But then what happens at A and at B? Well, the limit of F of X as X approaches A from the right hand side, is equal to F of A. And the limit of F of X as X approaches B from the left hand side is equal to F of B.
What is the intermediate value theorem?
So far, we've been selling continuity as something about nearness. Continuous functions preserve nearness, so nearby points gets into nearby points. But continuity isn't just about how small changes become small changes. Continuity has consequences for the global structure of the function as well. Here's one of them. I've graphed just some random looking continuous function. And I've picked a couple input points, input point a, input point b. Here is the point a, f of a, here's the point b, f of b. And on the y-axis I've got the value f of a and the value f of b. Now, in between f of a and f of b, I've just picked some random value. I'm calling it y. Here's a consequence of continuity. There has to be some corresponding input x so that if I plug in x, I get out y. This is the so-called intermediate value theorem. Let me write down a more precise definition now. So here's a statement of the intermediate value theorem. Theorem says the following. Suppose f of x is continuous in the closed interval between a and b, and that y is some point between f of a and f of b. Then, there's an x between a and b so that the function's output, when you plug in x, is equal to y. Here's a real world example, or if you like, a real world non-example of the intermediate value theorem. Is it possible for me to be standing here in one moment and over here in the next moment without actually occupying the points in between? What does the intermediate value theorem say? My position is a continuous function of time. So, if I'm standing here at say, time t equals zero seconds, and I'm over here at say, time equals three seconds, mustn't there be a time when I'm standing, say, right here? Yeah, maybe it's a time t equals two seconds. Who knows? What the intermediate value theorem says is that for a continuous function, all of the intermediate values are actually achieved at some point along the way.
How can I approximate root two?
So, I get a number like the square root of two. And square root of two is irrational. So, how am I going to figure out what it's approximately equal to? Well, it turns out that it's approximately, you know, 1.414 and a bit more. But how would ever figure that out? How would I know that that's approximately the value of the square root of two? Let's use the intermediate value theorem to try to pull it. I want to use the intermediate value theorem to approximate the square root of two. To do that, I'm going to use this function, f(x) = x^2 - 2. Notice something about this function. This function is equal to zero when x is the square root of 2. And how is that? f of the square root of 2 is the squared of two squared, which is two, minus two, which is zero. So, if I'm trying to approximate the square root of 2, what I'm going to think about this, is that I'm going to look for a positive value, that I can plug into this function to make it a square root of zero. How am I going to find such a value? You know, the other good thing about this function is that it's continuous. And then, I can plug in some values, like f(1). Now, what's f(1)? It's one squared minus two and that's minus one. And I can plot that on my graph right here, and it's this point right here. And take a look at, say, f(2). f(2) would be two squared minus two, which is two. And I can plot that on my graph, right here. It's 2, 2. Now, f is a continuous function and at one, it's negative.
And at two, it's positive. So, it's a continuous function. It's negative at one and it's positive at two. By the intermediate value theorem, there must be a point in between when the functions value is equal to zero. And you can see that on the graph. We're looking for that point. That is the square root of two where this graph crosses the x-axis, the positive x-axis. And that's the point that we're looking for, right? The intermediate value theorem promises me there's a value in between where the functions value equals zero. And I can see that there must be such a point on the graph. And now, I know that that value is between one and two. And I can do better. I can cut this interval between one and two in half and I get that f(1.5). Well, 1.5 squared is 2.25 - 2 is 0.25. Alright. So, that's, say, this point on my graph. Now, what do I know? I know that the squared of two, which is where this graph crosses the x-axis, is between one, where the function is negative and 1.5, where the function is positive. And I can do better, alright? I can pick some other point. Let's be brave and pick 1.4. And if I plug in 1.4, well, 14 squared is 196. So, 1.4 squared is 1.96 - 2, that makes this -0.4, alright? This is barely below the x-axis. And I can plot that point here. And now, I know that the square root of two, the positive input to this function, which is equal to zero, well, it must be between 1.4 and 1.5. Because the function's value at 1.4 is negative and the function's value at 1.5 is positive. So, this is not so bad. I mean, we, we could do this with a few calculations on the blackboard. Let's, let's bring out some sort of computation device so we can try to get an even better approximation to the square root of two. So here, I've got a function. The function is f(x) = x^2 - 2. And you can see that if I plug zero into the function, I get out -2 because 0^2 - 2 is -2. Now, this is a physical function. I've got this knob here. And as I spin this knob, the function's input value changes. So, for example, if I spin it up to f(05.), f(1/2), that's negative 1.75 and that's right because if I take a half squared, I get a quarter, and a quarter minus two is -1.75. Now, I could spin the input up to one. f(1) is -1 = because 1^2 - 2 is -1 or f(1.5), well, 1.5 squared is 2.25, 2.25 - 1 is 0.25. Now, this is positive or I could wheel it back down a little bit. And I see that when I get to 1.41, the function's output is negative again. Then, I could increase the function's input a bit. Oh, now the function's output is positive again, at f(1.145). And then, I could decrease the value a little bit. Oh, now, the function's output value is negative again and that would increase the output value a little bit. Oh, now, the function's output is positive and I could decrease the input a bit.
