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2015 AP Calculus AB BC 4cd AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | And we know dy dx is equal to m. We know this is m. And so there you have it, we have enough information to solve for m. We know that zero is equal to two minus m. So zero is equal to two minus m. And so we can add m to both sides and we get m is equal to two. So that by itself was quite useful. And then what we could ... |
2015 AP Calculus AB BC 4cd AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | Well we know that this right over here, dy dx, this is m. This is m. And it's equal to two. So we could say that two is equal to two x minus y. Two is equal to two x minus y. And let's see, if we solve for y, add y to both sides, subtract two from both sides, we get y is equal to two x minus two. And there we have our ... |
2015 AP Calculus AB BC 4cd AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | And let's see, if we solve for y, add y to both sides, subtract two from both sides, we get y is equal to two x minus two. And there we have our whole solution. And so you have your m right over there. That is m. And then we also have our b. This one was a tricky one. Anytime that you have to do something like this and... |
2015 AP Calculus AB BC 4cd AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | That is m. And then we also have our b. This one was a tricky one. Anytime that you have to do something like this and it doesn't just jump out at you, and if it wasn't obvious it didn't jump out at me at first when I looked at this problem, I said well let me just write down everything that they told us. So they wrote... |
2015 AP Calculus AB BC 4cd AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | So they wrote this before. And then we say okay, this is going to be a solution. And so let me see if I can somehow solve, so let's see what I didn't use. I didn't use that. I did use this. I absolutely used that. I did use that. |
2015 AP Calculus AB BC 4cd AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | I didn't use that. I did use this. I absolutely used that. I did use that. I did use that. And I did use that. So this was a little bit of a fun little puzzle where I just wrote down all the information they gave us and I tried to figure out, based on that, whether I could figure out m and b. |
2015 AP Calculus AB BC 4cd AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | I did use that. I did use that. And I did use that. So this was a little bit of a fun little puzzle where I just wrote down all the information they gave us and I tried to figure out, based on that, whether I could figure out m and b. And this is pretty neat. This is a solution, 2x minus 2. If we go to our slope field ... |
2015 AP Calculus AB BC 4cd AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | So this was a little bit of a fun little puzzle where I just wrote down all the information they gave us and I tried to figure out, based on that, whether I could figure out m and b. And this is pretty neat. This is a solution, 2x minus 2. If we go to our slope field above, it wouldn't have jumped out at me, but if you... |
2015 AP Calculus AB BC 4cd AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | If we go to our slope field above, it wouldn't have jumped out at me, but if you think about, so 2x minus 2, its y-intercept would be negative 2 like that. Let me do this in a different color. And so the line would look something like this. The line would look something like this. And you can verify that any one of the... |
2015 AP Calculus AB BC 4cd AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | The line would look something like this. And you can verify that any one of these points, at any one of these points, the slope is equal to 2. If we're at the point 2, 2, well it's going to be 2 times 2 minus 2 is 2. 1, 0, 2 times 1 minus 0 is 2. Negative 2, 0, negative 2, well 0 minus negative 2, that's 2. So you see ... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | And you learn multiple notations for this. For example, if you know that y is equal to f of x, you might write this as y prime. You might write this as dy dx, which you'll often hear me say is the derivative of y with respect to x, and that you could use the derivative of f with respect to x, because y is equal to our ... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | But then later on, especially when you start getting into differential equations, you see people start to treat this notation as an actual algebraic expression. For example, you will learn, or you might have already seen, if you're trying to solve the differential equation, the derivative of y with respect to x is equa... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | You'll see this technique where people say, well, let's just multiply both sides by dx, just treating dx like as if it's some algebraic expression. So you multiply both sides by dx, and then you have, so that would cancel out algebraically. And so you see people treat it like that. So you have dy is equal to y times dx... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | So you have dy is equal to y times dx, and then they'll say, okay, let's divide both sides by y, which is a reasonable thing to do. Y is an algebraic expression. So if you divide both sides by y, you get one over y dy is equal to dx. And then folks will integrate both sides to find a general solution to this differenti... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | And then folks will integrate both sides to find a general solution to this differential equation. But my point on this video isn't to think about how do you solve a differential equation here, but to think about this notion of using what we call differentials, so a dx or a dy, and treating them algebraically like this... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | And so to just feel reasonably okay about doing this, this is a little bit hand-wavy. It's not super mathematically rigorous, but it has proven to be a useful tool for us to find these solutions. And conceptually, the way that I think about a dy or a dx is this is the super small change in y in response to a super smal... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | And that's essentially what this definition of the limit is telling us, especially as delta x approaches zero, we're going to have a super small change in x as delta x approaches zero, and then we're gonna have a resulting super small change in y. So that's one way that you can feel a little bit better of, and this is ... |
