Applied Math

Quite some time ago I put out a request for recommendations on books with which to refresh my pretty much entirely forgotten calculus knowledge. Several readers provided some very helpful suggestions, and as a result, I ended up with a few new books on the shelf, a couple of which I've been slowly working through ever since (given my habit of having several books of different sorts going at any given time, my pace is at times glacial).

However, I was excited the other day to finally (ten years after finishing the Saxon Calculus book in high school) manage to use calculus at work. I believe the following is sufficiently scrubbed of any company specific or confidential information to be sharable, and for those others out there with an interest in how things work (and/or mathematics in particular -- though I fear any true math afficianados will find my efforts rather simplistic) I thought I'd share the result.

Much of what I do at the moment consists of setting the list price for products, and trying to determine what is the best price at which we will sell a large number of units (usually done by cutting price) while still booking enough profit on each unit to make money for the company.

If you track the results of this experimentation over time, you can plot the data points out on a graph of price (Y axis) versus unit volume (X axis) such as this:



This line conforms to the equation: [quantity] = [slope]*[price]+[intercept]

You can then use the equation, and the values of slope and intercept for the line, to calculate estimated quantities at additional price points.

Now by using these now known values to solve for revenue and profit (assuming that you know cost), you can calculate the likely amounts of revenue and profit at various price points, as well as looking at their historical values.

You can plot these values on a graph, and see that there is a curve for revenue and one for profit.



Now say that you want to find the price at which the most profit would be realized. You take the equation for the profit curve and take its derivative. When the derivative of the equation equals zero, you have the point of the curve where it is flat, as it turns over from ascending to descending. Solve for the price that will produce that point equal to zero, and you have the price that will result in the maximum profit -- or at least as close to it as the correlation of your data to reality will allow.

Fun, eh?