Estimating derivatives with two consecutive secant lines

The table shows the number of stores of a popular US coffee chain from 2000 to 2006. The number of stores recorded is the number at the start of each year on January 1. So in 2000, there was 1,996 stores, in 2005, 6,177, so on and so forth. Determine a reasonable approximation for the instantaneous rate of change, in coffee stores per year at the beginning of 2003-- so we care about 2003-- by taking the average of two nearby secant slopes. So let's visualize this. So this right over here, I've plotted all of the points. Now let me make sure that the axes are clear. This horizontal axis, this is my t-axis that tells us the year. And then the vertical axis is the number of stores. And we could even say that it is a function of time. So you see in the year 2000, there was 1,996 stores-- 2003, 4,272. 2003-- let me do that in that blue color. 2003, 4,272 stores. Now if you could imagine that they're constantly adding stores, you could even imagine minute by minute they're adding stores. So this is just sampling the number of stores they had on January 1. But if you were to really plot it as a more continuous function, it might look something like this. I'll do my best to approximate it. It might be more of some type of curve that looks something like this. And once again, I'm just approximating what it might actually look like. So when they're saying the instantaneous rate of change in coffee stores per year, so this is the change, the instantaneous rate of change of stores per time. They're really saying, we need to approximate the slope of the tangent line in 2003, when time is 2003. So the tangent line might look something like that. I want to draw it so you see that this line right over here is tangent. Now, they say approximate. We don't have the information to figure it out exactly. But we have some data around it, and we can figure out the slopes of the secant lines between this point and those points. And then we can take the average of the slopes of the secant lines to approximate the slope of this tangent line. So for example, we could find the slope of this secant line right over here as we go from 2002 to 2003. And then we can find the slope of this secant line as we go from 2003 to 2004. And if we average those, that should be a pretty good approximation for the instantaneous rate of change in 2003. So let's do that. So the slope of this pink secant line, as we go from 2002 to 2003, that's going to be the number of stores in 2003 minus the number of stores in 2002. So that's the change in our number of stores over the change in years, or the change in time. So this is going to be 2003 minus 2002. And so what is this going to be equal to? Let' see, in 2003 we have 4,272 stores. In 2002, we have 3,501. It's going to be that over-- well, the denominator is just 1. So this is going to simplify to, let's see. I can do a little bit of math on the side here just so I don't make a careless mistake. Minus 3501, 2 minus 1 is 1, 7 minus 0 is 7, 42 minus 35 is 7. So this is equal to 771. So their average rate of change from 2002 to 2003 was 771 stores per year. Now let's do the same thing for this red secant line between 2003 and 2004. The slope of that secant line is going to be the number of stores in 2004 minus the number of stores in 2003 over 2004 minus 2003, which is equal to, let's see. They had 5,239 stores in 2004. They have 4,272 stores in 2003. And then only one year changes, so 2004 minus 2003 is 1. So this is equal to-- I'll do the subtraction problem up here. So 5,239 minus 4,272. So this, let's see. 9 minus 2 is 7. Let's see, it looks like we're going to do a little bit of regrouping here. So let's say we take one of the thousands from the 5,000, so it's 4. And then that becomes 10 hundreds. So this becomes 1,200. And now let's take one of those 100s and give it to the 10. So now this becomes 1,100, and we give that 10. So then this becomes 13. We took 100 from here, and that gave us 10 tens. So now we're ready to subtract. 13 minus 7 is 6, 11 minus 2 is 9, and then 4 minus 4 is 0. So it's 967. So this is equal to 967 stores per year. So we really just have to take the average of this number and that number to approximate the instantaneous rate of change. So let's do that. So let's take 967 plus 771, and then we can divide by 2. So let's actually figure out what that is. So let me add 771 over here. 7 plus 1 is 8. 6 plus 7 is 13. And then 1, plus this is 17 right over here. And then we're going to divide that by 2. So 2 goes into 1,738, it goes into 17 8 times. 8 times 2 is 16. Subtract, we get a 13, it goes six times. 12. Subtract, we get an 18. It goes nine times. And then we don't have any remainder. So 18, no remainder. So our approximation for the instantaneous rate of change for the slope of our approximation for the slope of the tangent line right over here is 869 stores per year.