Find relative rates of change of x & y axis?

Questions
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1

Hi All,

The following question is a Calculus - relative rates problem.
Given the following data enclosed & plot, I am trying to find the :-
(i) - Rate of change of degrees - on the x axis …&
(ii) - Rate of change of value - on y axis at each data point plotted.

Ideally I am looking for a numeric & symbolic function / solution that represents the rate of change of both these axis.
Please review attachment for more details.

Many thanks for your help attention.
Best regards,
Lea…

2

Given the data as

In[1]:= Clear[leDa]
leDa = {{0, 32400}, {513.354, 57600}, {1173.38, 90000}, {1980.08, 
  129600}, {2933.45, 176400}, {4033.5, 230400}, {5280.22, 
  291600}, {6673.61, 360000}, {8213.67, 435600}, {9900.41, 
  518400}, {11733.8, 608400}, {13713.9, 705600}, {15840.7, 
  810000}, {18114.1, 921600}, {20534.2, 1040400}, {23101., 
  1166400}, {25814.4, 1299600}, {28674.5, 1440000}, {31681.3, 
  1587600}, {34834.8, 1742400}};

the differences in X and Y between a value and its predecessor are

In[3]:=  {deltaX, deltaY} = (Rest[#] - Most[#]) & /@ Transpose[leDa]
Out[3]= {{513.354, 660.027, 806.7, 953.373, 1100.05, 1246.72, 1393.39, 
  1540.06, 1686.74, 1833.41, 1980.08, 2126.75, 2273.43, 2420.1, 
  2566.77, 2713.44, 2860.12, 3006.79, 3153.46}, {25200, 32400, 39600, 
  46800, 54000, 61200, 68400, 75600, 82800, 90000, 97200, 104400, 
  111600, 118800, 126000, 133200, 140400, 147600, 154800}}

What do you mean with a symbolic function about that? Should deltaX and deltaY considered separately?

3

Hi Udo,
Many thanks for your helpful response.
Your reply above looks like a good start.
But I was wondering if it was possible to produce a function[time] on the x axis that would give the correct value at each time point.
Assuming the starting point is function[0t] = 32400. Then for example the following would look like :-
function[1t] = 57600,
function[2t] = 90000,
function[3t] = 129600

4

You could use e.g. Interpolation, giving for the sake of definitness arguments to the deltaX and deltaY data again

In[7]:= Clear[fX, fY]
fX = Interpolation[Transpose[{Range[0, Length[deltaX] - 1], deltaX}]];
fY = Interpolation[Transpose[{Range[0, Length[deltaY] - 1], deltaY}]];

In[10]:= GraphicsRow[{Plot[fX[o], {o, 0, Length[deltaX]}], Plot[fY[o], {o, 0, Length[deltaY]}]}]
Out[10]= (* do it yourself *)

In[11]:= fX[\[Pi]]
Out[11]= 974.14

In[13]:= fY[GoldenRatio^3] // N
Out[13]= 55699.7

if you prefer a formula, see e.g. FindFit
, requesting to have a candidate formula to get fitted in.

5

Many thanks. All very helpful. Really appreciate your help.
Lea…