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# Estimating derivatives with two consecutive secant lines

Video transcript

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.