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 • Cost – A cost function specifies the cost C as a function of the number of items x. Thus, C(x) is the cost of x items and has the form: Cost = variable cost + fixed cost, where the variable cost is a function of x and the fixed cost is constant. • A cost function of the form 𝐶 𝑥 = 𝑚𝑥 + 𝑏 is called a linear cost function: – The variable cost is mx, – The fixed cost is b, – The marginal cost, the cost for one more, is m. Definition • Revenue – The revenue resulting from one or more business transactions is the total payment received, sometimes call the gross proceeds. • If R(x) is the revenue from selling x items at a price of m each, then revenue is the linear function 𝑅 𝑥 = 𝑚𝑥. • The selling price m can also be called the marginal revenue. Definition • Profit – The profit is the net proceeds, or what is left of the revenue when costs are subtracted. • If the profit depends linearly on the number of items, the slope m is called the marginal profit. • Profit, revenue, and cost are related by: Profit = Revenue – Cost, or 𝑃 𝑥 = 𝑅 𝑥 − 𝐶(𝑥) Definition • Break-even – The break-even point is the number of items x at which profit is zero. • Note that if 𝑃 𝑥 = 0, then 0 = 𝑅 𝑥 − 𝐶(𝑥) and so revenue is equal to cost. 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 The demand for your hand-made skateboards, in weekly sales, is 𝑞 = −3𝑝 + 700 if the selling price is $p. You are prepared to supply 𝑞 = 2𝑝 − 500 per week at the price $p. At what price should you sell your skateboards so that there is neither a shortage nor a surplus? Example 6 Worldwide quarterly sales of a brand of cell phones were approximately 𝑞 = −𝑝 + 156 million phones when the wholesale price was $p. a. If the cellphone company was prepared to supply 𝑞 = 4𝑝 − 334 million phones per quarter at the wholesale price of $p, what would have been the equilibrium price? b. The actual wholesale price was $93 in the fourth quarter of 2004. Estimate the projected shortage or surplus at that price.