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About this Lesson
 Type: Video Tutorial
 Length: 18:15
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 Posted: 06/26/2009
Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UTAustin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, padic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
About this Author
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Founded in 1997, Thinkwell has succeeded in creating "nextgeneration" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technologybased textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.
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Calculus in 20 Minutes
Calculus in 20 Minutes Page [1 of 4]
All right, so, now we're going to try to do the impossible. We're going to try to do all of calculus in under twenty minutes. So, we have to work really, really fast, go through the whole course. Let's begin.
The first two basic issues of calculus, two big questions, what are they? How do you find instantaneous rate of change? How do you find how things have changed instantly? And, then completely separate from that, how do you find areas under curves? Two completely different questions, turns out answers are completely related. Let's see how they go.
Question one, instantaneous rate of change, what are we going to do? Well, what you do there is you first remember, what does rate mean? Well, rate is just change in distance over change in time. It's as easy as that, distance equals rate times time, not a big deal. Okay, now what do you do with that? Well, if you graph a function that represents, sort of, distance against time, then what do you notice? If you want to look at the change in time and the change in distance, what do you got? You actually got a slope of a line, right? This is slope, rise over run, change in distance over change in time. Distance over time, you've got a slope. So, automatically, we see a really neat thing. We see that the average rate of change between two points is equal to the slope of the lines connecting them. Well, that's really cool. By the way, what about lines? Maybe you forgot about lines? We'll remind you about lines. Okay, no problem, we can do lines. We can do lines. Lines, y  y[1] = m(x  x[1]). This is the point slope form. All you need to give me is a point on the line, x1y1 and a slope, m. You give me those two pieces of information, I can always write down the line uniquely. Always, always, always. Never forget it. Okay, fine, now we're back to here. Now, we can find average rate and what do we see? Well, in fact, a line that touches the curve at two points, is sometimes called a ces line. So in fact, we just discovered that the average rate change in distance over change in time is equal to the slope of the ces line, cool. So, all you have to do, want to find average rate? Connect the two points of the line, find the slope, you got the average rate. No biggie. But, that's not what we want, we want instantaneous rate, so how do you do that? Well, what would that be? Well, if I [inaudible] instantaneous rate here, what would I do? I'd make that ces line closer and closer and closer, bring those points together and look what I'm converging to? I'm coming to a tangent line. Wow. Instantaneous rate of change equals the slope of the tangent line. So, what we want is the slope of tangent line. Well, what's the slope of the tangent line? It's change in y over change in x, change in distance over change in time, but now, the change in time from that point to itself is 0. 0/0, so I'm getting 0/0, which is complete garbage and that is a big problem. Our first major problem of the course. Okay, so, what do we do?
So, we can that. Distance equals rate times time, we're going to can that. Instantaneous rate of change, I really want that, so what do we do? Well, how do we get that 0/0 problem to go away? The answer is, we inch up to it. We just approach the 0 point, and what would that look like? Well, I'm going to remind you what you've done when you were a little teeny kid. And when you were a little teeny kid, this is what you were looking at, you were looking at a value of a function at a point. Value of a function at a point. Okay, not a big deal, there it is. F(a), it's that point. But you don't know anything else but the function. You don't know what's going around around there, because all you're looking at is f(a), you open it up, whew hoo, the function could be quite interesting, who knows? Okay, but, now what I invite us to do and what calculus invites us to do, is to look at the function this way, cover up that point and look at everything else. Open the window look outside. That's what calculus is. And what you see here, is we can see what things are approaching, and we can actually determine the idea of a limit. The limit is what things are approaching. We don't care about what actually happens at that point, only what things approaching. Armed with the idea of a limit, what can we do? Well, now we can return to the question and figure out, let's take the limit as delta t goes to 0. What do we get? We get 0/0: that is called an indeterminate form when you get 0/0. So what do you do? You got to do some algebraic gymnastics. You got to factor the top and cancel with the bottom. You can try to multiply by the cognizant. You can try to combine the fractions. There's always tricks of the trade to actually reduce this to something that you can actually find. So, you find the limit, okay. Well, once you find the limit, and you take the limit as delta t goes to 0, what do you get? You get the answer to the question, how do you find instantaneous rate of change, the answer is what we call, the derivative, and what's the derivative? It's the limit as delta x goes to 0, or f(x) plus delta x minus f(x)/delta x. Looks pretty confusing, doesn't it? It's just rate, it's distance over time, but now I'm letting time go to 0. Not a big deal. So, that's the derivative, bingo, we're done with the first question.
