d Applications of Partial Derivative i E 0.25 Lecture a l l u - PowerPoint PPT Presentation

Section 17.2 d Applications of Partial Derivative i E 0.25 Lecture a l l u d b Dr. Abdulla Eid A . College of Science r D MATHS 104: Mathematics for Business II Dr. Abdulla Eid (University of Bahrain) Applications 1 / 6

The Marginal Cost d i E Definition a Given the cost c = c ( x , y ) as a joint function of producing x quantities of l l u item A and y quantities of item B . d 1 The partial marginal cost with respect to x is given by ∂ c b ∂ x . A 2 The partial marginal cost with respect to y is given by ∂ c ∂ y . . r D Dr. Abdulla Eid (University of Bahrain) Applications 2 / 6

Example If c = 7 x + 0.3 y 2 + 2 y + 400. Find the marginal cost with respect to y d for x = 20 and y = 30. i E a Solution: l l u ∂ c d ∂ y = 0.6 y + 2 b A ∂ c ∂ y ( 20, 30 ) = 20 . r D Dr. Abdulla Eid (University of Bahrain) Applications 3 / 6

Competitive and Complementary Products d i E Sometimes two products may be related such that change in price of one a of them affect the demand for the other. For example, Milk and yoghurt. l l u d q 1 = f ( p 1 , p 2 ) Demand function for the first product b q 2 = g ( p 1 , p 2 ) A Demand function for the second product . r D Dr. Abdulla Eid (University of Bahrain) Applications 4 / 6

Definition 1 If ∂ q 1 > 0 and ∂ q 2 > 0 d ∂ p 2 ∂ p 1 i E Then we say that the two products are competitive , i.e., an increase a of the price in the second item, increase the demand for the first item l l u and vice versa. Example, milk and yoghurt. d b 2 If A ∂ q 1 < 0 and ∂ q 2 < 0 ∂ p 2 ∂ p 1 . r D Then we say that the two products are complementary , i.e., an increase of the price in the second item, decrease the demand for the first item and vice versa. Example cars and gasoline. Dr. Abdulla Eid (University of Bahrain) Applications 5 / 6

Example If q 1 = 1500 − 40 p 1 + 3 p 2 and q 2 = 900 + 5 p 1 − 20 p 2 . Determine whether these two products are complementary or competitive or neither. d i E Solution: a l l ∂ q 1 u = 3 > 0 d ∂ p 2 b A ∂ q 2 = 5 > 0 ∂ p 1 . r D Hence the two products are competetive. Dr. Abdulla Eid (University of Bahrain) Applications 6 / 6