Now, it's positive, could decrease the input a little bit. Now, it's negative again. Now, Look at what happened, right? What's going on? I'm looking at the function f(x) = x^2 - 2 and I'm walking back and forth across the point where this function is equal to zero. And as I walk to the left and to the right, the function is positive and the function is negative. And I'm getting closer and closer to a, to the actual place where the function is equal to zero and what is 1.41421356? It's awfully close to the square root of two. And if you think the square root of two and square it and subtract two, you get zero. What we just saw with the knob that I was turning is actually an incredibly general technique for figuring out where a continuous function crosses the x-axis. And we can do it very efficiently. Let's take a look at how we might do this a little bit more carefully. Here, I've drawn a graph with some random looking continuous function and I've labeled some points along the x-axis. Imagine that we couldn't see this whole graph. Just imagine that the only thing you can do is interrogate this function. You can ask this function at some input point, are you positive or negative? We're trying to figure out where, approximately, this continuous function crosses the x-axis. We're looking for a zero of this continuous function. How might the game begin? I'll let it begin by asking the function, are you positive or negative at zero? And the function is positive at zero. And then, maybe I'd ask at sixteen. At sixteen, is this function positive or negative? And I'd see the function is negative. The function's value is negative at sixteen. Because it's a continuous function and the function's value at zero is positive and the function's value at sixteen is negative, I know that there's a zero in between. There's some point in between that if I plug it into the function, I get zero out. Now, I could cut that interval in half. I could plug in eight and I could ask the function at f(8), are you positive or negative? And the function f(8), the value of the function at eight is positive. Now, look. I know the function's value at eight is positive, the function's value at sixteen is negative. So, by the Intermediate Value Theorem, there must be a value in between, right there, where the function's value is zero. And now, I could take that interval and cut it in half again. I could take a look at twelve, which is exactly between eight and sixteen. And I could ask the function, at f(12),12), are you positive or negative? And the function's value at twelve is negative. So now, I know by the intermediate Value Theorem, the function zero zero of this function must land between eight and twelve and I could cut that interval in half again. I could look at ten, and I could ask the function, well, if I plug in ten, are you positive or negative? And the function's value with ten is negative. And now, I know that the zero must land between eight and ten, because the function's value at eight is positive and the function's value at ten is negative, so there must be a zero in between, a point where the function's value is equal to zero. Now, I could cut that interval between eight and ten in half again. I could look at nine. And I could ask the function, at nine, are you positive or negative? And if I plug in nine, it looks like the function is positive. And that means that a value that I can plug into the function to make the output zero lies between nine and ten. And this is really quite remarkable. I, I, I could cut the interval between nine and ten and half again and see if it was between nine and 9.5 or 9.5 and ten. And I could keep doing this every time I'm getting a little bit more precision as to the exact value of the input that would make the function zero. The other really neat thing about this is that I could have just tried all of the values between zero and sixteen. I could have plugged all seventeen of those numbers into the function to figure this out. But by doing this clever cutting in half method, this bisection method, I managed to only ask six questions, right? I asked the function's value at zero and sixteen, at eight and twelve, at ten and at nine, and by only asking six questions, I was able to narrow down a region where a zero resided. Now, that's pretty quite efficient, much better than asking seventeen questions. So, this is a very general technique for figuring out where zeroes of continuous functions lie. And I encourage you to pick your favorite function and see if you can approximate a root, a zero of that function.
Why is there an x so that f(x) = x?
Here's an application of the intermediate value theorem I'm very fond of. Here's how it goes. Suppose you've got a function f. f is continuous in the closed interval between zero and one. And whenever x is between zero and one, f of x is between zero and one. So that's what you've got to suppose when you get out of this. Then, there is a point x, x between zero and one, and f of x = x. It's telling you that there exists an input. So that if you apply f to that input value, you get the output which is the same as the thing you plugged in. You call these things fixed points. You imagine that f is moving around the points between zero and one, and you're finding a point x which doesn't move a fixed point. Why is something like this true? Let's see. Let me use the Intermediate Value Theorem but I'm not going to apply the Intermediate Value Theorem to f but I'm going to make up a new function. g(x) = f(x) - x. What do I know about g? g's continuous. f is continuous by assumption. The identity function x is continuous and differences of continuous functions are continuous. So, g is a continuous function and the closed interval 01. I also know some of the values of g. What's g of zero? Well, g(0) is by definition f(0) - 0. f(0) is between zero and one. So I've got a number between zero and one minus zero, that number's at least zero. I also know something about g(1). g(1) is by definition f(1) - 1 f(1) is between zero and one, it could be zero, could be one. But, a number between zero and one minus one is less than or equal to zero. What do I know altogether? I've got a continuous function g, its value at zero is bigger than equal to zero. Its value at one is less than equal to zero. By the intermediate value theorem, this gives me a point x so that g(x) = 0. Why do I care about finding a point x so that g(x) is equal to zero? Well, if I take a look at what g(x) is equal to, g(x) is f(x) - x. So, I found a point x so that f(x) - x is equal to zero. I found a point x so that f(x) = x, I found the fixed point. So, we've seen why this is true. Let'd try to apply this to a specific example life f(x) equals cosine x.
Now, here I've got a graph of cosine and that the graph of y equals cosine x. Now, what do I know about cosine? I know cosine's continuous and cosine of a number between zero and one is a number between zero and one. So, our statement applies to it. There's a fixed point for cosine. There's some value of x between zero and one so cosine of x equals x. Let's see if we can find that on a computer.
Okay. I've got cosine loaded up on to my computer. Cosine of one is about 0.54. Let me decrease the input. And I notice that cosine of 0.68 is 0.77. Here the input is smaller than the output. Let me increase the input. Now, notice what happened. Now the input is 0.74 but the output is smaller than 0.74. Cosine of 0.74 is just 0.73. Now, it decreased the input. Oh, the opposite thing happened. Now the input is smaller than the output. Now, I'll increase the input. I'll decrease the input.
Increase the input. And you can see that I, I can sort of zoom in or narrow in on the correct value of of x so that cosine of x is equal to x. Sees in the computer I found that this point here is at about 0.739. So, without a number between zero and one, cosine of that number is itself. It's a fixed point for cosine. These kinds of fixed point theorem statements that assert the existence of fixed points, they pervade mathematics. Here's another example. I've got a map of Ohio right here. Let me apply a continuous function to this map. I'm going to throw it down to the ground, stomp on it. And now let's take a look at it.
