AP®︎/College Calculus BC
Course: AP®︎/College Calculus BC > Unit 2Lesson 1: Defining average and instantaneous rates of change at a point
- Newton, Leibniz, and Usain Bolt
- Derivative as a concept
- Secant lines & average rate of change
- Secant lines & average rate of change
- Derivative notation review
- Derivative as slope of curve
- Derivative as slope of curve
- The derivative & tangent line equations
- The derivative & tangent line equations
Newton, Leibniz, and Usain Bolt
Why we study differential calculus. Created by Sal Khan.
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- why did they name this subject calculus?(359 votes)
- It's from the Latin meaning reckon or account (like calculate), which came from using pebbles (calx) to count with.(553 votes)
- People say that we see math in our everyday lives -- and while I understand how this concept applies to beginning math, pre-algebra, algebra, and trig, how does this apply to calculus? At1:01, Sal says that differential calculus is all about finding instantaneous rate of change, but is that the only "everyday use"? Or is calculus simply a concept that is used in other subjects, or even professions, like engineering?
- I think the best way to find a good answer to this question is to just keep watching the videos! If you attend college for any engineering discipline, you have to learn calculus before you even begin learning the specifics of your discipline (whether it be mechanical, electrical, civil, computer, computer science, etc...). The best way to understand what "every day" things calculus will enable you to do is to learn calculus and start doing incredible things every day :-)(220 votes)
- Who invented calculus?(41 votes)
- Lots of people were involved in the development. Archimedes approached the idea in his iterative computation of the value of pi, but he was hampered by a poor numeric system and lack of algebraic notation. Ultimately, Leibniz and Newton should get the credit for actually formulating it, and we still use the Leibniz notation of dy/dx for most purposes.(95 votes)
- what exactly is a derivative?(11 votes)
- What a great question......A derivative calculates the slope of a line.
Well then,, why not just call it the slope?
But what if the line is curving? Crap. Now the slope keeps changing.
So draw a curve,,,any curve,,,,,,,,,,,,,now draw a tangent line at any point on that curve . THE DERIVATIVE FUNCTION WILL CALCULATE THE SLOPE AT ANY POINT ON THE CURVE.(43 votes)
- why is it called "differential" calculus(13 votes)
- The name comes from the fact that we're dealing with rates of change, which are calculated by dividing the difference in one quantity by the difference in another quantity (for example, speed is the difference in position divided by the difference in time).(42 votes)
- what is the secant and tangent line?(7 votes)
- secant line passes between two points on the curve, and tangent line never goes through the curve, it barely touches a point on the curve and goes straight.(11 votes)
- So calculus is all about finding instantaneous rate of change, or is there a better, simpler way of describing it.(10 votes)
- That is essentially what differential calculus is all about. You will later learn that there are actually two primary types of calculus: differential and integral. Differential calculus focuses on, as you pointed out, the instantaneous rate of change of a function at any point on that function. This is necessary because, as Sal pointed out in the video, the rate of change of a function can vary wildly on any given interval. The other type, integral calculus, focuses on finding the area underneath a curve (for example, finding the area underneath a standard y=x^2 parabola on the interval x=1 and x=4). While these two may seem like completely separate topics, you would be surprised by how closely related the two are, and how you can apply each in a variety of situations.(19 votes)
- Was it Newton or Leibniz, who invented the form of notation that we use today in calculus?(6 votes)
- We actually use a mixture of notations, though mostly Leibniz and Lagrange.
For differential calculus:
The notation with the lowercase letter d is from Leibniz.
The notation involving the primes as in f'(x), is from Lagrange.
And there are still some other notations by a variety of mathematicians, mostly for more advanced calculus.
Newton's notion uses dots placed over the variable. I've never seen anyone use that notation other than to say "this is Newton's notation", but I suppose there must be somebody, somewhere that uses it.