And so now, what do we see? What we see now is we come back to here, and the derivative, in fact, gives us the slope of the tangent line. Well, that's really cool, if you want to find the slope of a tangent line ever in life, you just take the derivative, and that gives you the slope of the tangent once you evaluate at the point you want. Great, but for free, we answer the first question, because remember the derivative also represents the instantaneous rate of change. Do you want to find out how things are changing? No big deal, you take a derivative, plug in and that would tell you how things are changing at that instant. Okay, great, well, now we know all about how to graph these things, how to look at these things, the derivative, velocity, boom, boom, boom, we got all the way. Now, let's take a look at some applications. What can we do with this?
Well, how would you take derivatives of complicated functions? Well, if you've got a product, use the product rule. Remember the product rule, don't memorize the formula; memorize the chant. First, times the derivative of the second, plus the second times the derivative of the first. So, the derivative of a product is this, it's not the product of derivatives, you got to use the product rule. We've got five minutes, listen up here folks, I've got to move fast today. I'll move faster. What if you have a quotient, well, then use the quotient rule. So, what's the quotient rule say? If you had a quotient, you take the bottom times the derivative of the top minus the top times the derivative of the bottom, over the bottom squared. That's the quotient rule, that's what you use when you have a derivative of a quotient. Okay, great, no problem. Now, what about if I had a really complicated function? What if you got a function that looks like this? It's got insides; it's got guts right in there. See, you got something this, and you want to take the derivative of that, what do you do? Well, you got to use the chain rule, folks. This is a thing that you can chain together, there's an inside, there's a blop right here, there's a whole big blop there, and then you got an outside. So, take the derivative of the outside, the derivative of sin(blop) is actually cos(blop), so it's cos(blop) and what's the blop? The blop is going to be 3x^3 + 1, and then you multiply that by the derivative of the inside, and the derivative of that turns out to be just, let's see, 9x^2 + 0. So, there's the derivative using the chain rule. The idea is to peel off, like an onion, just peel off, keep peeling off the outside until you get to the inside, always remember though, when you take the derivative, if you have sin(blop), the derivative is cos of the blop. Don't put the derivative in there, put the blop and multiply by the derivative of the inside. That's the key to the chain rule, now we got the chain rule.
Okay, so what about when you have functions that aren't functions? Like what if you have things that are relations? Like x^2 + y^2 = 1? Like a circle. How do you differentiate that? How do you find dydx there? Well, the answer is we use something called implicit differentiation. Implicit differentiation, how does that work? Well, you've got to remember that dydx, that's an object, that's a noun, and ddx is a verb, it's a commandment. Take the derivative with respect to x. So, you differentiate with respect to x, and what do you see? Well, you see something that looks like this. What you do is, you say okay, I'll take dx of x^2 + y^2 = 1, the derivative of x^2, with respect to x, is just 2x, not a big deal. The derivative of y^2, remember how I think about this. I think about this as clumping all this together, and I see this is a blop squared, so I actually used the chain rule, which we just developed, and the chain rule says the derivative of blop squared is 2 blop, and then I multiply that by the derivative of the blop, which is the derivative of y with respect to x, that's called dydx, folks. The derivative of 1 is 0. And now, you can actually solve this for dydx, by bringing this to the other side, that would be a 2x, you divide by the 2y, and you see dydx = x/y, and there's the answer. That's implicit differentiation, just go right through and differentiate implicitly, when you have a relationship, you can still find the derivative. And we're making progress here, folks. We are cutting through this stuff.
Well, now that you have derivatives, what can you do with that? Well, if you think about as velocity, you get instantaneous velocity. If you take the derivative of velocity, you actually get the change in velocity, which is acceleration. So, we get acceleration now, we get velocity. Acceleration is just a second derivative of the position. So, then you have to take derivatives upon derivatives upon derivatives as many derivatives as you want. So, it's great, we can do that, so how do I do derivatives? No problem. What can you use these things for? Okay, we know it's true for velocity, we can use if for velocity, what else we use it for? Well, it turns out you can use it for linear approximation. Suppose you got some wacko function, you got the wacko function like this. Woo, and you actually want to figure out the value right here. Right here and you don't know, you don't know what that value is. But you know near by there's a point that you can actually compute. So, what do you do? You find the tangent line approximation, because you remember that the tangent line closely emulates the activity of the function. The tangent line closely emulates the activity of what the function's doing, and it looks the same there, so you find the equation of the tangent line, and then plug in the mysterious point then you're all set because you can approximate the value by plugging in the tangent line. So, how does that look? Well, that's call linear approximation, here's the formula, but don't bother memorizing it, just think about it, what you've got to do is you've got to find the equation of the line, that's tangent at the known point which is x, here in this case. And this is going to be x + delta x, the known point plus a little teeny off set, so it's approximately equal to the derivative times the change in x plus the function. So, that's it that's the whole thing right there, linear approximation. Allows you to actually compute things, computers actually help this way, computers know calculus, everyone knows calculus, you guys know calculus.