What does lim f(x) = infinity mean?
People have thought, people have wondered about infinity for thousands of years. We're going to continue that tradition of thinking about infinity, but we're going to focus in on one specific instance of infinity in mathematics. What does it mean to say that the limit of a function equals infinity? Here's what it means. The limit of F of X, as X approaches A equals infinity means that F of X is as large as you like. Provided x is close enough to A. As a bit of a warning, I'm not saying that this number limit F of X, X approaches A is equal to this number, infinity is not a number. What I am doing here is attaching a precise meaning to the entire statement that the limit of F of X is infinity. So, that's what it means precisely, but what does it mean practically. Let's go to the board. This is an example, the limit of one over X squared as X approaches zero. Now this limit is not equal to a number. One over X squared is not getting close to a number when X is approaches zero. But one over X squared is as big as I want it to be if I'm willing to put X close to zero. For instance, if X is within a 1000th of zero, what happens? Well to say X is within a 1000th of zero is to say that X is between minus.00 and.001. Now, X is also non zero here because I don't want to divide by zero. So I've got a non zero number within a 1000th of zero. What does that mean about X squared? . Well that means that X squared is, it's bigger than zero because if I square any non zero number, it's positive and it's less than, a millionth. So if I take a number within a thousandth of zero and I square it, that was within a millionth of zero. And what does that mean about one over X squared? Well if X squared is within a millionth of zero, one over X squared is now bigger, than a million. What happened, right? I made one over x squared very large by making x close enough but not equal to zero. And there's nothing particularly special about these numbers. If I wanted one over x squared to be bigger than a trillion, I would just need x to be within a millionth of zero. This example is at once maybe too complicated because I've got some explicit numbers in here. And maybe this formula is not complicated enough to, to really get a sense of what's going on. Let's look at a slightly more complicated function.
So what's the limit of X plus two over X squared minus 2x plus one as X approaches one. And this is a limit of a quotient. So your first temptation is to think the limit of a quotient the quotients the limits. But what's the limit of the denominator? The denominator is a polynomial. Polynomials are continuous. So I can evaluate the limit of the denominator just by plugging in one. And if I plug in one, I get one minus two plus one. That's zero. The limit of the denominator is zero. So I can't simply say that the limit of the quotient is the quotient of the limits in this case. Because the limit of the denominator is zero. And that limit law does not apply in this case. What am I going to do? Well I could also notice the denominator factors. X squared minus 2X plus one, I can rewrite that. X squared minus 2X plus one is X minus one squared. So instead of evaluating this limit, I could try to look at this limit, the limit of X plus two over X minus one squared as X approaches one. Now what do I know? The numerator is X plus two and if X is close to one, I can make X plus two close to three. What about the denominator? A number close to one, minus one squared, is a number close to zero. But not just a number close to zero. It's a positive number close to zero. So let me write this down. Alright. The numerator is about three. And the denominator is about what? Well, it's some small but positive number. . And what happens if I take a number, like, near three, and divide it by a number which is small and positive? That number can be as big as I like. So I can make this quantity as large as I like, if I make x close enough to one. Because I can make the numerator close to three, and I can make the denominator as close to zero and positive as I like. And if I take a number close to three and divide it by a number close to zero, I can make a number as large as I want. So this limit is equal to infinity. The homework includes even trickier situations. For instance, limit problems where x approaches negative infinity, instead of infinity. If you get stuck I encourage you to contact us. We're here to help you.
What is the limit f(x) as x approaches infinity?
. A lot of calculus is about understanding the qualitative features of functions. Nobody really cares about f of 17. But you might care about f of a big number, qualitatively. We're going to try to make this concept of big number a bit more precise. Well, here's how we're going to get around talking about a big number. We're going to talk about the limit. The limit of f of x as x approaches infinity equals to l. Means that f of x is as close as you want it to be to l, provided x is large enough. So instead of talking about just plugging in a big number, I'm going to say F at some big number, so to speak, is equal to L if I can make the output of F close to L by plugging in big enough numbers. At this point we've seen a bunch of different definitions of limits. But they're all united by a common theme. What if someone had asked us to cook up a definition of the limit of F of X equals infinity as X goes to infinity? Could we come up with a definition for this? Yeah, absolutely we can, here we go. The limit of F of X as X approaches infinity equals infinity means that I can make F of X as large as you want it to be provided X is large enough. Consistently when we're talking about infinity in limits we're never actually talking about a specific value. We're just talking about a value which is as big as you want it to be. Let's go do an example at the blackboard. There is a question. With the limit, of 2x over x+1 as x approaches infinity. Before we dive into this analytically, lets get some numeric evidence. This is my function, again fx)= of 2x over x+1.1. I want to know qualitatively, what happens when I plug in big numbers in this function? So, lets say f100). of 100. Well, that's not too hard to figure out. 22 times 100 is 200 and 100 plus one is 101. Now, 200 divided by 101 is pretty close to 2. And there's nothing too special about 100, right? If I'd done this four million I would have gotten two million over a million in one, which would be even closer to 2. Numerically, it looks like this limit's two, but we just figured that out by plugging in some big numbers. I want a more rigorous argument, some analytic that is limit is actually equal to two. How I'm going to proceed? My first guess would be to use the limit law for quotients. This is the limit of a quotient which is the quotient of limits provided the limits exist and the limit of the denominator is none zero. Bad news here is the limits don't exist. The limit of the numerator is infinity which isn't a number. So I can't use my limit law for quotients here. Instead I'm going to sneak up on this limit problem by wearing a disguise. I'm going to multiply by a disguised version of one. I'm going to multiply by one over x divided by one over x. Now this is just one. Admittedly, I have changed the function. The function's not defined anymore at zero. But for large values of X, this doesn't affect anything. And I'm taking the limit as X goes to infinity. So I only care about agreement at large values of X. I'll do some algebra. This limit has now the limit of 2X times one over X. Which is two divided by the limit of X plus one times one over X. Which is one plus one over X. Maybe it doesn't look like I've made a lot of progress here, but this is a huge progress. The limit of the numerator is now a number, it's two. And the limit of the denominator is also a number. And a non zero number, at that. So I can use my limit law for quotients. This is the limit of a quotient, which is the quotient of a limit. It's the limit of two, the numerator divided by the limit of the denominator, as x approaches infinity. The limit of the numerator is just the limit of a constant, which is 2. The limit of the denominator is a limit of a sum, which is the sum of the limits provided the limits exist, and they do. The limit of 1 is just 1. And what's the limit of 1 over x as x approaches infinity? Well that's asking, what is 1 over x close to when x is very large? Well I can make 1 over x as close to zero as I like if I'm willing to make x large enough. So the limit of 1 over x as x approaches infinity is zero. That means my original limit is 2 over 1 plus zero, which is 2.
