In this segment, we will talk about how we can estimate the second derivative of a function,
which is given at discrete data points.
So, if we are given, let's suppose, x naught, y naught, x one, y one,
all the way up to x n, y n. So, let's suppose we're given n plus one data points
and if they're not given in ascending order of x, you can put them in that particular fashion. So,
we are assuming that this data set which is given to us is in ascending order of x values.
And so what we want to do is, we want to be able to find out what the value of the
second derivative of a function will be at a particular point, so then the
second derivative is approximately given by the value of the function at x of i plus 1, minus 2,
the value of the function x sub I, plus the value of the function at x
sub i minus 1, divided by the difference between
the… Let's suppose x of i plus 1 and x of I, whole squared. Now, keep in mind that this is the same
formula which you use for the second derivative of a function of a continuous function. But, you got
to understand that when we are finding the second derivative of a continuous function the value of
the function is given at all the points which we need in order to find the second derivative.
So, what is the difference? Is that now since the value of the function is given
at discrete data points you have to have the value of the function at this point,
this point, and the point where you are trying to find the second derivative.
Those are have to be available to you. Also, what you got to have is that the difference between
the point ahead with the point where you are looking for the second derivative,
and the diff and the difference between the point where you're looking for the second derivative,
and the point behind it, that distance has to be the same. Only then we can apply this
particular central divided difference formula to find the second derivative of a function.
Let's uh look at an example and I think some of the things which I mentioned here
will be clear when we look at those… Look at the example. So, here is an
upper velocity of a rocket which is given and you are given the velocity. So, v of t is given
and you want to find the jerk of the of the rocket. So, jerk off the rocket is given as the
first derivative of the acceleration, which will be of course then the second derivative of the
velocity. So, if we want to find the jerk of this rocket at 15 seconds that means we have to find
the second derivative of the velocity, and it is asked that hey we find the uh this jerk at
15 seconds. If that is the case, now, first we are given these data points in ascending order,
so we don't need to re, we don't need to sort them out. And then let's see now we have 15
is where we want to find the jerk, and so what we have to figure out is that hey do we have a point
behind and a point ahead? Which we do. 10 and 15. 10 is a point behind and 20 is in front
of it. But then we also have to see that whether the distance between the two points here is same
as the distance between these two points, which it does seem to be the same because 15 minus 10 is 5,
and 20 minus 15 is 5. So, having said that, let’s go ahead and see that how we can find the jerk
at 15 seconds. So, the jerk of 15 seconds will be equal to, so based on the formula which we have,
will be the velocity at 15, plus five, so this is your term x of i plus one, or t sub i plus one,
minus two times the velocity at the point where we are seeking to find the jerk, plus v at 15
minus 5, which is the point at the point behind. So, now here we have, what is the difference
between 20 minus 15 squared, because that is the difference between the two, and here we'll have
velocity at 20, minus 2 times the velocity at 15, plus the velocity at 10, divided by 5 squared. So,
what do we get from here? Velocity at 20 minus is 517.35, minus 2 times the velocity at 15 is 362.78
plus the velocity at 10 is 227.04, and we divide it by 5 squared. So, having said that,
all we have to do is to do these calculations and be able to find out what the velocity is.
So, in this uh sorry the what, what the jerk is. So, in this case, it turns out to 0.7532
meters per second cubed. Because it is the first derivative acceleration with respect to time,
so the units are meter per second cubed. And that is the end of uh this segment.