Mean value theorem for integrals AP Calculus AB Khan Academy.mp3 | So the slope between the two endpoints is gonna be your change in y, which is going to be your change in your function value, so f of b minus f of a over, over b minus a. And once again, we do this, we go into much more depth in this when we covered it the first time in differential calculus, but just to give you a vis... |
Mean value theorem for integrals AP Calculus AB Khan Academy.mp3 | Remember, what we saw for the average value of a function, we said the average, the average value of a function is going to be equal to one over b minus a, notice, one over b minus a, you have a b minus a in the denominator here, times the definite integral from a to b of f of x dx. Now, this is interesting, because he... |
Mean value theorem for integrals AP Calculus AB Khan Academy.mp3 | So I encourage you to try to do that. So once again, this is, let me rewrite all of this. This is going to be equal to, this over here is the exact same thing as the definite integral from a to b of f prime of x dx. Think about it. You're gonna take the antiderivative of f prime of x, which is going to be f of x, and y... |
Mean value theorem for integrals AP Calculus AB Khan Academy.mp3 | Think about it. You're gonna take the antiderivative of f prime of x, which is going to be f of x, and you're going to evaluate it at b, f of b, and then from that, you're going to subtract it, evaluate it at a, minus f of a. These two things are identical, and then you can, of course, divide by b minus a. Now this is ... |
Mean value theorem for integrals AP Calculus AB Khan Academy.mp3 | Now this is starting to get interesting. One way to think about it is there must be a c, if we, there must be a c that takes on the average value of, there must be a c that when you evaluate the derivative at c, it takes on the average value of the derivative. Or another way to think about it, another way to think abou... |
Mean value theorem for integrals AP Calculus AB Khan Academy.mp3 | So there exists a c, where g of c is equal to one over b minus a, times the definite integral from a to b of g of x, g of x, dx. F prime of x is the same thing as g of x. So another way of thinking about it, and this is actually another form of the mean value theorem, it's called the mean value theorem for integrals. M... |
Mean value theorem for integrals AP Calculus AB Khan Academy.mp3 | Mean value theorem for integrals. Let me, so this is the mean, I'll just write the acronym, mean value theorem for integrals, or integration, which essentially, and to give it a slightly more formal sense, is if you have some function g, so if g is, let me actually go down a little bit, which tells us that if g of x is... |
Mean value theorem for integrals AP Calculus AB Khan Academy.mp3 | There exists a c where g of c is equal to the average value of your function over the interval. This was our definition of the average value of a function. So anyway, this is just another way of saying, you might see some of the mean value theorem of integrals, and just to show you that it's really closely tied, it's u... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | Now ready for part B, and I've wiped all the salt off of my fingers, and I've had a glass of water, so now I'm ready for some business. Find the second derivative of W with respect to T. In terms of W, use this second derivative to determine whether your answer in part A is an underestimate or an overestimate for the a... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | Let's find the second derivative in terms of W. And we already have the first derivative over here, and in part A I rewrote it just with slightly different notation, but I'll just use this just because I need to write down here and I can still refer to this thing. So let's just take the derivative of both sides of this... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | And let me make that clear. This is the same thing as 1 over 25 W as a function of T minus 12. 1 25th of 300 is 12. And you take the derivative of this, you get 1 25th times the first derivative of W, and then the derivative of this with respect to T is just 0. A constant obviously does not change with respect to T. An... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | And you take the derivative of this, you get 1 25th times the first derivative of W, and then the derivative of this with respect to T is just 0. A constant obviously does not change with respect to T. And so we get this right over here. Now this is the second derivative in terms of the first derivative. But the questi... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | But the question asks us, write the second derivative in terms of W. But lucky for us, we know how to express this as a function of W. They gave that to us in the problem. This was given. This is just me rewriting it in a different notation. So this is going to be the same thing as 1 over 25 times the derivative of W. ... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | So this is going to be the same thing as 1 over 25 times the derivative of W. And the derivative of W is this. The differential equation literally tells us the derivative of W is this over here. So 1 over 25 times 1 over 25 WT, or the function W as a function of T, minus 300. And so we can say the second derivative of ... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | And so we can say the second derivative of W as a function of T is equal to 1 over 625 times W as a function of T, which is a function of T, minus 300. So we've done the first part. We've found out the second derivative of W in terms of just W. Now let's try to address the second part of their question. So we did the f... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | So we did the first part. Now use what we just figured out to determine whether your answer in part A is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time 1 fourth. So in part A, we found the slope of the tangent line at time equals 0. And we use that slope to extrapola... