Section 1.1 Notes. May 29, 2018

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1 Section 1.1 Notes May 29, 2018 Mathematical Models Goal: Recall the following facts about lines: 1) Equation: 2) Slope: 3) x-intercept: 4) y-intercept: Definition 1. Let D and R be two collections of objects. A function is The collection D is called the R is called the We usually represent elements of D and R by symbols (for efficiency). reprsenting elements of the domain (often x) the We call the symbol and the symbol representing elements of the range (often y) the 1

2 In this case, we will write to indicate that the value of y depends on the value of x. Additionally, we will often write (x, f(x)), called an ordered pair. This is convenient to convey information and for graphing. Example. The amount Westin spends on cat toys depends on how many cats he owns. For each cat he owns, he spends $15 on cat toys each year. a) Determine an appropriate choice for domain and range. b) Write an equation describing the dependent variable in terms of the independent one. c) How much does Westin spend on cat toys if he owns 3 cats? Cost, Revenue, and Profit We consider linear cost/revenue models. There are two kinds of costs: Variable cost can we written as Therefore cost can be written: For simplicity, we assume the number of units produced is the same as the number of units sold. Thus, we may represent revenue as. 2

3 The profit is defined as Example. A pencil maker has a monthly operating cost of $6,500 and makes pencils at a cost of $0.03 each. She sells them at a price of $0.10 each. a) Find the cost function. b) Find the revenue function. c) Combine the two to find the profit function. d) Find her monthly profits if her revenue is $9,500. 3

4 Example. Fred can order a product domestically or internationally. If he orders it domestically, it costs $12 per item and shipping is free. If he orders it internationally, it is $2 per item but shipping is a flat $10,000. a) Write the cost functions for each situation. b) Which is the better option if he buys 800 items? 1,200 items? c) If Fred buys 1,200 items, what should he set the price at to guarantee he breaks even (regardless of whether he buys the items domestically or internationally)? d) From (c), what is Fred s profit when he chooses the cheaper purchasing option? 4

5 Supply and Demand Availability and demand for a good affects the price that the good can be sold at. For convenience, we assume the relationships are linear. We expect an increase in products available for sale to correspond to a Below we illustrate the demand equation: Likewise, for the supply equation, we expect an increase in products demanded to correspond to a On the supply equation, the y-intercept represents the lowest price a supplier is willing to sell and is called the Remark: in the following example, we are able to choose how to represent our independent variable. There are multiple choices that all give the correct answer, but it is worth considering a convenient choice. Example. Diamonds sell for an average of $87 each when the demand per year is 230,000 diamonds. If the demand increases to 290,000 diamonds, then the price increases to an average of $91 per diamond. a) Find the supply equation for the diamonds. 5

6 b) Estimate the cost of a diamond if the demand increases even further to 295,000 diamonds per year. Linear Depreciation Many things become less valuable on the market over time.( For convenience, we again assume that this decay of value is linear.) In this case, their value is said to The object often has a minimum value, called the scrap value that represents We expect the slope of the line to be Example. A new car has a value of $15,000 and a 15-year life expectencey with a scrap value of $1,250. a) Find an equation modelling the value depreciation. b) How much is the car worth after 7 years? 12 years? 6