Why is infinity not a number?
Infinity is not a number like seventeen. Don't treat it as a number. Let, let me show you something that people do all of the time. Here it is. People write down the interval from zero to infinity including zero, which is fine but including infinity, which is not okay. Infinity's not a number. What people should be writing down is this. The interval from zero to infinity including zero but not including infinity. Because infinity's not a number. Why am I freaking out about this? What's so bad about thinking of infinity as if it were some number? Now, if you start trying to do arithmetic with infinity, you're liable to walk straight into a trap. Let me demonstrate for you by walking into one of these traps myself. Let's treat infinity as if it were a number. Well, what's infinity plus one in that case? I got infinitely many things and I add one more. That's the same as having infinitely many things. So, infinity plus one would equal infinity. Now, what else would I know about infinity as a number? Well, what would infinity minus infinity be equal to? This would be a number minus itself, and a number minus itself is zero. So, you're going to treat infinity as a number, you're going to be believing these two statements. You're going to believe infinity plus one is infinity and you're going to believe infinity minus infinity is equal to zero. This is how these numbers would work. This is actually very bad. Let's just look at infinity plus one minus infinity, and let's subtract infinity from both sides. And you said infinity minus infinity is equal to zero. But, what's infinity plus one minus infinity? Well, that would be infinity minus infinity plus one. Infinity minus infinity is zero. This would be zero plus one. That side would be one. So, if you're going to treat infinity like a number, you're going to end up telling me that one equals zero. That's ridiculous. The upshot here is that you can't do arithmetic with infinity. Infinity's not a number like seventeen. But something comes to save the day. We can do limits. For instance, consider this problem where I'm doing a calculation involving infinity. But instead of working with infinity directly, I'm phrasing it as a limit question. What's the limit of x * x - 1 as x approaches infinity? Well, x is as large as I like.
By making x big enough, I can make x as big as I like. x - 1 is also as large as I like, by making x big enough, I can make x - 1 as big as I like. And what happens if I multiply together two numbers which are as large as I like? Well, the product of two numbers that are as big as I want them to be can be as big as I want them to be. So, in that sense, the limit of x * x - 1 as x approaches infinity is equal to infinity. So, what about our original example? What's infinity minus infinity? Well, it depends. Here's one possibility. Let's consider the limit of x^2 - x as x approaches infinity. Now, this is a limit of a difference. You might remember back, the limit of a difference is a difference of the limits provided the limits exist. Ignoring that last part about whether the limits exist, you might just blindly start writing down the limit as x approaches infinity of x^2 minus the limit of x as x approaches infinity. This is no good, right? The limit of x^2 as x approaches infinity, that's equal to infinity. And the limit of x as x approaches infinity, that's infinity, right? I can make x^2 as large as I like if x is big enough, and I can make x as large as I like if x is big enough. So, what just happened? I'm running the limit of a difference as the difference of the limits, but these limits don't exist, right? They're equal to infinity and infinity is not a number. Now, I'm left with infinity minus infinity. I don't know what to do, right? Who knows what that's equal to? That's exactly the sort of thing I'm not permitted to think about. Instead, if I wanted to know the limit of x^2 - x as x approaches infinity, I could rewrite this limit as the limit as x goes to infinity of, what's another way of writing x^2 - x? Could write it as x * x - 1. And we saw just a minute ago that the limit of x * x - 1 as x approaches infinity is equal to infinity. So, in this case, it seems that infinity minus infinity is infinity. On the other hand, infinity minus infinity could be seventeen. Let's see how. Here again, I've got a limit of a difference of two things. Something minus something as x approaches infinity. Now, if I blindly just apply the limit of a difference as the difference of limits without remembering that that's only valid if the limits exist, let's see what happens. And then, I get the limit as x approaches infinity of the first thing, which is x + 17, minus the limit of the second thing which is just x. Now, what's the limit of x + 17 as x approaches infinity. Well, I can make x + 17 as large as I like if x is big enough, so that's infinity. And what's the limit of x as x approaches infinity? Well, I can make x as large as I like if x is big enough, so that's also infinity. So I've again walked into the trap of writing down infinity minus infinity, right? I applied the limit of a difference equals the difference of the limits without remembering that that's only valid if the limits exist. And in this case, the limits only exist in the weak sense that I can write down that they equal infinity. Anyway, I don't want do that. Now, how could I figure out what this limit's equal to? Well, this is a limit of something and I could rearrange the something, right? This is the same as the limit as x approaches infinity. What's another way of writing this? x + 17 - x, I could have just written seventeen. Now, what the limit of seventeen is as x approaches infinity? Well, what the limit of seventeen as x approaches anything at all? That's just seventeen. So, in this admittedly contrived case, infinity minus infinity ended up equaling seventeen.
