Applications of partial derivatives Solve the following problems using multivan'able calculus. 1. Wi1 x skilled workers and 1,! unskilled workers, a manufacturer can produce [1 [3, v} = 312v units per dav. llIurrentIIpr there are 20 skilled and 30 unskilled workers. a]: How manvr units are produced each dav? b} Ev how much will the dailvr production level change if an unskilled worker is added to the cu rrent workforce? c] Ev how much will the daily production level change if a skilled worker is added to the current workforce? d} Ev how much will the dailv production level change if a skilled and an unskilled worker are added to the current workforce? 2. In economics it is said that two items are substitute items it the [11 demand for the rst increases when the price p2 of the seoond increases, and it the [12 demand tor the second increases Men the prioe p1 of the first increases. a} IGive an example of a pair of substitute items. 3 1 2 b} If two articles are substitutes, what should happen to the partial derivatives 4% v % ? 3. A paint store sells two brands of plastic paint. The sales calculations indicate that if the first brand sells 1: pesos per gallon and the second brand sells v pesos per gallon, the demand tor the rst brand will be Di {x, v] = 200 1x + Ev gallons per month, and the demand for the second brand will be D2 [1, v] = 1m + 5:: 1[Iv gallons per month. a} Express the total monthlvr income of the paint store in the sale of the paintings as a function of the prices 3: and v. Income is the prise times demand, so the total income function is: c] Calculate the income from the previous paragraph if the rst brand is sold at 10 pesos per gallon and the second at 15 dollars per gallon. 1 1 4. In a certain tactorv, the dail',r output is = K'lLunits, where K represents the capital invested measured in thousands of pesos and L the size of the labor foroe measured in worker hours. Determine the marginal product functions of labor and capital