Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

eRbIviFanbNþak;edIm,Irk

dz

dt

eTAtamtémøtEdleK)anbBa¢ak;.

a. zxyxtytt=+ = = =23; , 5;^22 b. zxyx t yt t=^22 ;31,=+ =− =1; 1

c. zxy x ty tt===12 13;2,2;^2 = 2

11 Using x skilled workers and y unskilled workers, a manufacturer can produce

()

f xy,10= xy^12 units. Currently the manufacturer uses 30 hours of skilled labor

and 36 hours of unskilled labor and is planning to use 1 additional hour of skilled

labor. Use calculus to estimate the corresponding change that the manufacturer

should make in the level of unskilled labor so that the total output will remain the

same. (Answer: −2.4)

edayeRbIkmμkrCMnajcMnYn xem:ag nigkmμkrminCMnajcMnYn yem:ag plitkrmμak;Gacplit)an

f()xy,10= xy^12 Ékta. bc©úb,nñenH plitkreRbIkmμkrCMnaj 30 em:ag nigkmμkrminCMnaj 36

em:ag. plitkreRKagnwgeRbIR)as;kMlaMgBlkmμCMnaj1em:agbEnßm. cUr):an;sμanbrimaNERbRbYl

EdlRtUveFVIeLIgcMeBaHkMlaMBlkmμminCMnaj y:agNaedIm,I[plitplsrubenArkSadEdl.

12 At a certain factory, output Q is related to inputs x and y by the function

. If the current levels of input are xൌ20 and yൌ10, use

calculus to estimate the change in input x that should be made to offset an increase

of 0.5 unit in input y so that output will be maintained at it current level.

Qx xyy=+ + 23323

3

(answer:−0.21)

enAkñúgeragcRkmYy output QCab;Tak;TgeTAnwg input x nigytamry³GnuKmn_

. brimaNbc©úb,nñrbs; input xKW xൌ20nig input yKW yൌ10. eRbIrUbmnþ

témøRbEhl edIm,I):an;sμanGMBIbrimaNERbRbYlrbs; input x edIm,ITUTat;nwgbrimaNkMeNInrbs;

input y kñúgkMri 0.5Ékta y:agNaedIm,I[brimaN outputsßitenAkñúgkMritdEdl.

Qx xyy=+ + 2332

13 Suppose the utility derived by a consumer from x units of one comodity and y
units of a seccond commodity is given by the utility function
)( )y+^2. The consumer currently owns ݔൌ25 units of the first
commodity and ൌ8ݕ units of the second. Use calculus to estimate how many
units of the first commodity the consumer could substitute for 1 unit of the second
commodity without affecting total utility. ( about 2.6).

Uxy()(,1=+x

]bmafakareRbIR)as;GtifiCnmñak;ecjBIrbs;eRbIR)as;TI1cMnYnxÉkta nigrbs;eRbIR)as;TI2cMnYn

yÉktaRtUv kMNt;edayGnuKmn_kareRbIR)as; Ux( ,y)=(x++ (^1) )(y (^2) ). GtifiCnbc©úb,nman
rbs;eRbIR)as;TI1 cMnYn ݔൌ25Ékta nigrbs;eRbIR)as;TI2cMnYn ݕൌ8Ékta. eRbIrUbmnþtémø
RbEhl cUr):an;sμanfaetIbrimaNrbs;eRbIR)as;TI1cMnYnb:unμanÉktaEdlGacCMnYsbrimaNrbs;eRbI
R)as;TI2 cMnYn 1 Ékta edIm,IkMu[mankarERbRbYldl;brimaNkareRbIR)as;srub?
15 The output at a certain plant is Qxy( , )=++0.08x^22 0.12xy0.03yunits per day,
where x is the number of hours of skilled labor used and y is the number of hours
of unskilled labor used. Currently 80 hours of skilled labor and 200 hours of
unskilled labor are used each day. Used the total differential of Q to estimate the