What is the difference between potential and actual infinity?
Let's consider this limit. The lim (2x-x), as X → ∞ = ∞. That's a true statement, but there's two totally different ways to think about this statement, and it really hinges on the distinction between potential and actual ∞. To have a potentially infinite pile of fish is to have an endless supply of fish, as many fish as you'd like to have, and that sense really goes will with how we're thinking about infinity in these limits. To say the limit of 2x minus x equals infinity, as x approaches infinity, is to say that I can make 2x minus x as big as I like, as long as x. Big enough. Contrast that with actual infinity. To have an actually infinite pile of fish would be to have, right now, a pile of fish that contains infinitely many fish at this very moment. Is it possible to combine that way of thinking with infinity, with these kind of limit statements? Let me share with you a fable, to see one of the paradoxes that results. Once upon a time, there was a house. I lived in that house, with my cat. And we lived, near a lake, and this lake is full of fish, so every day we went fishing. And each day I caught two fish. The first day I caught fish labelled 1 and 2. Now, my cat prefers eating the lowest number of fish in our stockpile, so my cat ate the fish numbered 1. The next day, I went fishing again, and I caught fish labelled 3 and 4. And my pet, still preferring to eat the lowest numbered fish. Eats fish number 2. Another day another fishing expedition, I go fishing again, I get 2 more fish, label 5 and 6, my cat preferring to eat the lowest number fish in our stock pile eats fish number 3. I go fishing the next day and I get 2 more fish, fish label 7 and 8, my cat preferring to eat the lowest numbered fish in our stock pile eats fish number 4. And so it goes forever. Each day our stockpile gets bigger. There's more fish in my pile every single day. And yet, at the end of time, do any fish remain. On the Nth day my cat ate the Nth fish. So which fish survives my cat's appetite? Human beings want to understand infinity, but reasoning about actual infinity is liable to walk us straight into those kinds of paradoxes. Instead, limits by focusing our attention on potential infinity provide a way to reason about infinity that avoid those kinds of paradoxes. It provides a way for mere human beings to think about infinity in a precise way.
What is the slope of a staircase?
And today, we're going to measure the slope of some staircases, namely, this staircase here and this even larger staircase. You may think that this larger staircase is steeper than this staircase because, well, this staircase is harder to climb. But let's actually find out. And we're going to find out by measuring how far the staircase goes up by how wide each step is. When I measure this step, I see that it goes over twelve inches and it rises about six and a quarter inches. On the other hand, we have this larger staircase here. Let's see what its slope is. So, when I measure, I get 36 inches for the width of each step. On the other hand, I find that the height of each step is about 18-3/4 inches. Now, that we've made the measurements, let's go compute the slope.
So, we've measured our stairs and we found out that the small steps went up by 6.25 inches and they were 12 inches wide, okay? This gives us a slope for the small steps of 6.25 all over twelve. But there were also large steps. Let's draw those in. Here we go. So, this was one of the large steps and then, we had another one. It goes off and it builds, it climbs over there. It goes down like this. This is a large step. Alright. The height of the large step was this place right here. 18.75 inches, that's the height of the large step. And the width of the large step was 36 inches. Okay, great. So, the slope of the large steps is 18.75 all over 36. These two slopes are the same, because we can simplify, we can reduce this fraction to be dividing the numerator by three and dividing the denominator by three, we get 6.25 / 12. And we can see, if we connect the tips of these stairs with a line or this makes a line of the desired slope and see how the large stairs and the small stairs all meet at the corner of this line. The fact that the staircase with the large steps and the staircase with the small steps have exactly the same slope is evidenced by the fact that they have the same railing. There's the railing for the staircase with the large steps and here's the railing for the staircase with the small steps. And they're exactly at the same angle. So, we've seen that the staircase with the larger steps and the staircase with the smaller steps actually have exactly the same slope. Now, let's use the slope to estimate how tall the staircase is after you've gone a certain distance in the horizontal direction. How high up am I now? Here, you can see that we have a tape measure where we're measuring the width or the horizontal distance that I traveled here on the staircase. And if we get to the point where I was standing, we'll find out that it's about 23 feet. Now, how high was I standing when I was up there? So, now that we've made our calculation, let's check our calculation. Okay. Now, we have a question. We went to a certain part on the staircase, say, right here. And then we measured. We, we, we, we marked, we had a point in mind and we measured the distance on the ground to that point. We found out that it was 23 feet, okay? So, if you remember, the slope of a staircase was 6.25 / 12, which is approximately 0.52. We want to know this height, x. So, we went x / 23 to be approximately 0.52.
Solve for x. Multiply both sides by 23 and we get x is approximately 0.52 times 23. 0.52 times 23 is 11.96. So, x is approximately 12 feet. Now, let's go check our answer.
So here we are, checking our measurement. And I can see, it's almost exactly twelve feet.
What is the official definition of limit?
