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Introduction to average rate of change

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So I have two different definitions for the function D over here over here d of t is equal to 3 t plus 1. We could imagine the d could represent distance as a function of time. Distance measured in meters and time measured in seconds. So over here when time is 0 right when we're starting our distance is going to be 1 meter. After 1 second has gone by our distance is now going to be, so 3 times 1 plus 1 is going to be 4. Our distance now is going to be 4 is going to be 4 meters. After 2 seconds our distance is going to be 3 times two is 6 plus 1 it's going to be 7 it's going to be 7 meters. So given this definition of D of T this function definition. What is the rate of change of distance with respect to time and let me write it this way. What is the change in distance the rate of change of distance with respect to time which people sometimes called speed. This triangle right over here. That's the Greek letter Delta. Just means change in distance over change in time Well, what is this going to be? Let's just take two points Let's just say the change in distance over change in time when time goes from time equals 0 to time equals 1 So over here our change in time is equal to 1 our change in time is 1 and what's our change in distance? Well our change in distance when our time increased by 1 our distance increases by 3 it goes from 1 meters to 4 meters. So our change in distance is equal to 3. So it's going to be equal to 3 over 1 or just 3. If we wanted the units it would be 3 meters, every 1 second. Now let's think about it, does that change if we pick any other two points? what if we were to say between 1 second and 2 seconds so between 1 second and 2 seconds our change in time is 1 second and then our change in distance is our distance increases by 3 meters. So once again our change in distance over change in time is 3 meters per second. This is all review and you might recognize, we pick any two points on this line here and we're going to have the same rate of change of distance with respect to time. In fact that's what defines a line. Or one of the ways to think about a line or a linear function is that the rate of change of one variable with respect to the other one, is constant. In this particular one we're talking about the rate of change of the vertical variable with respect to the horizontal one. We're talking about the slope of the line. This is the slope. This line has a slope of 3. That's what defines a line or one of the things that defines a line is the slope between any two points is going to be exactly 3. Just as a little bit of review from other Algebra you've seen before. You can even pick it out in the function definition. It's actually written in slope-intercept form right over here where the slope is this 3 and our D intercept, when t is equal to 0, is going to be equal to one. All of this is review and if any of this sounds foreign to you, I encourage you to watch the Khan Academy videos on slope and slope intercept form and things like that. But this is all a background to get us to this curve over here because this is interesting. Because here we're no longer dealing with a line. We actually have a curve and this curve right over here. It's a quadratic, it's a parabola. Let's just say distance as a function of time was defined this way instead of it being 3t plus 1 it is t squared plus 1 and what's interesting here is that the rate of change you can have visualized it is it's always changing for example: If you were to pick an arbitrary point on this curve and if you think about a tangent line to it. A tangent line, a line that has the same slope for just that one moment, it just touches on it, over here it seems pretty positive, while over here it still is positive, but it seems a lot less positive. So it looks like the rate of change. It looks like the rate of change is actually changing here. In this case it looks like the line is getting steeper and steeper and steeper as time goes by. Or another way to think about it is, our rate of change of distance with respect to time is Increasing its not just constant. It's not a constant slope. The slope is increasing as time increases. So for something like this, how do we think about rate of change? How do we think about rate of change of one variable with respect to another, in particular distance with respect to time? Well later on in your math careers you'll find out that this is actually what most of differential calculus is all about and you will get to differential calculus. But for our purposes, we have another tool at our disposal and this is actually a good foundation for the calculus that you will learn in the future. This is the notion of average rate of change. Let me write this down average average rate of change I already hopefully gave you an argument Why it's very hard you'll need actually calculus to figure out the Instantaneous rate of change the rate of change Right at that point the slope of the tangent line that you're going to need a calculus But to figure out the average rate of change between two points We can use very similar tools that we use to figure out the slope of a line. So for example: We could figure out the average rate of change between any two points on this curve. So for example: We could say the average rate of change from when we go from t equals 0 to t equals 3 That's going to be the slope of this secant line So let me draw that. So let me do this in a more fun color. I'll do it in this color All right So as I said the actual Instantaneous rate of change is constantly changing. In this case it's increasing, and we'll need calculus for that. But now we could think about average rate of change which would be the slope of the line that connects these two points. And once again, it's only an average rate of change. As we see, the actual curve its rate of change is lower earlier on, and then it's rate of change is higher as we get closer and closer to 3 seconds. But the average rate of change is going to be the slope of this line right over here. And we could think about that. This is going to be our change. So the slope of this line, is going to be a change in distance over change in time. An average rate of change especially when you're talking about a curve like this it depends on what starting and ending point This is the average rate of change for the first 3 seconds, is going to be- Well what is our distance right at the third second? Well, it's going to be d of 3. Now what was our initial distance? It's going to be d of zero. So this expression right over here. It's going to give me my change in distance. It's going to give me my change in distance right over here Let me write. It actually maybe I could do it this way So I could do it over here. So our change in distance is this. Change in distance which is d of 3, d of 2 minus d of zero. So it gives us this height and then we want to divide it by our change in time Well, we finished it 3 seconds, and we started at zero seconds. And notice, all this is, is change in your vertical axis over change in your horizontal axis So it is just the slope of this line right over here. So what is this going to be? Well, d of 3 after 3 seconds. 3 squared plus 1, we added a distance of 10 meters, and at Time zero T zero squared is zero plus 1. We had a distance of 1 meter, so our change in distance is 10-1 It's 9 meters, and you see that right over here. Change in distance is 9 meters We go from 1 meter to 10 meters and and we increase 9 meters So that's why it's a positive value. And our change in time? We see time increased by 3 seconds, so divided by, change in time is 3 seconds, and so this is going to be equal to 3. If we care about our units, which is often useful to care about, we see that the numerator here that's in meters. So we'll have a meters in the numerator and down here our change in time that's going to be in seconds. So our average rate of change over the first 3 seconds is 3 meters per second. Once again, it's not the exact instantaneous rate of change at any moment That is actually changing but it comes as sort of an average rate of change. We could also think about our average rate of change over other durations. We could think about what is our average rate of change between the second and the third second. So this one is going to be the slope of this Secant line. So over there it's going to be our distance at 3 seconds, minus our distance at 2 seconds, over our change in time. We finish at 3 seconds, we started at 2 seconds So D of 3 is 9 we already figured that out. D of 2? 2 squared plus 1 is 5. This right over here is at 5. We see that. So, 9-5. We have an increase in distance of 4 meters And you see that right over here, increase in distance of 4 meters. And our change in time? This happens over 1 second So we have this is equal to 4 meters per 1 second. Or we could say, 4 meters per second. So this is interesting. Notice, our average rate of change here, the slope of this secant line, is 4 meters per second. Which is faster than the average rate of change of this bigger Secant line, that we started with. Which is 3 meters per second. Points that are further as time goes by, that is increasing. Which gives you a sense of what might be happening. As you get smaller and smaller intervals, and t is larger and larger. That's actually a nice little little tidbit for differential calculus