Applications of Differentiation in Economics

Applications of Differentiation - Slope of tangent line - Rate of change - Example Determine the slope of the tangent line of the curve ? = ?2+ 2 at ? = 3

Solution: ? would give equation for slope ? = 2? thus the slope at x = 3 is 2 3 = ?.

Exercise Find the equation of the tangent line to the graph of ? = 3?2 2? at the point (2,8). Find slope first. Given m and (??,??) find equation of the tangent line. ? ?1= ?(? ?1) To find slope, ? = 6? 2 ?? ????? ?? ? = 2 ?? 10

?1,?1 = 2,8 ? ?1= ? ? ?1 ? 8 = 10 ? 2 ? 8 = 10? 20 ? = 10? 20 + 8 ? = 10? 12

Exercise Find the equation of the tangent line to the graph of ? = (?+1)at ? = 1. Solve for m (gradient) ?? ??= Therefore m at x=3 is 4 (1 + 1)2=4 4? 4 (? + 1)2?.? (?????? ? ??) ? = 4= 1

m=1 and ??,?? = (?,?) ? ?1= ? ? ?1 ? 2 = 1 ? 1 ? 2 = ? 1 ? = ? 1 + 2 ? = ? + 1

Application to Economics (marginal analysis) Marginal cost : Suppose C = ?(?) is the cost of producing x units. Then the derivative ? ? ?? ?????? ? ? ???????? ????. Marginal cost is the change in the total cost that arises when the quantity produced has an increment by a unit. That is, it is the cost of producing one more unit of a good.

Example Suppose that the cost in dollars for a weekly production of x tons of steel is given the formula: ? = 0.1 ?2+ 5? + 1000 a) Find the marginal cost function b) Find the cost and marginal cost when ? = 1000 tons c) Interpret ? (1000)

a) ?? = 0.2? + 5 b) ? 1000 = 0.1 (1000)2+5(1000) + 1000 = $ 106, 000 ? 1000 = 0.2 1000 + 5 = $205 c) ? 1000 = $205 means that the cost of producing the next ton (ie. the 1001th ton) would be $205

Marginal Revenue Marginal Revenue : Suppose R = ?(?) is the revenue of selling x units. Then the derivative ? ? ?? ?????? ? ? ???????? ???????. The marginal revenue would give the revenue from the sale of one additional unit.

Example Suppose that the revenue for a weekly sale of x units of T-shirts is given by the formula ? ? = 4?2 2?. a) Find the marginal revenue b) Find the revenue and marginal revenue when 10 T-shirts are sold. c) Interpret ? (10)

a) ?? = 8? 2 b) ? 10 = 4(10)2 2 10 = 380$ ? 10 = 8 10 2 = $78 c) ? 10 = $78 means that the revenue from selling the next T-shirt (11thunit) would be 78$

Maximum Profit attainable Given the C(x) and the R(x), maximum profit is achieved when ? ? = ? ? A firm maximizes profit by operating where marginal revenue equal marginal costs. http://en.wikipedia.org/wiki/Profit_maximization http://en.wikipedia.org/wiki/Profit_maximization

Example The cost of producing a certain commodity is given by ? ? = 120 + 5?2dollars, where x is the number of items produced in thousands. Each item is sold for $300. How many items need to be produced to maximize profit?

? ? = 120 + 5?2, ? ? = 300? Max profit is attained when ? ? = ? ? ? ? = 10 ? ? ? = 300 ; 10? = 300 ? = 30 thousand units

Exercise The cost of producing x number of cars is given by ? ? = 45?2 3? + 300 whilst the revenue function is given by ? ? = 2000? + 1500 a) How many cars should the company produce to maximize profit? b) What is the maximum profit attainable?

a) Max profit is when C(x) = R(x) 90? 3 = 2000 ? = 22.19 = 22 ???? b) We need to find ?(?) ?????? = ??????? ???? = ? ? ?(?) P(x) =(2000? + 1500)-( 45?2 3? + 300) = 2000x+1500 - 45?2+ 3? 300 = 45?2+ 2003? + 1200 P(22) = 45 222+ 2003(22) + 1200 = $23, 486

To be continued .