How to convert cumulative to incremental for plot

Question

Trying to show a plot of a power function $Y = a X^b$, where $Y$ is cumulative cost and $X$ is cumulative shipments where we'd like to see the incremental cost.

For $X_1$ the cost is $Y_1$, for $X2$ the cost is $Y_2 - Y_1$ for just the second unit shipped, for $X3$ the cost is $Y_3 - Y_2$ for just the third unit shipped... and so on.

I have a model of $\text{CumulativeCosts} (Y) = a \text{CumulativeShipments} (X) ^ b$ where $a$ is $8.11\times10^5$ and $b$ is $0.95$.

I'm having trouble creating the data give the above function such that we see a decreasing individual unit cost as more units are shipped. (like a decreasing exponential curve...)

I tried:

ListPlot@Table[
 (811014.9219997985` x^0.9495224913459213`)-
 (811014.9219997985` y^0.9495224913459213`), 
 {x, 1, 15}, {y, 0, 14}
]

The result is a dot plot of 15 $y$ values for each of the 15 $x$ values

Ideally I'd like a smooth curve showing the decrease in cost as more units ship.

There has to be a way to create a table of number of units shipped and the cost of the $i^\text{th}$ unit shipped?

Answer 1

You can use Differences and ListPlot:

c[x_] := 8.11 10^5 x^(0.95);
tc = Table[c[x], {x, 0, 15}];
dif = Differences@tab;
Column[{Panel@ ListPlot[tc, Joined -> True, PlotLabel -> "Total Cost", 
    DataRange -> {0, 15}, ImageSize -> 300],
  Panel@ListPlot[dif, Joined -> True, PlotLabel -> "Incremental Cost",
     DataRange -> {0, 15}, ImageSize -> 300]}]

Or, take the derivative of the total cost function c[x] and use Plot:

Column[{Panel@Plot[c[x], {x, 0, 15}, PlotLabel -> "Total Cost", ImageSize -> 300], 
  Panel@Plot[c'[x], {x, 0, 15}, PlotLabel -> "Incremental Cost", ImageSize -> 300]}]