Math 1320 Chapter 1: Linear Functions 1.2 Functions and Models

Modeling To mathematically model a situation means to represent it in mathematical terms. The particular representation is called a mathematical model. Mathematical models do not always represent a situation perfectly or completely. Some represent a situation only approximately, or only some of the time, whereas others represent only some aspects of the situation.

Type of Models Analytic models are obtained by analyzing the situation being modeled. Curve-fitting models are obtained by finding mathematical formulas that approximate observed data. Continuous models are defined by functions whose domains ae intervals of the real line. Discrete models have only set points as the domain. Discrete models are used extensively in probability and statistics.

Example 1 The amount of free space left on your hard drive is 50 GB and is decreasing by 5 GB per month. Find a mathematical model for this situation.

Example 2 My square orchid garden is right off the side of the house so that the house itself forms the northern boundary. The fencing for the southern boundary costs $4 per foot, and the fencing for the east and west sides cost $2 per foot. Find the total cost function as a function of the length x of a side.

Definition

Example 3 Your college newspaper has fixed production costs of $70 per edition and marginal distribution costs of 38¢ per copy. The paper sells for 48¢ per copy. a. Write down the associated cost function C(x) in dollars. b. Write down the revenue function R(x) in dollars. c. Write down the profit function P(x) in dollars. d. What profit (or loss) results from the sale of 100 copies of the paper? What about 500 copies? e. How many copies should be sold in order to break even?

Example 4 The Metropolitan Company sells its latest product at a unit price of $5. Variable costs are estimated to be 30% of the total revenue, while fixed costs amount to $7,000 per month. How many units should the company sell per month in order to break even?

Definitions Demand – A demand equation or demand function expresses demand q (for quantity) as a function of the unit price p (price per item). Supply – A supply equation or supply function expresses supply q as a function of the unit price p. Equilibrium – Occurs when supply = demand. It is usually the case that demand decreases and supply increases as the unit price increases.

Example 5

Example 6