Presentation on theme: "1.2 Linear functions & Applications"— Presentation transcript:
1 1.2 Linear functions & Applications Linear Functions: y=mx+b, y-y1=m(x-x1)Supply and Demand FunctionsEquilibrium PointCost, Revenue, and Profit FunctionsBreak-even Point (quantity, price)
2 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 decreasesIf price of an item increases, producers see profit and supply of item increases.
3 Linear Function f defined by (for real numbers m and b)x=independent variabley=dependent variable
4 Cranberry example and explanation of quantity (x-axis), price (y-axis) See Page 18 of e-textCranberry example of late 1980’s early 1990’s.Explanation of why price is on the y-axis.
5 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.
6 EXAMPLEIfis 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?
10 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.
11 EXAMPLEIf 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?
13 Example 2 page 22Part 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.
15 Equilibrium PointThe 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.
16 To find the equilibrium quantity algebraically, set the supply and the demand functions equal and solve for quantity.
17 ExampleUsing demand function p = q supply function p = .02q + 3 to find… (a) the equilibrium quantity (b) the equilibrium price (c) the equilibrium point
18 Answer a) 74 – 0.08q = 0.02q + 3 71 = 0.10q 710 = q c) (710, $17.20) b) p = 0.02q + 3p = 0.02(710) + 3p = 17.2
19 Fixed costs (or overhead) costs that remain constant regardless of the business’s level of activity.Examplesrental feessalariesinsurancerent
20 costs that vary based on the number of units produced or sold. Variable Costscosts that vary based on the number of units produced or sold.Exampleswagescost of raw materialstaxes
24 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.
25 ExampleThe 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.
26 Answer Use and slope = 15, point (40, $2000) y – 2000 = 15 (x - 40)
27 Revenue, R(x) R(x) = p x p is price per unit x is number of units money from the sale of x unitsR(x) = p xp is price per unitx is number of units
28 Profit, P(x)the difference between the total revenue realized and the total cost incurred: P(x)= R(x) – C(x)
29 ExampleIf 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.
31 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.
32 Finding breakeven quantity If R(x) = C(x), then P(x) = 0. Where this happensis the breakeven pointTo 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.
33 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.