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Introduction to average rate of change

Video transcript
so I have two different definitions for the function D over here over here D of T is equal to 3t plus one and we could imagine the D could represent distance as a function of time distance measured in meters and time measured in seconds and so over here when time is zero right when we're starting our distance is going to be one meter when time after one second has gone by our distance is now going to be so three times one plus one is going to be for our distance now is going to be four is going to be four meters after two seconds our distance is going to be three times two is six plus one it's going to be seven it's going to be seven 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 call speed and this this triangle right over here that's the Greek letter Delta just means change in distance over change in time well what's what is this going to be well wait let's just take two points let's just say the change in distance over change in time when time goes from time equals zero to time equals one so over here our change in time is equal to one our change in time is one and what's our change in distance well our change in distance when our time increased by one our distance increases by three it goes from one meters to four meters so our change in distance is equal to three so it's going to be equal to 3 over 1 or just 3 and if we wanted the units it would be 3 meters 3 meters every every one second and let's think about it does that change if we pick any other two points what if we were to say between between one second and two seconds so between one second and two seconds our change in time is one second and then our change in distance is our distance increases by by three meters so once again our change in distance over change in time is three meters per second and this is all review and you might recognize well it's 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 and 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 slope of three that's what defines a line or one of the things that defines a line is the the slope between any two points is going to be exactly three and 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 three and our d intercept when T is equal to zero is going to be equal to one and 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 and let's just say distance as a function of time was defined this way instead of it being 3 T plus 1 it is T squared plus 1 and what's interesting here is that the rate of change you can visualize 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 is 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 it actually in this case it looks like the line is getting steeper and steeper and steeper as time goes by as time goes by or another way to think about it is a rate of change of distance with respect to time is increasing it's not just constant it's not a constant slope hope 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 can have we have another tool at our disposal and this is actually a good foundation for the calculus that you will learn in the future and 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 to 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 any two any two points on this curve so for example we could say the average rate of change from when when we go from T equals zero to T equals three that's going to be the slope of this secant line so let me draw that so let me do this in a more 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 but 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 and as we see the actual curve it's rate of change is slower earlier on and then it's rate of change is higher as we get closer and closer to three 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 and an average rate of change especially when you're talking about a curve like this it depends on what's starting and ending point this is the average rate of change for the first three seconds is going to be well what is our what is our distance at the end of the right at the third second well it's going to be D of three now what was our initial distance it's going to be D of zero so this is 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 three D of three 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 three 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 three after three seconds three squared plus one is your we added a distance of 10 meters and at time zero T is zero squared is zero plus one we had a distance of one meter so our change in distance is ten minus one it's nine meters and you see that right over here our change in distance is nine meters we go from one meter to 10 meters and and we increase nine meters so that's why it's a positive value and our change in time we see time increased by three seconds so divided by change in time is three seconds and so this is going to get B equal to three and 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 a change in time that's going to be in seconds so our average rate of change over the first three seconds is three meters per second once again it's not the exact instantaneous rate of change at any moment that is actually changing which tells us our average rate of change and 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 of this secant line and so over there it's going to be our distance at three seconds minus our distance at two seconds over our change in time we finish at three seconds we started at two seconds so D of three is nine we already figured that out D of two two squared plus one is this right over here is at five we see that so nine minus five so we have we have an increase in distance of four of four meters and you see that right over here increase in distance of four meters and our change in time this happens over one second so we have this is equal to four meters four meters per one second or we could say four meters per second four meters per second and so this is interesting notice our average rate of change here the slope of this secant line is four meters per second which is faster than the average rate of change of this of this law of this bigger secant line that we started with which is three meters per second and that hopefully gives you it gives you a sense that look even the average rate of change if we go between 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 and that's actually a nice little little tidbit for differential calculus