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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# Category: Applied Calculus

## Undo a Rate with the Substitution Method

## Death and Piecewise Linear Functions

## Rates and Rational Models

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?

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.

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