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | And we use that slope to extrapolate out to 3 months, or time equals 1 fourth, 1 fourth of a year. Now, if the function's W's slope over that fourth was exactly the same as the slope of the tangent line, or if that slope did not change, then our extrapolation would be exactly right. If W's slope is increasing over that... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | And if the slope is decreasing over that time, then our estimate would be an overestimate of the actual amount in the landfill. And to figure out whether the slope is increasing or decreasing, we just have to look at the value of the second derivative. If the second derivative is positive, that means our slope is incre... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | Let me just be clear here. Let me draw this. Let me make it very clear. So let me just draw a random function. So let's say that's W. So this is a case where W's slope is increasing faster, or W's slope is increasing from that starting point. So our starting point, this was our slope, and then W's slope keeps increasin... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | So let me just draw a random function. So let's say that's W. So this is a case where W's slope is increasing faster, or W's slope is increasing from that starting point. So our starting point, this was our slope, and then W's slope keeps increasing from there. And in this case, our estimate is going to be an underesti... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | And in this case, our estimate is going to be an underestimate of where W actually is after a fourth of the year. If W's slope is exactly the same as our function over the course of the year, over the course of the first 3 months, so it would look something like that. Maybe it diverges later on. And in this case, our a... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | And in this case, our approximation would be really good. It would probably be exact for W. And if W's slope, for whatever reason, goes negative after that point, and they already tell us this is an increasing function, so that is not likely. Well, it doesn't have to even go negative. If W's slope decreases, it could s... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | If W's slope decreases, it could still stay positive. And the way I drew it doesn't, let me draw it like this, just to make it clear. So let's say that, let's say we find out that the slope looks something like this. This is the slope of the tangent line at time equals 0. If W's slope increases from that point, then W ... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | This is the slope of the tangent line at time equals 0. If W's slope increases from that point, then W might look something like that. And then, our answer to part A would be an understatement, would be an underestimate of where W actually is after a fourth of a year. Let me make it clear to you what I'm doing. So this... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | Let me make it clear to you what I'm doing. So this is my W axis. So this is right at our initial condition of 1400. And this right over here is our time axis. And this is at 1 fourth. So in the last video, we said, hey, this is sitting right at 1411. If W's slope increases from that point, then this is an underestimat... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | And this right over here is our time axis. And this is at 1 fourth. So in the last video, we said, hey, this is sitting right at 1411. If W's slope increases from that point, then this is an underestimate. If W's slope stays the same, then this is actually a very good estimate. Because then we're going to hit that poin... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | If W's slope increases from that point, then this is an underestimate. If W's slope stays the same, then this is actually a very good estimate. Because then we're going to hit that point directly with W. And if W's slope decreases, so you can imagine maybe W looks something like this. It has that slope of the tangent l... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | It has that slope of the tangent line when it starts, but then the slope decreases. And it's still an increasing function, but the slope is decreasing. And in that case, we would have an overestimate. And this is a situation, so this first situation, slope is increasing, that means W prime prime is positive. The slope ... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | And this is a situation, so this first situation, slope is increasing, that means W prime prime is positive. The slope is increasing. The second derivative is positive. This means, or this would be a byproduct, or this would cause the second derivative to be negative. Our slope is decreasing. And this would be our seco... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | This means, or this would be a byproduct, or this would cause the second derivative to be negative. Our slope is decreasing. And this would be our second derivative. Our second derivative is 0. If your slope isn't changing, if your slope is constant, if your first derivative is constant, your second derivative is going... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | Our second derivative is 0. If your slope isn't changing, if your slope is constant, if your first derivative is constant, your second derivative is going to be 0. So let's just see what our second derivative is at our initial condition, and then we'll have a pretty good sense of whether we have an overestimate or an u... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | So let's just figure out what our second derivative is at time 0. It's going to be equal to 1 over 625 times W of 0, which we know, minus 300. Well, W of 0, the amount of waste we have at time 0, they told us in the problem, is 1400 tons. 1400 minus 1300 is 1100. And then 1100 divided by 625 is a small number, it's one... |
2011 Calculus AB free response #5b AP Calculus AB Khan Academy.mp3 | 1400 minus 1300 is 1100. And then 1100 divided by 625 is a small number, it's one point something, but it is a positive number. And that's the important thing here. So this thing, all of this business, it is positive. So the second derivative is positive, which means the slope is increasing, at least right at our start... |