Okay, now what else do you do? Well, suppose that a derivative were to be 0? Well, how could that possibly happen? That could possibly happen because maybe the function goes like this, and I see the tangent over here has slope 0. Or maybe the function goes like this, and I see the tangent has slope 0. In particular, if the tangent equals 0, maybe we have a max or min. Also, maybe the slope or the tangent doesn't exist? Like if we have a wave kind of thing. A cusp, very pretty cusps like a wave. Well, that might be a max, that might be a min. So, in particular you could find out when objects are maximized or minimized. You can find the maxima or the minima very easily by using calculus. What do you do? You take a derivative and you see where equals 0, or where the derivative doesn't exist, but the function does. Those, give you candidates for possible max and min. And what can you use that for? Well, you can do all sorts of max and min problems. You want to maximize profits; you want to minimize costs? You gotoh my God! Five minutes, so there's five minutes left, they are trying to fool me here folks, but I'm not going to go for it. Five minutes left, okay, so you want to maximize cost you want to minimize cost, you want to maximize profit? You want to maximize area; you want to minimize volume? Whatever it is, set up the problem really carefully, figure exactly what you want to optimize, take the derivative, set it equal to 0, solve, find out where the derivative's undefined and you've got it made. Really, not, not, not a big deal. However, you should always remember, and never forget the fundamental method of solving problems. So remember how you solve all of life's problems? The first thing you have to do is understand what you're being asked. You can't answer a question that you don't understand. The next thing you do after you understand what you're suppose to find, what you're being asked, is figure what you know, list every single thing that you know, every single fact; maybe it's frivolous, maybe you don't use it. Who cares, write it down, understand it make it your own. And the last thing is, take the information that you know and see a relationship between that and the thing that you seek. Try to find a connection. Once you got the connection, then you're on the road to actually finding the solution. That is the simple method to finding any single answer to any single problem.
Another application if you think about derivatives is the rate. We related rate. Suppose for example that you actually have a ladder, for example, it's falling down. The ladder is falling and you don't want to be sued, but the only thing you do know is how fast the bottom is falling. You want to know how fast the top is falling. What you need to do there is if you know this rate, you can find that rate by linking them up with a connection. And in this case, the connection would actually by the Pythagorean theorem. You can take the derivative with respect to time, because here you see the variable, the thing that's independent that's always changing, is time. Time keeps on ticking into the future. So you can find this, you actually solve this, take the derivative using implicit differentiation, differentiate with the respect to time, and plug in what you know, how this is changing and that tells you how this is changing. Pretty cool. That's called related rate. Suppose for example, you drop a stone into a very, very still pool? You have a ripple effect, those make consensus circles, and they are getting larger, and you know how fast the radius is changing, you can find how fast the area is changing because you have a connection between area and rate, area equals pr^2, so there you go, related rate. You know how one rate is changing; you can find how the related rate is changing. Okay, what else can you use a derivative for?
Well, the other thing you can do is actually graph really, really accurate pictures of functions. Finally, you can figure out that a parabola looks really pretty and bowl like, like this, and it's not something real exotic, that's just a nice pretty bowl. How do you do it? Well, you just start taking derivatives and analyzing things. First, you find the critical point. Those are the points where the derivative either equals 0 or the derivatives is undefined but the function is defined. So, for example, we can do these examples right here, you'll notice that the derivative equals 0 here, the tangent is horizontal right there, the tangent is horizontal here, and this example, the tangent is horizontal here, and then where is the derivative of nonexistence? The derivative doesn't exist here and the derivative doesn't exist there. Those are tangents for max or min; those are called critical points. Then what do you do with those things? Well, you set them up on a little number line on the xaxis and you look at the integrals all around it and you see whether the derivative is positive or negative. If the derivative positive, then that means slopes are positive, so the function must be increasing. If a derivative is negative, then that means the function must be decreasing. So, you can see that the function is, for example here, is decreasing, decreasing, decreasing, then increasing, then you can see it's increasing, increasing, increasing, then decreasing, then increasing. So, you can see where it's going up or it's going down by the sign of the first derivative. That also determines whether you have max's or min's anywhere, and that's called the first derivative test. Now, how do you figure out the curvature? The curvature is given by the rate of change of the derivatives. How the derivative is changing, so what you do there is you take the derivative of the derivative. So, you look at the second derivative, and with the second derivative and with the second derivative of 0, are potential points of inflexion, points where the concavity changes. This is concave up; it's curving upward. This is concave down; this is concave up. So, the cup is sitting up, the cup is sitting down, concave down. So, here you would see the second derivative is positive and the second derivative is negative, it changes here, now it's sitting up, positive, positive, we're concave up. Here, we're concave up, the second derivative is positive. Now you here we see concave down, second derivative is negative and second derivative here's also negative. This is a cusp point, the derivative doesn't exist there, this is a point of inflexion, this is a point of inflexion, this is a point of inflexion, and that's a minimum. So, just by taking derivatives and second derivatives, you can actually figure out and graph a very accurate sketch of even very complicated almost scary looking functions. By the way that fractional exponent, expect cusps. That's my warning for the day.
