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- Thread starter jasonlr82794
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If I may be pedantic, I believe you mean ##\frac{d\vec{x}}{dt}##. :tongue:

You seem to be looking at it in a Zeno's Paradox-like way. Instantaneous velocity is a lot like the average velocity between two infinitesimally distant points in time.

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If I may be pedantic, I believe you mean ##\frac{d\vec{x}}{dt}##. :tongue:

You seem to be looking at it in a Zeno's Paradox-like way. Instantaneous velocity is a lot like the average velocity between two infinitesimally distant points in time.

You may be :P

Totally agree with this, it turns out that if you look at the average velocity over two points and you keep making the points closer to each other, the average velocity keeps getting closer and closer to a specific number. If you make the points infinitely close, the average velocity gets infinitely close to this number. This number is the instantaneous velocity.

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I like to consider myself a pure mathematician for the most part, so I prefer using covectors to define derivatives so that no one knows what I'm talking about except people who know stuff. But that's messy.

For the more common usage, we consider the derivative of a scalar function of one variable by finding a limit of the difference quotient, given by

$$\lim_{\Delta x\rightarrow 0}\frac{y(x+\Delta x)-y(x)}{\Delta x} = \frac{dy}{dx}$$

To get 2x using the limit method,

$$\frac{d}{dx}[x^2]=\lim_{\Delta x\rightarrow 0}\frac{(x+\Delta x)^2-x^2}{\Delta x} = \lim_{\Delta x\rightarrow 0}\frac{x^2+2x\Delta x +(\Delta x)^2-x^2}{\Delta x} = \lim_{\Delta x\rightarrow 0}(2x+\Delta x) = 2x$$

Velocity is different, because it's a vector, but in one dimension (forward and backward movement only) we can treat it as a scalar.

Zeno's paradoxes are a bunch of famous paradoxes that were conceived by a Greek philosopher.

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