Limits of composite functions external limit doesn't exist AP Calculus Khan Academy.mp3 | And what I want to do is figure out what is the limit of g of h of x as x approaches one. Pause this video and see if you can figure that out. All right, now let's do this together. Now the first thing that you might try to say is, all right, let's just figure out first the limit as x approaches one of h of x. And when... |
Limits of composite functions external limit doesn't exist AP Calculus Khan Academy.mp3 | Now the first thing that you might try to say is, all right, let's just figure out first the limit as x approaches one of h of x. And when you look at that, what is that going to be? Well, as we approach one from the left, it looks like h of x is approaching two. And as we approach from the right, it looks like h of x ... |
Limits of composite functions external limit doesn't exist AP Calculus Khan Academy.mp3 | And as we approach from the right, it looks like h of x is approaching two. So it looks like this is just going to be two. And then we say, okay, well maybe we can then just input that into g. So what is g of two? Well, g of two is zero, but the limit doesn't seem defined. It looks like when we approach two from the ri... |
Limits of composite functions external limit doesn't exist AP Calculus Khan Academy.mp3 | Well, g of two is zero, but the limit doesn't seem defined. It looks like when we approach two from the right, we're approaching zero. And when we approach two from the left, we're approaching negative two. So maybe this limit doesn't exist. But if you're thinking that, we haven't fully thought through it. Because what... |
Limits of composite functions external limit doesn't exist AP Calculus Khan Academy.mp3 | So maybe this limit doesn't exist. But if you're thinking that, we haven't fully thought through it. Because what we could do is think about this limit in terms of both left-handed and right-handed limits. So let's think of it this way. First, let's think about what is the limit as x approaches one from the left-hand s... |
Limits of composite functions external limit doesn't exist AP Calculus Khan Academy.mp3 | So let's think of it this way. First, let's think about what is the limit as x approaches one from the left-hand side of g of h of x. All right, when you think about it this way, if we're approaching one from the left, right over here, we see that we are approaching two from the left, I guess you could say, or we're ap... |
Limits of composite functions external limit doesn't exist AP Calculus Khan Academy.mp3 | And so the thing that we are inputting into g of x is approaching two from below. So the thing that we are inputting into g is approaching two from below. So if you approach two from below, right over here, what is g approaching? It looks like g is approaching negative two. So this looks like it is going to be equal to... |
Limits of composite functions external limit doesn't exist AP Calculus Khan Academy.mp3 | It looks like g is approaching negative two. So this looks like it is going to be equal to negative two, at least this left-handed limit. Now let's do a right-handed limit. What is the limit as x approaches one from the right-hand of g of h of x? Well, we can do the same exercise. As we approach one from the right, it ... |
Limits of composite functions external limit doesn't exist AP Calculus Khan Academy.mp3 | What is the limit as x approaches one from the right-hand of g of h of x? Well, we can do the same exercise. As we approach one from the right, it looks like h is approaching two from below, from values less than two. And so if we are approaching two from below, because remember, whatever h is outputting is the input i... |
Limits of composite functions external limit doesn't exist AP Calculus Khan Academy.mp3 | And so if we are approaching two from below, because remember, whatever h is outputting is the input into g. So if the thing that we're inputting into g is approaching two from below, that means that g once again is going to be approaching negative two. So this is a really, really, really interesting case where the lim... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | And that's what we're going to cover in this video. What you see here is a flow chart developed by the team at Khan Academy. And I'm essentially going to work through that flow chart. It looks a little bit complicated at first, but hopefully it'll make sense as we talk it through. So the goal is, hey, we want to find t... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | It looks a little bit complicated at first, but hopefully it'll make sense as we talk it through. So the goal is, hey, we want to find the limit of f of x as x approaches a. So what this is telling us to do is, well, the first thing, just try to substitute what happens when x equals a. Let's evaluate f of a. And this f... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Let's evaluate f of a. And this flow chart says, well, if f of a is equal to a real number, it's saying we're done. But then there's this little caveat here, probably. And the reason why is that the limit is a different thing than the value of the function. Sometimes they happen to be the same. In fact, that's the defi... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | And the reason why is that the limit is a different thing than the value of the function. Sometimes they happen to be the same. In fact, that's the definition of a continuous function, which we talk about in previous videos. But sometimes they aren't the same. This will not necessarily be true if you're dealing with so... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | But sometimes they aren't the same. This will not necessarily be true if you're dealing with some function that has a point discontinuity like that, or a jump discontinuity, or a function that looks like this. This would not necessarily be the case. But if at that point you're trying to find the limit towards the, if a... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | But if at that point you're trying to find the limit towards the, if as you approach this point right over here, the function is continuous, it's behaving somewhat normally, then this is a good thing to keep in mind. You could just say, hey, can I just evaluate the function at that, at that a over there? So in general,... