I have some data in Seconds (X axis) and % (y axis). Lets say for ten seconds the data below:
S = 0 % = 0.1
S = 1 % = 0.2
S = 2 % = 0.3
S = 3 % = 0.4
S = 4 % = 0.5
S = 5 % = 0.6
S = 6 % = 0.7
S = 7 % = 0.8
S = 8 % = 0.9
S = 9 % = 1.0
S = 10 % = 1.1
I know i can calculate the rate of change, lets say between 0 and 1 seconds with: ((0.2 - 0.1 )/0.1)*100 which would give ROC of 100%
So the maximum rate of change in these 10 seconds is actually 100%. So for my first question, if i wanted the max rate of change for 3 minutes from this data (180 seconds) would i use something like:
100% / 10 = 10% (per second average)
10% * 180 = 1800% (per 180 seconds)
1800 / 60 = 30 average per min
30 * 3 = 90% average ROC per min over 3 minutes
Is that correct?
My second question: After plotting the Second and Percent onto a graph, how would i work out what the First Derivative plot points would be? I am a bit lost with this part as i can't seem to get the first derivative calculation right.
I think im confused by the formula: f(x) - f(a) / x - a
because this function (f(x)) is not telling me what the function is to do with x. For example :
f(x) = x + 1
tells me that the output = input + 1 , so what would be the output of f(x) ??
The $x$ are your seconds, and $f(x)$ are the % on your y-axis. I don't see how the rate of change is 100%, as the value on the y-axis changes by 100% from $S=0$ to $S=1$, then by 50% from $S=1$ to $S=2$, then by 33.333..%, etc. The rate of change is constant, and is the value
$\displaystyle \frac{f(0+1) - f(0)}{(0+1) - 0} = 0.2 - 0.1 = 0.1$
for $S=0$, then it is
$\displaystyle \frac{f(1+1) - f(1)}{(1+1) - 1} = 0.3 - 0.2 = 0.1$
for $S=1$, etc.
This value is your first derivative as well, as the function is linear (given the discrete points you have listed). If you don't get this I suggest picking up a book on calculus. There are many good ones out there which will give you a good introduction to the derivative within the first chapter