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**1.2 Linear functions & Applications**

Linear Functions: y=mx+b, y-y1=m(x-x1) Supply and Demand Functions Equilibrium Point Cost, Revenue, and Profit Functions Break-even Point (quantity, price)

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**Linear functions - good for supply and demand curves.**

If price of an item increases, then consumers less likely to buy so the demand for the item decreases If price of an item increases, producers see profit and supply of item increases.

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**Linear Function f defined by**

(for real numbers m and b) x=independent variable y=dependent variable

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**Cranberry example and explanation of quantity (x-axis), price (y-axis)**

See Page 18 of e-text Cranberry example of late 1980’s early 1990’s. Explanation of why price is on the y-axis.

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**Demand Function defined by p = D(q)**

The function that gives the relationship between the number of units (q) that customers are willing to purchase at a given price (p). The graph of a demand function is typically decreasing.

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EXAMPLE If is the relationship between p, the price per unit in dollars and q, the quantity demanded, what is the demand when the price is $50 per unit?

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ANSWER p = -0.04q + 72 50 = -0.04q +72 -22 = -0.04q 550 = q

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EXAMPLE: Find the price when the level of demand is 500.

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Answer: p = -0.04q + 72 p = (500) +72 p = p = 52

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**Supply Function defined by p = S(q)**

gives the relationship between the number of units (q) that suppliers are willing to produce at a given price (p). The graph of a supply function is typically increasing.

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EXAMPLE If p = q is the relationship between the price (p) per unit and the quantity (q) supplied, When the price is set at $73 per unit, what quantity will be supplied?

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Answer p = q 73 = q 68 = 0.04q 1700 = q

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Example 2 page 22 Part c shows (6, $4.50) as the intersection of the supply and the demand curve. If the price is > $4.50, supply will exceed demand and a surplus will occur. If the price is < $4.50, demand will exceed supply and a shortage will occur.

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Graph of example 2

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Equilibrium Point The price at the point where the supply and demand graphs intersect is called the equilibrium price. The quantity at the point where the supply and demand graphs intersect is called the equilibrium quantity.

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To find the equilibrium quantity algebraically, set the supply and the demand functions equal and solve for quantity.

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Example Using demand function p = q supply function p = .02q + 3 to find… (a) the equilibrium quantity (b) the equilibrium price (c) the equilibrium point

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**Answer a) 74 – 0.08q = 0.02q + 3 71 = 0.10q 710 = q c) (710, $17.20)**

b) p = 0.02q + 3 p = 0.02(710) + 3 p = 17.2

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**Fixed costs (or overhead)**

costs that remain constant regardless of the business’s level of activity. Examples rental fees salaries insurance rent

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**costs that vary based on the number of units produced or sold.**

Variable Costs costs that vary based on the number of units produced or sold. Examples wages cost of raw materials taxes

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Cost Function Total cost C(x)= variable cost + fixed cost

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Example A company determines that the cost to make each unit is $5 and the fixed cost is $1200. Find the total cost function

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Answer C(x) = 5x

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Marginal Cost is the rate of change of cost C(x) at a production level of x units and is equal to the slope of the cost function at x (in linear functions) It approximates the cost of producing one additional item.

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Example The marginal cost to make x capsules of a certain drug is $15 per batch, while it cost $2000 to make 40 batches. Find the cost function, given that it is linear.

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**Answer Use and slope = 15, point (40, $2000) y – 2000 = 15 (x - 40)**

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**Revenue, R(x) R(x) = p x p is price per unit x is number of units**

money from the sale of x units R(x) = p x p is price per unit x is number of units

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Profit, P(x) the difference between the total revenue realized and the total cost incurred: P(x)= R(x) – C(x)

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Example If the revenue from the sale of x units of a product is R(x) = 90x and the cost of obtaining x units is (a)determine the profit function. (b)find the profit from selling 300 units.

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**Answer a) P(x) = R(x) – C(x) P(x) = 90x – (50x + 800) P(x) = 40x – 800**

b) P(300) = 40 (300) – 800 P(300) = $11,200

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**Review of Profit, Revenue, and Cost**

P(x)= R(x) – C(x) when R(x) > C(x) then P(x)> 0 or a gain. If R(x) < C(x) then P(x) < 0 or a loss.

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**Finding breakeven quantity**

If R(x) = C(x), then P(x) = 0. Where this happens is the breakeven point To find the breakeven quantity (x-value of the break even point) either use a or b below. Set R(x)=C(x) and solve for x. Set P(x)=0 and solve for x. Always round the breakeven quantity up to the next whole number.

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**A manufacturer can produce x units for (240 + 0. 18x) dollars**

A manufacturer can produce x units for ( x) dollars. They can sell the product for $3.59 per unit. (a) find the cost function (b) find the revenue function (c) find the profit function (d) the break-even quantity (e) the profit from producing 250 units. (f) number of units for profit of $1000.

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