 So something else we can talk about is the net change and the average rate of change of a function. So functions are usually used to describe the changing values of a quantity over some interval. And so what this suggests is there are two new quantities we can try to compute. One of them is the net change of a quantity over some interval. And this is going to be just the difference between the amount at the end of the interval and the amount at the beginning of the interval. And it's important to remember difference corresponds to a subtraction, but very specifically in this case this is always end minus beginning. And that's a useful idea to carry through for the next five, six, ten, twenty math courses that you take. The other thing that we might be interested in is what we call the average rate of change of a quantity. And this is going to be the ratio between what the net change is and the length of the interval. And this average rate of change is fairly important. If I go from the third floor of a building down to the first floor of a building, I have a net change of height of maybe about 30 feet. But if it takes me 10 minutes to undergo that change, that's an average rate of change of 30 feet in 10 minutes. If it takes me one second to undergo that change, that's an average rate of change of about 30 feet per second. And one of them is bad, one of them is good. So we do care about this average rate of change. And one important note when you are dealing with these quantities, you must include the units when giving the net change and the rate of change. If you omit the units, your answer is at best incomplete. Now the good news is, units act just like algebraic variables. And we'll see what happens, what we mean by that in a few examples. So for example, suppose that I know that in the year 2000, the value of a stock portfolio was $130,000. Yay! In 2005, however, the value was $25,000. And so what we might care about is what is the net change and the rate of change. So let's go ahead and figure that out. Again, the net change is the difference between the value at the end and the value at the beginning. So the net change value at the end of the interval, which was $25,000, minus the value at the start, $130,000. So the net change is going to be the difference between those two values. That's going to be a negative number, $105,000. And this value is in dollars, this value is in dollars. And here's the thing that's worth keeping in mind. Units are like algebraic variables. If I had 25,000x minus 130,000x, my ability to combine these terms, they're like terms. So I'd get the difference of the coefficients, 25,000 and 130,000. And the variable doesn't change. It's still a variable x. So my units of my net change are also going to be in dollars. Now the fact that there's a negative sign there is that my net change corresponded to a decrease in the value. And again, we saw that was 130 is $25,000 at the end of the interval. Now the interval itself ran from the year 2000 to the year 2005. So we need to know what the length of that interval is. And so we start here, we end here, and the length is going to be end minus beginning. And in general, that's going to be true anytime we talk about change. It's always end minus beginning. And that's going to be five. And these are both years, so the difference is going to also be in years. So my rate of change is the ratio between the net change and the length of the interval, 105,000 over five years. And numerically, that's 105 over five, that's 21. And the units, well again, if this were an algebraic variable x and this were an algebraic variable y, my value would be negative 21,000 x over y. And so my units are going to be dollars over years. And so there's my net average rate of change minus $21,000 per year. What if I have the formula for a function? So here I have the velocity of an object where the electric field is given by V of t equals 25 minus 10 t meters per second. That's t seconds after the experiment begins. And so I want to find the net change and the rate of change of the object's velocity during the lighting of the first 10 seconds of the experiment. Now, importantly, we want to determine what interval we're looking at here. And so we're interested in the first 10 seconds of the experiment. And that suggests that the interval is going to go from t equals zero from the start of the experiment, because t is our number of seconds after the experiment begins, from the start to 10 seconds after. And that counts as the first 10 seconds of the experiment. So next I want to calculate the net change. So I need to know how much I have at the end of the interval at t equals 10 and how much I have at the beginning of the interval at t equals zero. So I need to know these values. At t equals 10, I have this nice formula that gives me the velocity. So I'll substitute t equals 10 into that formula. That's 25 minus 10 times 10. And after all the dust settles, that's 75. And don't forget the units. The units are these meters per second. So that's not an answer 75. That is an answer of negative 75 meters per second. And there's my value at the end. What about the start? Well, that's going to be at t equals zero. So I substitute that into my formula for velocity, the zero, 25 minus 10 times zero, just going to be 25. And again, I want to make sure I include those units still in meters per second. So there's my value at the start. I want the value at the end minus the value at the start. That's 25 meters per second. And again, the units act just like algebraic variables. These are meters over seconds. So are these. And so these are like terms because they have the same variables, raised to the same powers. And so I can add the coefficients minus 75 minus 25. That's going to be minus 100. And the variables don't change meters per second. And so there's our net change. Now to find the average rate of change, I need to know how long the interval was. So the interval went from t equals zero to t equals 10. So the length of that interval, n minus beginning, 10 minus zero is going to be 10 seconds. And our units per seconds, we're going to abbreviate that s. So I'm going to divide the net change by the length of the interval, 10 seconds. And that's going to give me the average rate of change. So the average rate of change, net change over the length, that's going to be minus 100 meters per second. That's our net change. The length of the interval is 10 seconds. That's going to be this. Numerically, this is going to be minus 100 over 10. That's minus 10. And the units, I can simplify this expression. It's a compound fraction. So maybe the easiest way of simplifying this is by multiplying numerator and denominator by s. That gets rid of the s in the numerator. It puts another factor of s in the denominator. So my units are going to be meters over seconds squared. And so my average rate of change, minus 10 meters per second squared.