Or is there some other function?

This is the string to be simplified.

So if we want to be really cute about it, we would say non negative. rational numbers.

Well all of the coefficients on and I want to be careful with the term coefficients, because traditionally we view coefficients as always being constants but here we have functions of x as coefficients.

So what if p of x is equal to, well, the constant term is f of 0, and then it's plus the derivative of this function, so the slope of this function at that point, f prime of 0 times x.

So, x 2, our function is equal to 1.

And its solution isn't a number, its solution is a function, or a set of functions, or a class of functions.

The point at which we break is when i becomes greater than n.

And this is what kind of leads us to the formal definition.

And it keeps going in either direction.

I'll do it as a dotted line.

When we multiply these two functions, when we multiply this times this times the delta function, this is let me write this.

And this is a new thing that we talk about technology on semi log curves.

So I could also write this as psi, as x and y, which is a function of x.

The inverse Laplace transform of this thing is going to be equal to we can just write the 2 there as a scaling factor, 2 there times this thing times the unit step function.

Let's go ahead and get these fields populated.

I'll just show you some Python syntax that I've been using that is pretty helpful.

Well, I have to raise it to the second power.

So let's solve for x.

The domain of a function, you'll often hear it combined with domain and range.

I'm just gonna define that as 5x minus y.

In general, if you have f(x) x n, then this is odd if 'n' is odd, and this is an even function if 'n' is even.

But, it turns out that the cost function for gradient of cost function for linear regression is always going to be a bow shaped function like this.

And then the hash itself is designed to be a very slow hash function.

So the idea here is just to show you that exponential functions are really, really dramatic.

And don't get confused with the term even function and the term even number.

So just like that, using a kind of visual evaluation of the integral, we were able to figure out the Laplace transforms for a bunch of different situations involving the Dirac delta function.

So it can generate a whole bunch of lists and the reason why this is useful is that we can use these inside of loops.

But the neat thing about the completing the square is it will always work.

And in that domain, 2 is sitting there, you have 3 over there, pretty much you could input any real number into this function.

So this is also an odd function.

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So we need to find a y where 1 times its second derivative, plus 5 times its first derivative, plus 6 times itself, is equal to 0.

If this point essentially almost becomes this point, then our slope is going to be the slope of our tangent line.

And so you're probably saying, Hey Sal, this is very abstract, how does this relate to the functions that I've seen in the past?

If it were x 3, that is odd.

And we know that in the real world, everything isn't these nice continuous functions, so over the next couple of videos we're going to talk about functions that are a little bit more discontinuous than what you might be used to even in kind of a traditional calculus or traditional

What we really want here is to take n divided by 5 and round it up.

Here's how a capture replay tool works.

But let's say that psi, and I won't write that it's a function of x and y.

If you haven't seen this before in Python, don't worry, but it is possible to make 1 function inside another.

Lets move on to the next one and as spoiler each of the remaining functions is exactly one line

But the domain of a function is just what values can I put into a function and get a valid output.

This is a badly structured function.

You're intersecting the graph twice.