So now we learn what the official definition of limit is. To say that the limit of f(x) = L as x approaches a, means the following. It means that for all epsilon bigger than zero, this backwards three is the real number epsilon or the Greek letter, a variable. So for all epsilon greater than zero, there's a delta greater than zero, this is the Greek letter delta. So, for all epsilon greater than zero, there's a delta greater than zero. So that if, this, if the absolute value of x - a is between zero and delta, then, the absolute value of f(x) - L is less than epsilon. When you say it like that, I think it's really hard to see how this has any relationship to what a more intuitive description of this limit statement might be. I mean, what's this trying to get at? It's trying to say f(x) is as close as I want to L by making x sufficiently close to a. So, how, how to reconcile those, those two perspectives, right? How does this have anything to do with things being close. The key, take a look at this absolute value of the difference, right? The absolute value of x - a is the distance between x and a. So to say that the distance between x and a is between zero and delta is to say that x is within delta of a, alright? The distance from x to a is less than delta. And to say that the distance between x and a is bigger than zero is just to say that the distance between x and, and a, you know, isn't zero, right, x isn't a. So I can rewrite that, maybe in a little bit easier way. So instead of saying that, it's the same thing to say, if x is not equal to a, so the absolute value of x - a isn't zero, and x is within delta of a. So the distance between x and a is delta. And I can do the same thing to this absolute value of a difference, alright? The absolute value of f(x) - L, that's the distance, between f(x) and L. And the sum of the distance between f(x) and L is less than epsilon. Well, that just means that f(x) is within epsilon of L. So, I'll rewrite that as that. Here we go. Then f(x) is within epsilon of L. So, I think when, when you write it like this, it makes a little bit more sense, right? To say that the limit of f(x) = L as x approaches a, means that for all numbers epsilon, epsilon is measuring how close I want f(x) to L, then, there's some corresponding number delta, which is how close x has to be to a. So that whenever x is that close, delta, within delta of a, then f(x) is really within epsilon of L. And to say that the limit of f(x) = L means that no matter which epsilon I choose, there's some corresponding delta, so that whenever x is within delta of a, then f(x) is within epsilon of L. Now, how this actually gets played out in, in more concrete situations can be, you know, kind of complicated, but this is really the official definition of what it means to say that the limit of f(x) equals L as x goes to a. And we're going to be trying to unpack this definition to see what it might mean in some specific cases.
Why is the limit of x^2 as x approaches 2 equal to 4?
So, what does it mean to say that the limit of x^2, as x approaches 2, is 4? Strictly speaking, it means that you can make x^2 as close as you want to 4 by making x sufficiently close to 2, but talking in that way can be a little bit confusing. So I'm going to sort of format it here as, as if it were a dialogue. Alright? So you're going to make some sort of demand. You're going to demand that x^2 be close to 4. Maybe you're going to demand that x^2 be within 1/10 of 4. Alright? So that means you're, you're asking that x^2 be between 3.9, 3.9 is a 1/10 less than 4, and 4.1, which is a 1/10 more than 4. So you're going to make some demand that the output be close to 4 and I have to satisfy your demand by making x sufficiently close to 2. Now, what does that mean? I'm going to satisfy your demand by stipulating that x be within some small distance of 2. So, how close is sufficiently close? Well, in this case, let's make let's make x be within a 1/100 of 2 and see what happens. So x is within a 1/100 of 2 and that means that x is bigger than 1.99 and smaller than 2.01. And if x is bigger than 1.99, then x^2 is bigger than 3.9601, and if x is smaller than 2.01, then x squared is smaller than 4.0401. And, and look at these numbers, 3.9601, that is bigger than 3.9, and over here, 4.0401, that's smaller than 4.1. So notice what happened here. If x is within a 1/100 of 2, then x is between 1.99 and 2.01. But if x is between 1.99 and 2.01, then x^2 is between 3.9601 and 4.0401, and if x^2 is bigger than 3.9601, it's bigger than 3.9, and if x^2 is smaller than 4.0401, it's smaller than 4.1. So that means demanding that x be within a 1/100 of 2, in fact, forces x^2 to be between 3.9 and 4.1. In other words, it forces x^2 to be within a 1/10 of 4. So if your demand is that x squared be within a 1/10 of 4, I can satisfy that demand by simply requiring x to be within a 1/100 of 2. Now, I can do the same thing for other demands that you might make. You might have demanded that x^2 be within a 1/100 of 4 and I can do that as well. So that would mean that x^2 is between 3.99 and 4.01. Well, if x^2, if you want x^2 to be between 3.99 and 4.01, I'm going to have to make some condition on how close x have to be to 2. So let's try a 1/1000. So if x is within a 1/1000 of 2, that means that X, it'll be between 1.999 and 2.001. And if x is bigger than 1.999, then x^2
is bigger than 3.996001. And if x < 2.001, then x^2 < 4.004001. Now, look at these numbers, 3.996001, that is bigger 3.99, and 4.004001, that is smaller than 4.01. So if x is within a 1/1000 of 2, that means x is between 1.999 and 2.001. That means that x^2 is between 3.996 and 4.004, a little bit more. That makes x^2 be, between 3.99 and 4.01, which is exactly what you demanded. So this is the structure of, of what it means to say that the limit of x^2 is 4 as x approaches 2. It means no matter close you demand x ^ 2 to be to 4, even if this epsilon were replaced by a very tiny number. I'd be able to find some other distance, some number to put here, so that as long as x is that close to 2, then x^2 is as close as you wanted to the number 4. So the exercise, below asks you to take a look at another situation like this to see if you can figure out number to put here to satisfy some condition that someone might make on you.
Why is the limit of 2x as x approaches 10 equal to 20?
