Getting the speed of known x,y coordinates in a graph

Question

Good day mathematicians,

I'm a computer engineer working on an automation project. Right now I have measured an area surface of cm2 which I am plotting on the y-axis. On the x-axis I have the time which starts at 0 and increments by 1.

With my data, I want to determine the speed (area/s) at all given x-intervals and plot that on a new graph.

What I have tried: Keep adding every single y-value and divide it by the current x-value. So:

x: 1, y: 5 x: 2, y: 8 x: 3, y: 10

My speed at x3 would be (5 + 8 + 10)/3 but it does not seem quite right. The graph I am plotting is slowly nearing 1 (I forgot the name of this phenomenon, but I know it has a name) even though it the y values are far and few between. What is going on, and how can I solve this? Some colleague talked about using differentials.

Answer 1

I'm not sure I totally understood your problem, but let me try it.

From what I got, you have a growing area you keep mesuring at regular intervals, and from this data, you want to plot the speed¹ at which it grows.

Speed, in this context can mean 3 things

1. Instant speed. This doesn't really make sense, because you have discrete data, and can't calculate an instant rate from this.
2. Average speed from the beginning. To get this you have to use the formula: $y/x$. So your speed at $x_3$ would be $10/3$
3. Average speed since the last measure. And I thnik you meant this one. You can find it by using: $$\frac{y_n-y_{n-1}}{x_n-x_{n-1}}$$ Your speed at $x_3$ is now: $$\frac{y_3-y_{2}}{x_3-x_{2}}=\frac{10-8}{2-1}=2$$

Please note that this is the average speed between t=2 and t=3. So you have to decide whether to place this data at 2 or 3 in your new graph.

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¹ Disclaimer: I'll avoid using "speed" for anything other than distance/time, but "growth rate" seems economy-conoted, any suggestion is welcomed.