For Integral Calculus, the integral symbol was invented by Leibniz, but the notation for the definite integral, based on Leibniz's
∫symbol was invented by Fourier.(23 votes)
- So, was calculus originally invented to find out the instantaneous speed of an object?(7 votes)
- Calculus for the most part was rooted in the ancient obsession with trying to square the circle (reduce the area of a circle to a simple x*y formula) as well as computing areas and tangents to other curves. Since there are a lot of curves in position-speed-acceleration calculations, calculus proves an extremely valuable tool with which to analyze and predict behavior but it wasn't really the precipitating factor in its creation, for one main reason that there really wasn't the ability to precisely measure time on small scales.
For a brief introduction:
And if you are further curious:
Although this book is specifically centered about the evolution of "e", the author does a great job of tying the discovery of logarithm properties to modern calculus methods by building up the history of calculus from its earliest ancient dabbling to its explosion in the 17th and 18th centuries.
Here’s a great video series for the history of mathematics, from Pythagoras to calculus and beyond:
- How useful is calculus in regard to chemistry?(6 votes)
- If you study reaction kinetics in second semester college chemistry or the second half of AP chemistry, you might find that the integrated rate laws are derived from the integration of the differential rate laws. This is usually just very basic calculus as it applies to rate of change, so you aren't likely to need to use any calculus when learning about rate laws, but it's good to be aware of it.
Later on in college or grad school calculus may appear more often in chemistry especially when it overlaps with physics and probably not so much in organic chemistry.(8 votes)
This is a picture of Isaac Newton, super famous British mathematician and physicist. This is a picture of a Gottfried Leibnitz, super famous, or maybe not as famous, but maybe should be, famous German philosopher and mathematician, and he was a contemporary of Isaac Newton. These two gentlemen together were really the founding fathers of calculus. And they did some of their-- most of their major work in the late 1600s. And this right over here is Usain Bolt, Jamaican sprinter, whose continuing to do some of his best work in 2012. And as of early 2012, he's the fastest human alive, and he's probably the fastest human that has ever lived. And you might have not made the association with these three gentleman. You might not think that they have a lot in common. But they were all obsessed with the same fundamental question. And this is the same fundamental question that differential calculus addresses. And the question is, what is the instantaneous rate of change of something? And in the case of Usain Bolt, how fast is he going right now? Not just what his average speed was for the last second, or his average speed over the next 10 seconds. How fast is he going right now? And so this is what differential calculus is all about. Instantaneous rates of change. Differential calculus. Newton's actual original term for differential calculus was the method of fluxions, which actually sounds a little bit fancier. But it's all about what's happening in this instant. And to think about why that is not a super easy problem to address with traditional algebra, let's draw a little graph here. So on this axis I'll have distance. I'll say y is equal to distance. I could have said d is equal to distance, but we'll see, especially later on in calculus, d is reserved for something else. We'll say y is equal to distance. And in this axis, we'll say time. And I could say t is equal to time, but I'll just say x is equal to time. And so if we were to plot Usain Bolt's distance as a function of time, well at time zero he hasn't gone anywhere. He is right over there. And we know that this gentleman is capable of traveling 100 meters in 9.58 seconds. So after 9.58 seconds, we'll assume that this is in seconds right over here, he's capable of going 100 meters. And so using this information, we can actually figure out his average speed. Let me write it this way, his average speed is just going to be his change in distance over his change in time. And using the variables that are over here, we're saying y is distance. So this is the same thing as change in y over change in x from this point to that point. And this might look somewhat familiar to you from basic algebra. This is the slope between these two points. If I have a line that connects these two points, this is the slope of that line. The change in distance is this right over here. Change in y is equal to 100 meters. And our change in time is this right over here. So our change in time is equal to 9.58 seconds. We started at 0, we go to 9.58 seconds. Another way to think about it, the rise over the run you might have heard in your algebra class. It's going to be 100 meters over 9.58 seconds. So this is 100 meters over 9.58 seconds. And the slope is essentially just rate of change, or you could view it as the average rate of change