Okay, now, if you've got really exotic functions, that have denominators, then actually you may have asymptotes. So, don't forget that a vertical asymptotes is where the function after you simplify it, the bottoms equal to 0. So, whatever the bottom equals 0, after you simplify and reduce, those are going to give you, your vertical asymptote. Horizontal asymptotes, you take the limit as the x goes off to infinity, as you go off into the horizon, and you see what y value you are going to try to land to, if you are landing somewhere, then you know you've got a horizontal asymptote, and it's y equals that value. So, you can put in the asymptotes and you can do all the other calculus, get the curvature, see exactly what the beautiful picture looks like. And that was the end of differential calculus. Great, not problem, now what?
Well, now we move and look at the exact same thing we just did backwards. So, we look at math jeopardy. Oh, my goodness, we only have five minutes left. So, math jeopardy, here we go, so the idea is here we go, if I tell you what the derivative is, how can you find the function whose derivative is that? Well, this is a notion of an antiderivative. So, how do you find that antiderivative? Well, we set up formulas for that. If you want to find the anti derivative of x^n, it would be x^n + 1/n + 1. Take the derivative of that and see you get x^n, unless n = 1, then you're looking at the integral of 1/x, and what's the integral of 1/x? Well, it's a natural log of the absolute value of x because the derivative of natural log, we already saw was 1/x, so great. Now, how can you find exotic integrals? Well, remember that ddx represents differentiating with respect to x, and so therefore, the integral with respect to x, represents to integrate with respect to x. Now, if you were very complicated thing there, with an inside and outside, you might be able to untangle that, which potentially was made by the chain rule, by using substitution. Let u equal some big blop and the big blop derivative should appear somewhere else in your integral. If you got that, it sounds like a good candidate for udu substitution. And then you've got to change the dx to du by taking derivatives and seeing what the u equals in terms of dx. So, that's the udu substitution. You got that going on here, and now what can you do? You can take that and we can study the motion again.
Now, I can give you acceleration. If you integrate, you get velocity, if you integrate again, you get position. So, vertical motion, not a big deal. Anything that moves, anything at all that moves, we can now analyze. It's not a problem anymore folks. In this movement, we can antidifferentiate and figure out what it is. Not a problem. Now, where does this leave us, it brings us back to the very first question of the course, which was how do you find the areas under curves? We have to answer that; we haven't done it yet. It turns out the surprising answer is that it's the fundamental theorem of calculus. And the idea is if you want to find the area under this curve from a to b, right here, you want to find that area, then how do you do it? It turns out that if that function is called, let's say f(x), then all you do is integrate from a to b, f(x)dx, because you are summing up little rectangles in here that are base, small change in x, base times height, which is a function you use from a to b. And this equals F(b)  F(a), well, what's F? That's the antiderivative. So, if you take the derivative of F, you actually get little f(x). So, just find the antiderivative, plug in the big point, plug in the small point, subtract that will always give you the area under the curve. You can look at areas and more exotic things. For example, if the thing actually goes like this, then actually this is actually not very xeasy, the rectangles aren't very clear, the rectangles change from being going from green to green, over to green to orange, then orange to orange. It's not very uniform, however, if you put the rectangles in this way, and stack them this way, now you're summing with respect to y. And so, here, you would actually sum this with respect to y. So, you would integrate this, dy, and put the rectangles in this way, and you would put the rectangle in that way, and stack you would stack from low to high and you would stack the rectangle like this.
Okay, that is all of calculus; I did it under twenty minutes. That's what it is, go back think about it, have fun with it, congratulations folks, you just finished Calculus I. Have a good time. Celebrate, good luck on the final, bye.
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This is a fantastic way to kick off my review of calculus for my students! A great reminder of concepts they "should" know.