# Undo a Rate with the Substitution Method

Suppose that the profit for a company is increasing at a rate of

where the company has been in operation for t years. What is the total change in profit over the first three years?

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# Death and Piecewise Linear Functions

Although this may seem a little gruesome, it is not uncommon for businesses to give discounts for volume sales. In this case, a mortician charges less per pound for bodies weighing more than a certain amount.

The local mortician charges by the pound for embalming according to the following table:

Find a piecewise linear function that models the cost as a function of weight.

# Rates and Rational Models

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