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | If you're dealing with some type of a function that has all sorts of special cases, and it's piecewise defined, as we've seen in previous other videos, I would be a little bit more skeptical. Or if you know visually around that point there's some type of jump, or some type of discontinuity, you've got to be a little bi... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | If you're dealing with plain vanilla functions that are continuous, if you evaluate at x equals a, and you get a real number, that's probably going to be the limit. But now let's think about the other scenarios. What happens if you evaluate it, and you get some number divided by zero? Well, that case, you are probably ... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Well, that case, you are probably dealing with a vertical asymptote. And what do we mean by a vertical asymptote? Well, look at this example right over here, where you're just saying the limit, we'll do that in a darker color. So if we're talking about the limit as x approaches one, of one over x minus one. If you just... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | So if we're talking about the limit as x approaches one, of one over x minus one. If you just try to evaluate this expression at x equals one, you would get one over one minus one, which is equal to one over zero, which says, okay, I'm falling into this vertical asymptote case. And at that point, if you wanted to just ... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | You could say, all right, I probably have a vertical asymptote here at x equals one. So that's my vertical asymptote. And you could try out some values. Well, let's see, if x is greater than one, the denominator is going to be positive. And so my graph, and you would get this from trying out a bunch of values, might lo... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Well, let's see, if x is greater than one, the denominator is going to be positive. And so my graph, and you would get this from trying out a bunch of values, might look something like this. And then for values less than negative one, or less than one, I should say, you're gonna get negative values. And so your graph m... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | And so your graph might look something like that. And so you have this vertical asymptote. That's probably what you have. Now, there are cases, very special cases, where you won't necessarily have the vertical asymptote. One example of that would be something like one over x minus x. This one here is actually undefined... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Now, there are cases, very special cases, where you won't necessarily have the vertical asymptote. One example of that would be something like one over x minus x. This one here is actually undefined for any x you give it. So it would be very, you will not have a vertical asymptote. But this is a very special case. Most... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | So it would be very, you will not have a vertical asymptote. But this is a very special case. Most times, you do have a vertical asymptote there. But let's say we don't fall into either of those situations. What if when we evaluate the function, we get zero over zero? And here is an example of that. Limit as x approach... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | But let's say we don't fall into either of those situations. What if when we evaluate the function, we get zero over zero? And here is an example of that. Limit as x approaches negative one of this rational expression. And let's try to evaluate it. You get negative one squared, which is one, minus negative one, which i... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Limit as x approaches negative one of this rational expression. And let's try to evaluate it. You get negative one squared, which is one, minus negative one, which is plus one, minus two, so you get zero in the numerator. And then in the denominator, you have negative one squared, which is one, minus two times negative... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | And then in the denominator, you have negative one squared, which is one, minus two times negative one, so plus two minus three, which is equal to zero. Now this is known as indeterminate form. And so on our flow chart, we then continue to the right side of it. And so here's a bunch of techniques for trying to tackle s... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | And so here's a bunch of techniques for trying to tackle something in indeterminate form. And likely in a few weeks, you will learn another technique that involves a little bit more calculus called L'Hopital's Rule that we don't tackle here because that involves calculus, while all of these techniques can be done with ... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Try to see if you can simplify this expression. And this expression here, you can factor it. This is the same thing as x, let's see, x minus two times x plus one over, let's see, x, well, this would be x minus three times x plus one. If what I just did seems completely foreign to you, I encourage you to watch the video... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | If what I just did seems completely foreign to you, I encourage you to watch the videos on factoring polynomials or factoring quadratics. And so you can see here, all right, look, if I make the, I can simplify this because as long as x does not equal negative one, these two things are going to cancel out. So I can say ... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Sometimes people forget to do this part. This is if you're really being mathematically precise. This entire expression is the same as this one because this entire expression is still not defined at x equals negative one, although you can substitute x equals negative one here and now get a value. So if you substitute x ... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | So if you substitute x equals negative one here, even if it's formally, if we're formally taking it away to be mathematically equivalent, this would be negative one minus two, which would be negative three, over negative one minus three, which would be negative four, which is equal to 3 4ths. So if this condition wasn'... