Question

What is the difference between Delta x and dx?

Answer 1

infinity

Answer 2

\[\Delta x \] means the change in x where dx is the derivative of x

Answer 3

when Delta x reaches infinity it becomes dx; the ghost of a departed ratio

Answer 4

So when you multiply something by dx, what does that mean essentially?

Answer 5

it means that you are relating it back to when it had a real value

Answer 6

you might need to example that multiply by dx bit tho for clarity

Answer 7

So dx is a "ghost" ratio, basically a ratio with 0 in the denominator... but not quite... and multiplying that times something gives you a value based on... I mean, I know that's the integral but I'm still not sure I understand it.

Answer 8

2x dx for example is meaningless on its own we need more to the notation

Answer 9

{S} () dx is an operator on a derivative telling us that "x" was the important stuff

Answer 10

{S} (2x) dx means that we can undo this derivative to find out where it came from

Answer 11

or at least a family of functions from which it came;

{S} (2x) dx = x^2 + C where C is some arbitrary constant relating this to a family of functions

Answer 12

\[dy=\int\limits_{}^{}dx\]\[dy/dx = 1\]
Those are essentially the same thing, but why does it just work like dividing or multiplying?

Answer 13

\[\frac{dy}{dx}=1\]
becasue its a ration at heart can be split into the form
\[dy=1 dx\]
which is meaningless without the \(\int\) symbol

Answer 14

\[\int dy = \int 1dx\]
\[y=x+c\]

Answer 15

Hmm so why does there have to be an S symbol if it's multiplied by dx? I mean, I know you need some way of showing that there is a place to go from for definite integrals, but why have the symbol for indefinite integrals? What does the S symbol mean by itself with no dx around?

Answer 16

it represents an elongated "s" reminding us that an integration is the sum of the derivative parts - with respect to (*): d*

Answer 17

\[\sum_{0}^{\inf}f(x)\Delta x\implies\int_{0}^{inf}f(x)dx\]

the summation is for discrete functions; the integral is for continuous functions

Answer 18

Ok, that makes sense, so it is literally just the exact same thing as saying the limit of an infinite riemann sum of infinitely small values...?

Answer 19

yes, whereas the Reimann is a discrete count; the integration is a contiuous count

Answer 20

Ok, I think that makes sense, leibniz notation is kinda weirding me out still, I'll sit and think about it a little longer. Thanks.

Answer 21

good luck :)