Partiell derivering (praktisk bruk)

[Drillo]

OPPGAVE:

 

Suppose that Alpha AS and Beta AS manufacture competitive products, with the weekly sales of each product determined by the selling price of that product and the price of its competition. Suppose that Alpha sets a sales price of x dollars per unit for its product, while Beta sets a sales price og y dollars per unit for its product. Market research shows that the weekly profit made by Alpha is then

 

P(x) = -2x^2 + 12x + xy - y - 10

 

and that the weekly profit made by Beta is

 

Q(y) = -3y^2 + 18y +2xy -2x - 15

 

(both in thousands of dollars). The peculiar notation arises from the fact that x is the only variable under the control of Alpha and y is the only variable under the control of Beta.

 

Assume that both company managers know calculus and that each knows that the other knows calculus and has some common sense. What price will each manager set to maximize his company's weekly profit?

 

MITT FORSØK PÅ LØSNING:

 

I find the partial derivates dP/dx and dQ/dy, make them equal 0 and find x and y from the two equations y = 4x - 12 and x = 3y - 9.

 

This gives

 

x = 0,82 dollars

y = 3,27 dollars

 

 

However, the correct answer is supposed to be

 

x = 4,09 dollars

y = 4,36 dollars

 

HVILKEN FEIL HAR JEG GJORT?

[Janhaa]

Du har bare gjort en slurvefeil,

 

(I) y = 4x - 12 and x = 3y - 9 (II)

Sett (II) inn i (I)

 

y = 4*(3y - 9) - 12 = 12y - 48

11y = 48

y = 48/11 = 4,36

 

Og x = 3*(48/11) - 9 = (144/11) - 9 = (45/11) = 4,09

x = (45/11) = 4,09

[Drillo]

Hva med del B av oppgaven?

 

"Now suppose that the two managers enter into an agreement (legal or otherwise) by which they plan to maximize their TOTAL weekly profit. Now what should be the selling price of each product? (We suppose that they will divide the resulting profit in an equitable way, but the details of this intriguing problem are not the issue). "

 

What I tried:

 

R(x,y) = P(x)+Q(y)

Find the partials, find eq.s for x and y and solve them. This gives y=8,4, but is supposed to be 6,53. I don't think this is the same mistake as I did in part (a).

[Janhaa]

Har et forslag her, men orker ikke regne d ut, Jævli mye jobb.

Dessuten mangler forumet her TEX, slik at formler etc er et herk å skrive.

 

Men prøv Lagranges multiplikatormetode. Hørt om den....

 

Du har jo at P = P(x, y) og Q = Q(x, y)

 

1: (P_x) ' = a*(Q_x) '

 

2: (P_y) ' = a*(Q_y)'

 

3: Q = 0 eller evt P = 0

 

der a: konstant

(P_x)' og (Q_x)' er partiell deriverte av P og Q mhp x

og

(P_y)' og (Q_y)' er partiell deriverte av P og Q mhp y

 

Kombiner disse tre likningene og du får 2 likninger med 2 ukjent (x og y)

Løs 1. og 2. mhp a og sett disse like hverandre. Blir vel strengt tatt 2. gradslikninger mhp x og y.

Men jeg har ikke regna på noe, så jeg vet ikke sikkert. Tok det på strak arm...