between these two points. And you'll see, if you even just follow the units, it gives you units of speed here. It would be velocity if we also specified the direction. And we can figure out what that is, let me get the calculator out. So let me get the calculator on the screen. So we're going 100 meters in the 9.58 seconds. So it's 10.4, I'll just write 10.4, I'll round to 10.4. So it's approximately 10.4, and then the units are meters per second. And that is his average speed. And what we're going to see in a second is how average speed is different than instantaneous speed. How it's different than what the speed he might be going at any given moment. And just to have a concept of how fast this is, let me get the calculator back. This is in meters per second. If you wanted to know how many meters he's going in an hour, well there's 3,600 seconds in an hour. So he'll be able to go this many meters 3,600 times. So that's how many meters he can, if he were able to somehow keep up that speed in an hour. This is how fast he's going meters per hour. And then, if you were to say how many miles per hour, there's roughly 1600-- and I don't know the exact number, but roughly 1600 meters per mile. So let's divide it by 1600. And so you see that this is roughly a little over 23, about 23 and 1/2 miles per hour. So this is approximately, and I'll write it this way-- this is approximately 23.5 miles per hour. And relative to a car, not so fast. But relative to me, extremely fast. Now to see how this is different than instantaneous velocity, let's think about a potential plot of his distance relative to time. He's not going to just go this speed immediately. He's not just going to go as soon as the gun fires, he's not just going to go 23 and 1/2 miles per hour all the way. He's going to accelerate. So at first he's going to start off going a little bit slower. So the slope is going to be a little bit lot lower than the average slope. He's going to go a little bit slower, then he's going to start accelerating. And so his speed, and you'll see the slope here is getting steeper and steeper and steeper. And then maybe near the end he starts tiring off a little bit. And so his distance plotted against time might be a curve that looks something like this. And what we calculated here is just the average slope across this change in time. What we could see at any given moment the slope is actually different. In the beginning, he has a slower rate of change of distance. Then over here, then he accelerates over here, it seems like his rate of change of distance, which would be roughly-- or you could view it as the slope of the tangent line at that point, it looks higher than his average. And then he starts to slow down again. When you average it out, it gets to 23 and 1/2 miles per hour. And I looked it up, Usain Bolt's instantaneous velocity, his peak instantaneous velocity, is actually closer to 30 miles per hour. So the slope over here might be 23 whatever miles per hour. But the instantaneous, his fastest point in this 9.58 seconds is closer to 30 miles per hour. But you see it's not a trivial thing to do. You could say, OK, let me try to approximate the slope right over here. And you could do that by saying, OK, well, what is the change in y over the change of x right around this? So you could say, well, let me take some change of x, and figure out what the change of y is around it, or as we go past that. So you get that. But that would just be an approximation, because you see that the slope of this curve is constantly changing. So what you want to do is see what happens as your change of x gets smaller and smaller and smaller. As your change of x get smaller and smaller and smaller, you're going to get a better and better approximation. Your change of y is going to get smaller and smaller and smaller. So what you want to do, and we're going to go into depth into all of this, and study it more rigorously, is you want to take the limit as delta x approaches 0 of your change in y over your change in x. And when you do that, you're going to approach that instantaneous rate of change. You could view it as the instantaneous slope at that point in the curve. Or the slope of the tangent line at that point in the curve. Or if we use calculus terminology, we would view that as the derivative. So the instantaneous slope is the derivative. And the notation we use for the derivative is a dy over dx. And that's why I reserved the letter y. And then you say, well, how does this relate to the word differential? Well, the word differential is relating-- this dy is a differential, dx is a differential. And one way to conceptualize it, this is an infinitely small change in y over an infinitely small change in x. And by getting super, super small changes in y over change in x, you're able to get your instantaneous slope. Or in the case of this example, the instantaneous speed of Usain Bolt right at that moment. And notice, you can't just put a 0 here. If you just put change in x is zero, you're going to get something that's undefined. You can't divide by 0. So we take the limit as it approaches 0. And we'll define that more rigorously in the next few videos.