Here's a claim. The limit of 2x as x approaches 10, is 20. Now, to justify this claim using an epsilon delta argument I have to be able to prove that delta for every epsilon, alright? You're going to demand that 2x be within epsilon of 20. I'm going to satisfy your demand by saying x is within some number delta of 10. So proof, let epsilon be bigger than zero. I don't know how small a number epsilon you're going to choose. I'm going to set delta equal epsilon over two. I'm going to check now that this value of delta will work to satisfy your demand involving epsilon. So, if zero is less than x - 10 is less than delta, right? If x is within delta of 10, then if you multiply by 2, we see that 2 * x-10 is less than 2 delta. But what's 2 delta? Delta is epsilon over 2. 2 delta is just epsilon. So then, 2 times the absolute value of x-10 is less than 2 delta, which is epsilon. And so, 2 times the absolute value of x-10, that's the absolute value of 2x - 20, and that's less than epsilon. So look at what we've shown, alright? Some epsilon bigger than zero, a corresponding value of delta is going to be epsilon over 2. And then, I verify that if x is within 10, is within delta of 10, then 2x is within epsilon of 20. And that's exactly what it means to say that the limit of 2x is 20 as x approaches 10. For every epsilon, there's some delta so that whenever x is within delta of 10, then the function 2x is within epsilon of 20. You can imagine trying to do this with other functions, right? But I'll let you try that. You can try that in the exercise included below.
What comes next? Derivatives?
A major theme in calculus is change. You've got one thing affecting something else, you want to know, how does changing this one thing affect this other thing.
Often the relationship between these two things is given by a function, right? You've got some kind of function that has inputs and outputs. And you're wiggling that input point, and you want to know how the output value view changes. Very concretely. You might ask for the ratio of change in the output valve view to change in the input point and that ratio, that's the derivative. Alright. Derivatives measure how change in the import affects the out port. At least infinitesimally. And studying derivatives is exactly what we're going to do now.
What is the definition of derivative?
We want to capture precise information about how wiggling the input effects the output. Here's the difinition of derivative that will allow us to do exactly that. The derivative of f at the point x is defined to be this limit. The limit of f(x+h)-f(x)/h as h approaches 0. Now, when this limit exists, I'm going to say the function is differentiable, alright? If the derivative of f exists at the point x, I'm going to to say the function is differentiable at that point x. Sometimes, you'll see different definitions of the derivative. Here's an equivalent one. The derivative of f at x could equivalently be defined as this limit, the limit as w approaches x of f(w)-f(x) all over w-x. Now, how does this definition relate to our original definition of derivative? The derivative of f at x is this limit involving h. Well, look. In both cases, the numerators are measuring how the output is changing. Here, I'm plugging in a nearby input and I'm looking at how much the output changed compared to the output at x. And h is measuring exactly how much that input changed by. Here, I'm again measuring the difference of two output values. Here, w is my new output value, which is close to x. Alright, down here, h is just measuring how much I wiggled x by. Here, w is actually just some nearby value of x. And in both cases, the denominator is measuring how much the input was changed. Here, h is exactly how much the input was changed by. Here, w-x is measuring how much the input changed by. The other thing that makes these definitions sometimes a little bit tricky is that people will give you a definition that's not at the point of x. For instance, here's a definition of the derivative of f at the point a. It's the limit of f(x)-f(a)/x-a as x approaches a. You can compare the first and the third definitions here. This first one has a w and an x and w is approaching x to get the derivative at x. This bottom one, I'm trying to compute the derivative at a and x is approaching a, alright? So, the roles of w and x and the role of x and a are somehow analogous here. Now, think back to when we were talking about continuity last week. We started out with a definition of continuity at a single point and then we expanded that definition to be continuity on a whole interval. We played the same game with the derivative. Here we go. If the derivative of f exists at x, whenever x is between a and b, but not at a, or at b, we won't worry about that, just whenever x is between a and b. And if this happens, then, we say that f is differentiable on the interval (a,b).
So, as a little bit of a warning here, this is not a point. This is an interval. It's all the numbers between a and b, not including a, not including b. Now, contrast this with continuity, when we talked about continuity on an interval, I also had separate definitions for continuity and closed intervals or half-open intervals. But that doesn't really make so much sense for the derivative. here's why. The derivative is measuring how much wiggling x affects f(x). And if I'm standing in the middle of an interval, I can wiggle x. Even if I'm standing pretty close to b or pretty close to a, I can always wiggle just a little bit to the left and a little bit to the right, no matter how close I'm standing to a or how close I'm standing to b, unless I'm standing, say, at the point b. If I'm standing right at b, I can wiggle to the left. But I can't wiggle at all to the right without walking right outside of the interval. So, in light of this, I don't really want to talk about differentiability on closed intervals. I only want to talk about differentiablity when I can honestly talk about wiggling the input and that's only true on open intervals. There's plenty of other subtleties to this. If I differentiate a function which is differentiable on a whole interval, then I get a new function, f'. Specifically, the derivative of f at the point x will be written like this, f with this little tick mark, and we're going to pronounce that prime, f'(x). The point here is that I can define the derivative, right, as this limit of this quotient. And when you think of it this way, if this is really a function, right? This is a rule that defines some new function, so I can regard f'(x) not just as a specific values that I get by plugging specific values of x, but honestly as a function. Alright. This thing is the sort of thing I can plug any value of x into and see what I get out. It's a function that somehow derived from f. Maybe hence, the name derivative. There's a whole bunch of different notation that you're going to see for the derivative in the wild. Here are some of these. Alright, the derivitive of f at the point x might be written as f'(x), like we've just been seeing. Or it might be written d/dxf(x) or Dxf(x), or a bunch of other things, right? Lots of different people have their favorite notation for these. But really, this f' notation and this d/dx notation are what we're going to be using in this course. There are upsides and downsides to these various choices of notation. For instance, here's a huge upside to this d/dx notation. It's really emphasizing that the derivative is a ratio, right? It looks like df, the change in the output, dx, the change in the input, somehow revealing a little bit of how the derivative is actually defined. The f'(x) notation doesn't emphasize that the derivative is a ratio, but it does emphasize the derivative is a function. When we use the f'(x) notation, at least you can tell this thing is a function. You've clearly labeled the input, x. Differentiating gives a new function, even the name suggests that. The derivative is somehow derived from the original function. And if the derivative is now a function, you can then differentiate the derivative. And differentiate that, and keep on going. And dig deeper and deeper and deeper, trying to uncover more and more secrets about the original function, f. All of that is yet to come. But the point is just that there's so much yet to explore.
