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 1st so in 2000 there's 1996 stores in 2005 6000 177 so on and so forth determine a reasonable approximation for the instantaneous rate of change 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 and we 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 it is a function of time so you see in the year 2000 there was 1996 stores 2003 4270 to 2003 let me do that in that blue color 2003 four thousand two hundred and seventy two 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 what 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 my best to approximate it might be more of some type of curve that looks something like this it looks something like this and I'm once again I'm just approximating what it might 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 to approximate the slope of the tangent line in 2003 when time is 2003 so the tangent line might look something like might look something like that I want to draw it so you see that it's this line right over here is tangent now they say approximate we don't have the information to figure out it 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 could 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 is we go from 2002 to 2003 that's going to be the number of stores in 2003 2003 minus the number of stores in 2002 so that's the change in our number of stores over the change in years of the change in time so this is going to be 2003 - mm - and so what is this going to be equal to see in 2003 we have four thousand two hundred seventy two stores in 2002 we have three thousand five hundred and one it's going to be that over well the denominator is just one so this is going to simplify to let's see I could do a little bit of I can do a little bit of math on the side here just so I don't make a careless mistake - three five oh one two minus one is one 7-0 742 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 stores per 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 - 2003 which is equal to see they had five thousand two hundred and thirty nine stores in 2004 they have four thousand two hundred and seventy two stores in 2003 and then only one year changes so 2004 - mm this is equal to I'll do the subtraction problem up here so five thousand two hundred and thirty nine minus four thousand two hundred and seventy two so this let's see nine minus two is seven let's see it looks like we're gonna have to do a little bit of regrouping here so let's say we take up we take one of the thousands from the five thousand so it's four and then that becomes ten hundreds so this becomes twelve hundred and now let's take one of those hundreds and gives it to the ten so now this becomes eleven hundred and they give that ten so then this becomes thirteen that we took one hundred from here and that make made gave us ten tens so now we're ready to subtract 13 minus 7 is 6 11 minus two is nine and then 4 minus 4 is 0 so it's nine hundred sixty-seven so this is equal to nine hundred and sixty-seven stores per year 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 960 seven plus 771 771 and then we can divide by two so let's actually figure it figure out what that is so let me add 771 over here 7 plus 1 is 8 6 plus 7 is 13 is 13 and then 1 plus this is 17 right over here and we're going to divide that by 2 so 2 goes into 1738 1738 it goes into 17 8 times 8 times 2 is 16 subtract I get a 13 goes 6 times 12 subtract get an 18 goes 9 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 this of the tangent line right over here is 869 stores per year