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Hello, and welcome to the screencast about estimating the derivative.
So given a table of values for a function,
how can we estimate the value of the derivative at a certain point?
Recall that you saw before that the forward in backward different can be
calculated with the formula,
f'(a) is approximately equal to f(b) minus f(a) over b minus a.
So the example we're gonna look at today is we're gonna estimate f'(-3).
Using the different techniques, we're gonna do forwards, backwards,
as well as central.
And then, I've given a table of x values here as
well as some values of the function at those certain points.
Okay, so we're gonna be focusing in on negative 3.
So to do the backwards difference,
that means we're gonna wanna use the point before negative 3 as well as negative 3.
So then our backwards difference
is going to look like, so
f' at -3 is going to be approximately
f(-3) minus f(-4),
all over -3 minus a -4.
All right, so just following the formula up here.
Now I realize that the a value may not match with the a value here in
the formula, but remember the forwards and backwards can be used in the one before or
after, just depending on which way you're doing.
So this formula is just like I said kind of a rule to follow for that.
Okay, so f(-3) by their table is
4- f(-4) by our table is 1.5.
And when you do the algebra and
the dominator that ends up giving you the value of positive 1.
So when you go ahead and simplify that that gives you a value of 2.5, okay?
Forward difference, so
we're now gonna use the value forward all over front of negative 3.
So, in this case, that's gonna be negative 2.
So, we're gonna do f(-2) minus f(-3),
all over -2 minus -3.
And using the values off the table,
f(-2) is 0.5 minus, f(-3) is 4,
and that's all over a positive 1.
And that gives us a value of negative 3.5.
Okay, so you notice the forward and
backward's difference is quite different with this particular function.
So let's take a look at the central difference.
So central difference means you're gonna want to think of -3 as the center,
so f'(-3) is a value.
That means you want to use the values on both sides.
In this time we're not gonna be using the value at negative 3 and
let's see if this gives us a little estimate.
So f(-2) minus f(-4),
cuz those are gonna be the 2 values around the center of -3.
So negative 2 minus negative 4, my negatives will cooperate, there we go.
And when I go and plug and chug those that's gonna be 0.5
minus 1.5 all over 2.
So that gives us negative 1 over 2.
Or negative 0.5.
Now what's interesting is the central difference can also be calculated given
these two backwards and forwards differences.
So if you notice, this is also equal to,
I'll just kind of put this over here in brackets as an aside.
If you were to take the average of the backwards and
forwards, that would give you the central.
So remember, average means you're gonna add those two values together, And
if you divide by 2, you'll also get negative 0.5.
So I prefer obviously to do the calculation using the formula
only because just in case if one of these two numbers were off,
that means this number would also be off.
So I prefer to use the raw data in the table.
All right, thank you for watching.
[MUSIC]