What is a tangent line?
Here I've graphed some function. Notice what happens if I zoom in on one little piece of the graph, right? If I just focus on one part of the graph, well, that little bit of graph sort of looks like part of a straight line. Another way to think about this is as a limit of secant lines. By secant line, I mean, I'm going to pick two points on the graph and I'm going to put a line, the secant line, through those two points. But, I can do better. By taking this point and moving it closer to a, this red line is going to be a better approximation to the orange curve near the point a. So, instead of putting it through those two points, let me put the secant line through, say, these two points, this point at a and this point that's nearby. And now, the line segment through those two points is a much better approximation of the orange curve. And I can do better and better by taking a limit, by putting those two points closer and closer together, I can get my secant line to be a better and better approximation to the orange curve near the point a. What we're really calculating here is a limit. So here, I've got an input a and an input a+h, and here are the corresponding points on the graph of the function. I'm going to put a secant line through those points. What I want to know is what's the slope of that secant line because I'm going to take the limit as this a+h point is moved closer to a as h goes to zero in other words, and that'll make this secant line move closer and closer to the tangent line, to the curve. Okay, so what's the slope of the secant line? Well, the rise is f(a)+h-f(a). The run is h. So, the slope of that secant line is this, f(a+h)-f(a)/h.
If I take the limit as h goes to 0, I get this: the limit as h goes to 0 of this slope. We've seen this, this is the derivative of the function at the point a. Let's find the equation of the tangent line in a concrete example. Here's a graph of y=x^2. Let's figure out the equation of the tangent line to that graph at the point (3,9). So to do that, we're going to use the derivative and I know that the derivative of x^2 is 2x. So, the slope of the tangent line at the point 3 is 2*3=6, and that's the slope of the tangent line. I also know a point that the tangent line passes through. It should be passing through the point (3,9), right? The point (3,9) is on the tangent line. So, if I know the slope of the tangent line and I know a point on the line, I can write down an equation for the line using point slope form. So, y minus the y coordinate on the line is the slope time x minus the x coordinate on the line. So, this is an equation for this red tangent line.
Why is the absolute value function not differentiable?
. When I say a function is differentiable, what I really mean is that when I zoom in, the function looks like a straight line. This is the graph of the absolute value function. Let's zoom in on the origin.
No matter how much I zoom in, this graph doesn't look like a straight line. Consequently, the absolute value function is not differentiable at 0. We can also see this from the limit definition of derivative. So we just going to name the absolute value function f, for the time being. What I am trying to calculate is the derivative of f at 0. I want to know is this function differentiable at 0. By the definition the derivative is the limit. As h approaches 0, of the function, at |0 + h| - |0| / h. Now I can simplify that a bit. The absolute value of 0 + h, is just the absolute value of h, and the absolute value of 0, so I don't even need to subtract 0. And I"m dividing by h. Now what's the limit as it approaches 0 of absolute value h/h. That limit doesn't exist and consequently this function's not differential at zero. If you wonder why that limit doesn't exist, well think back to our 2 sided 1 sided limit discussion from before. What's the limit as h approaches 0 from the right-hand side of |h| / h? It's 1. Well what's the limit as h approaches 0 from the left-hand side of |h| / h? It's -1. And 1 is not equal to -1. Because the one-sided limits disagree The two sided limit doesn't exist. And this limit calculating the derivative means that this function is not differentiable at 0. Of course, that raises the question why should you care about differentiable functions at all? Here's some terrible looking function. But it's differentiable. So if I zoom in on some point, the thing looks like a straight line. Calculus is all about replacing the curved objects that we can't understand with straight lines, which we have some hope of understanding.
How does wiggling x affect f(x)?
Remember, f prime of x is encoding how wiggling x affects f of x. We can see this by thinking about slopes of tangent lines as well. Here, I've drawn in orange the graph of some random function y=f(x). And this point, a f(a) I've drawn in red the tangent line to the graph. I want to know how is the output f affected when input moves from a to a+h. I want to know what happens when I wiggle the input over by adding h. Let me zoom in on this picture a little bit. Here, I've got this tangent line, and the slope of that tangent line is the derivative of the function at the point a. That means that this triangle can be understood, right? This slope being f'(a), this base being h means this height is h*f'(a).
In order to get this slope as f'(a), rise over run better be equal to f'(a), this divided by this is f'(a). this is giving me some information about how wiggling the input will affect the output. If I move from a to a+h, well, on the tangent line m' moving up to this point which isn't so far off of the real value of the function up here. Now, if I made h really, really small, I'd be doing an even better job of staying close to the graph of the function when I follow the tangent line. Instead of starting with the function and trying to figure out the derivative, we can imagine that we know a little bit of information about the derivative and try to figure out something about the function. Let's make up a concrete example. Suppose that I've got some function f, and all I know is that its derivative is 3x, and its value at 2, is 4. Just knowing this information without a rule for the function at this point, can I say anything about the function's value at, say 2.01? Yes, I can. Right? f(2.01), well that's f(2) plus how much the output changes when I go from 2 to 2.01. Well, the output change is approximately something that I can compute from the derivative, right? The derivative is infinitesimally the ratio between output change and input change. So if I multiply by how much I change the input by the ratio of input change to output change, this should be approximately the true output change. Now, in this case, I know what these numbers are. 0.01 times the derivative of f(2) is 6. Which means that this is about 4+0.06 which is 4.06. And that's about what this function at 2.01 is equal to. One more cat moves into the neighborhood. What happens to the mouse population? Right? A small change to one thing will affect something else. That's what the derivative is encoding.
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