The cost per candle, 2.25, is the marginal cost (and also the slope) and the fixed cost is 22 (which is also the y-intercept).

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1 Section 1.4 Linear Models Lots of Vocabulary in this Section! Cost, Revenue and Profit Functions A simple cost function can be a linear function: C(x) = mx + b, where mx is the variable cost and b is the fixed cost (expenses that do not depend on the number of items). The slope m is called the marginal cost, the cost per item. And x is the number of items. Example: We are going to make candles. Each candle requires $2.25 worth of materials (wax, wick, scent, etc.) and in addition we need to purchase materials for making the candles (mold, stove, etc.) that will cost $22 but can be used over and over again. The cost per candle, 2.25, is the marginal cost (and also the slope) and the fixed cost is 22 (which is also the y-intercept). Our cost model is C(n) = 2.25 n + 22, with n being the number of candles made. Note, the marginal cost (slope) will be in units of y per units of x. In this example, Cost (C) is in dollars and n is in candles, so the marginal cost is $2.25 per candle. Revenue would be the money coming into our business. A simple revenue function would be R(x) = mx; which is the price we sell an item at, m, and x being the number of items sold. For this model m is the marginal revenue. How should we price our candles? Say each candle will sell for $5. Our revenue model is R(n) = 5n, n being the number of candles sold. The marginal revenue (aka slope) is 5, which is $5 per candle and the y-intercept is 0, there is no fixed revenue.

2 Profit would be what is left of our income after expenses are accounted for. Profit = Revenue Cost. Or P(x) = R(x) C(x). For our example, P(n) = [5n] [2.25n + 22], now we can pull in our algebra skills and simplify the function: P(n) = 2.75n 22. If profit is negative, for example P(2) = 2.75(2) 22 = -16.5, our business is in trouble as we have lost $ If profit is positive, for example P(10) = 2.75(10) 22 = 5.5, our business is profitable and we have made $5.50. If the profit is zero, P(8) = 2.75(8) 22 = 0, then our business has broken even. The break-even point is the number of items where we will break even. So 8 candles manufactured and sold is the break-even point for our example. Graphically, the break-even point is where the graphs of the revenue function and cost function intersect. {Sketch Graphs} (You should read all of section 1.4!! Especially the definition boxes and examples)

3 Demand and Supply Functions For many items, when prices are cut demand increases, as the item is more affordable. So demand functions usually have a negative slope,(or negative rate of change) as a function of price. (Remember to think about domain what is reasonable if you cut the price until the item is free you may not have a reliable model.) Also though, if prices are rising, more suppliers are interested in getting in on the action and offering more items for sale. So supply functions tend to have a positive slope (positive rate of change) as a function of price. So we manufacture our candles and go to the local Saturday market to sell them at $5. The following week we hit a craft fair and sell candles there also but raise the price to $6.50 (which changes our revenue and profit functions). Your spouse, who has been complaining about the mess in the garage, gets interested when s/he sees the profit potential with the higher prices. With only you manufacturing candles, your supply was limited to making 20 candles a week, but with assistance you could make 50 candles a week. Our Results: Price per candle $5 $6.50 Demand (candles sold) Supply (candles manufactured) To find models for the demand and supply function we will use algebra. We are just finding the equation of a line through two points. For the demand function our points are (5, 30) and (6.5, 18) and the demand function is q = -8p q represents the quantity demanded and p stands for the selling price. For the supply function our points are (5, 20) and (6.50, 50) and the supply function is q = 20p 80. Again, q is quantity but this time quantity supplied and p is the selling price

4 What do we have here? Demand: q = -8p + 70, so demand is decreasing as the price increases, and it is decreasing by 8 candles for each dollar increase in the price. 70 would be the demand if the candles were free (??) Supply: q = 20p 80, so the supply increases as the price increases, and it is increasing by 20 candles for each dollar increase in the price. So how does it all balance out? The equilibrium price is the price where supply matches demand; the value of price, p, when Demand = Supply. -8p + 70 = 20p 80 p 5.36, when the price of the candles is $5.36 the quantity demanded will match the quantity supplied. That quantity, or equilibrium demand, would be about 27 candles. {sketch graphs, show intersection} *Note, if the price is set less than the equilibrium price, you would sell all of your stock and there would be a shortage in the market. If you set your price higher than the equilibrium price you will have unsold stock to clear out of your warehouse (garage). When supply equals demand the market clears. **Also Note: page 72 Q and A: linear models are accurate for small ranges of the variables. See graph on page 74 as well.

5 Change Over Time Over time our reputation in the candle industry grows and our annual sales Year Annual Sales ($) increase. Letting t represent the number of years since 2000, and s represent our annual sales our model for sales as a function of time is S(t) = 500t 2500 Assuming the trend continues (here we are extrapolating!) when will our sales reach $4000? 4000 = 500t 2500 t = 13, so in the year 2013 we predict sales will reach $4000. Looking at our model, S(t) = 500t 2500, the slope 500 is measuring the rate of change, sales are increasing at $500 per year and the y-intercept is the initial quantity (which for our example doesn t make a whole lot of sense) that if we were making candles in the year 2000, sales would have been negative $2500. Note that the units for rate of change are the units for y per unit of x. Page 76: Linear Change over Time: q(t) = mt + b b = quantity at t =0 or initial quantity and the units are the same as for q. **Read FAQs on page 77!