Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

analysis to estimate the effect of an additional campital investment of $1,000 on

the daily output if the size of the labor force is not changed. (Answer: 10 units)

enAkñúgeragcRkmYy cMnYnplitplRbcaMéf¶kMNt;edayGnuKmn_QKL( ,60)= K L12 13 EdlKKWCa

vinieyaKmUlFn EdlxñatKitCaBaan;duløa ehIyLCaTMhMkMlaMgBlkmμ xñatKitCaem:agkmμkr. ]bma

favinieyaKmUlFnbc©úb,nñman 900000 duløa ehIyTMhMkMlaMgBlkmμman 1000 em:agkmμkr RtUv)an

eRbIR)as;ral;²éf¶. eRbIkarviPaKm:aCIn edIm,I)a:n;sμanGMBI\T§iBlénvinieyaKmUlFncMnYn 1000 duløa

bEnßm maneTAelIplitkmμRbcaMéf¶ ebITMhMkMlaMgBlkmμ minERbRbYl.

6 A grocer’s daily profit from the sale of two brands of orange juice is
Pxy()( ), =− −+ +− +−x30705 4( x y) (y40806 7)( x y)
cents, where x is the price per can of the first brand and y is the price per can of
the second. Currently the first brand sells for 50 cents per can and the second for
52 cents per can. Use marginal analysis to estimate the change in the daily profit
that will result if the grocer raises the price of the second brand by 1 cent per can
but keeps the price of the first brand unchanged. (Answer: $0.12)

R)ak;cMeNjRbcaMéf¶rbs;GaCIvkrmñak;BIkarlk;TwkRkUcBIrRbePT kMNt;edayGnuKmn_

Pxy()( )(, =− −+ +− +−x30705 4x y)(y40806 7)( x y)esn. x KWCaéfølk;kñúg

mYykMb:ugsRmab;TwkRkUcRbePTTI1 nig yKWCaéfølk;kñúgmYykMb:ugsRmab;TwkRkUcRbePTTI2.

bc©úb,nñenHTwkRkUcRbePTTI1 lk;éfø 50 esnkñúgmYykMb:ug nigTwkRkUcRbePTTI2 lk;éfø 52 esnkñúg

mYykMb:ug. eRbIkarviPaKm:aCIn edIm,I):an;sμannUvbrimaNERbRbYlénR)ak;cMeNjRbcaMéf¶EdlekIteLIg

ebIsinCaGaCIvkrbegáInéføTwkRkUcRbePTTI2 cMnYn 1 esnkñúgmYykMb:ugb:uEnþrkSaéføTwkRkUcRbePTTI 1

enAdEdl.

7 Compute all the second-order partial derivatives of the given function

KNnaedrIevedayEpñklMdab;2TaMgGs;rbs;GnuKmn_xageRkam

a. f()xy,5 2=+xy^43 xy (^) d. f()xy, = x y^22 +
b. (^) ()

1

,

1

x

fxy

y

+

=

−

e.^ ( )

f xy xye, =^2 x

c. (^) ()
2
f xy e, = xy
8 Use chain rule to find
dz
dt

. Check your answer by writing z as a function of t and

differentiating directly with respect to t. eRbIviFanbNþak;edIm,Irk

dz

dt

. epÞógpÞat;cemøIy

tamry³karsresrzCaGnuKmn_én t rYcrkedrIeveFobnwg t edaypÞal;.

a. zx yx ty t=+ = = +2; 3, 2 1 b. zxxyxt y=3;1,1^2 +=+=− 2 t

c. ()

2

zxyxty=+ = =23; 2, 3t

9 Use chain rule to find
dz
dt

for the specified value of t