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | And if I can just evaluate it at x equals negative one, I feel pretty good, I feel pretty good. So once again, we're now going and factoring. We're able to factor. We evaluate, we simplify it. We evaluate the expression, the simplified expression now and now we were able to get a value. We were able to get 3 4ths and s... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | We evaluate, we simplify it. We evaluate the expression, the simplified expression now and now we were able to get a value. We were able to get 3 4ths and so we can feel pretty good that the limit here in this situation is 3 4ths. Now let's, and I would categorize what we've seen so far as the bulk of the limit exercis... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Now let's, and I would categorize what we've seen so far as the bulk of the limit exercises that you will likely encounter. Now the next two, I would call slightly fancier techniques. So if you get indeterminate form, especially you'll sometimes see it with radical expressions like this, rational radical expressions, y... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | So for example, in this situation right here, if you just tried to evaluate it at x equals four, you get the square root of four minus two over four minus four, which is zero over zero. So it's that indeterminate form and the technique here, because we're seeing this radical and irrational expression, is hey, let's may... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Square root of x minus two over x minus four. When we say conjugate, let's multiply it by the square root of x plus two over the square root of x plus two. Once again, it's the same expression over the same expression, so I'm not fundamentally changing its value. And so this is going to be equal to, well if I have a pl... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | And so this is going to be equal to, well if I have a plus b times a minus b, I'm gonna get a difference of squares. So it's gonna be square root of x squared, which is, let me just write it, it's gonna be square root of x squared minus four over, well square root of x squared is just going to be x minus four. So let m... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | So it's x minus four over x minus four times square root of x plus two. Square root of x plus two. Well this was useful because now I can cancel out x equals four, or x minus four, right over here. And once again, if I wanted it mathematically to be the exact same expression, I would say well, now this is going to be e... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | And once again, if I wanted it mathematically to be the exact same expression, I would say well, now this is going to be equal to one over the square root of x plus two for x does not equal four. But we can definitely see what this function is approaching if we just now substitute x equals four into this simplified exp... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | And once again, you can feel pretty good that this is going to be your limit. We've gone back into the green zone. If you were to actually plot this original function, you would have a point discontinuity, you would have a gap at x equals four. But then when you do that simplification and factoring out that x minus, or... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | But then when you do that simplification and factoring out that x minus, or canceling out that x minus four, that gap would disappear. And so that's essentially what you're doing. You're trying to find the limit as we approach that gap, which we got right there. Now this final one, this is dealing with trig identities.... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Now this final one, this is dealing with trig identities. And in order to do these, you have to be pretty adept at your trig identities. So if we're saying the limit as, let me do that in a darker color. So if we're saying the limit as x approaches zero of sine of x over sine of two x, well, sine of zero zero, sine of ... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | So if we're saying the limit as x approaches zero of sine of x over sine of two x, well, sine of zero zero, sine of zero zero, you're gonna get zero over zero. Once again, indeterminate form, we fall into this category. And now you might recognize this is going to be equal to the limit as x approaches zero of sine of x... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | We can rewrite sine of two x as two sine x cosine x. And then those two can cancel out for all x's not equaling, for all x's not equaling zero if you wanna be really mathematical precise. And so there would have been a gap there for sure on the original graph if you were to graph y equals this. But now for the limit pu... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | But now for the limit purposes, you could say this limit is going to be the limit as x approaches zero of one over two cosine of x. And now we can go back to this green condition right over here, because we can evaluate this at x equals zero. It's gonna be one over two times cosine of zero. Cosine of zero is one. So th... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Cosine of zero is one. So this is going to be equal to 1 1⁄2. Now in general, none of these techniques work. And you'll encounter a few other techniques further on once you learn more calculus. Then you fall on the baseline, approximation. And approximation, you can do it numerically. Try values really, really, really,... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | And you'll encounter a few other techniques further on once you learn more calculus. Then you fall on the baseline, approximation. And approximation, you can do it numerically. Try values really, really, really, really close to the number you're trying to find the limit on. You know, if you're trying to find the limit ... |
Strategy in finding limits Limits and continuity AP Calculus AB Khan Academy.mp3 | Try values really, really, really, really close to the number you're trying to find the limit on. You know, if you're trying to find the limit as x approaches zero, try 0.00000000001. Try negative 0.0000001. If you're trying to find the limit as x approaches four, try 4.0000001. Try 3.999999999999. And see what happens... |
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