Let’s take a look at a problem that requires a bit of ingenuity to put into standard minimization form.
Problem – Garton’s Seeds has a seed mixture containing three types of seeds: bluegrass, rye, and Bermuda. The cost per pound of the three seeds are 16 cents, 14 cents and 12 cents. Bluegrass seed must be at least 25% of the each batch. The amount of Bermuda must be no more than 2/3 the amount of rye in each batch. To fill current orders, Garton’s must make at least 6000 lbs of the mixture. How many pounds of each seed should be in the batch so that the cost of the batch is minimized?
Many situations require us to take the derivative of a quotient. One situation like this is a model of a stocks price to earning ratio. In this situation the price is modeled by a function P(t) and the earnings by E(t).To find the rate at which the PE ratio is changing, you need to need to take the derivative of the model for the PE ratio.
The ratio is modeled by a quotient, P(t)/E(t). Since this is a quotient, we need to apply the quotient rule for derivative,
but apply this to our functions. This means u will be replaced with P and v will be replaced with E to give
As you may have noticed, some matrix operations are very tedious by hand. In particular, matrix multiplication and matrix inverses are much easier to do using a graphing calculator or WolframAlpha. In the Weekly Learning Plan, I put several handouts to help you use your calculator to do these problems.