Asif’s Backscratchers Inc.
Problem: Asif runs a big modern day company that is like any other. The company sells quantities of products that are plastic backscratchers. The company right now is failing severely because there are much more products being made then being sold and we need a way to find out how to find the number of production level that will maximize a company’s profit.
We are given that the Total money the company spends is 840 + 10.26x - 0.0541x2 + 0.0007x3 And the Profit we are making from this amount of quantities are 30.5 – 0.001x + 10.004x2
So what do we know so far? What kind of math can be used? The answer is derivatives. Usually you learn about derivatives in pre calc when you first learn about finding slopes of tangent lines right? Your teacher goes to tell you that derivatives are the regularly seen as a method finding the slope of a point for a function. But more importantly Derivatives runs our society. If you are thinking “oh the big companies and the rich white people run our society”, then you are right. These people know their calculus and use it to become rich. Derivatives are pretty simple for polynomials. To find the derivative of a polynomial you use the power rule. There are many other rules that can be used for other types of functions.
The proof for the power rules:
Historically the power rule was derived from Cavalieri’s quadrature formula which is when the area under Xn for any integer n≥ 0. But nowadays the power rule is derived first and integration considered as its inverse. For[pic], the derivative of [pic] is [pic]
That is [pic]
To prove the power rule for differentiation, we use the definition of the derivative as a limit. [pic]
Now you know what derivatives are; one important application of derivatives is on marketing, mostly on marginal revenue and marginal cost. Marginal revenue and Marginal costs are derivatives. The marginal...