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5 Homework Assignments —

page 8

14.1

a. Let

= + −+

2

(, )

2

3

f x y

x

xy

x

. Compute

f

(1, 2),

f

(0, 3),

f

(2,

-4).

b. The Wilson lot size formula in economics states that the optimal quant

ity Q of goods for a store to

order is given by:

=

=

2

(, ,)

CN

Q f CNh

h

, where C is the cost of placing an order, N is the number of

items the store sells per week, and h is the weekly holding cost for each item. Find the most

economical quantity of ten-

speed bicycles to order if it costs the

store $20 to plac

e an order,

$5 to hold

a bicycle for a week, and the store expects to sell 40 bicycles each week.

c. A car dealership estimates that the total weekly sales

(the number of cars sold)

of a particular model is

a function of the car’s list price, p (in

thousands

of dollars) and the interest rate in percent

(

not

decimal!

),

i

. The approximate weekly sales are given by

= −−

2

( , ) 32

2

.01

f p i

p

pi

p

. Find the

approximate weekly sales when price is $17,000 and interest rate is 7.5%

.

Hint: p = 17 an

d i = 7.5

14.2

a. Find all first and second order partial derivatives for:

=

+

(, )

1

x

f xy

y

b. Refer to the function in 14.1(c) above. Find and

interpret

the partial derivatives

f

p

and

f

i

, when

price is

$19,400 and interest rate is 8%.

c. The total weekly revenue in dollars of the Country Workshop associated with manufacturing and

selling their rolltop desks is given by the function:

=−− − ++

22

( , )

0.2

0.25

0.2

200

160

R x y

x

y

xy

x

y

where x denotes the number of finished units and y denotes t

he number of unfinished units

manufactur

ed and sold per week. Compute

∂∂

∂∂

and

RR

xy

when x = 300 and y = 250. Interpret your

results.

14.3

a. Find the critical points of the function and classify the nature of each of these points. I.e., use the

second partials test to decide if there is a local maximum, local mini

mum, or a saddle point here.

Finally, determine the local extrema (the functional values.)

= − − +−−

22

(,)442487

f x y

xy

x

y

x

y

b. The labor cost in dollars for manufacturing a precision camera can be approximated by:

= +−−− +

22

3

( , )

2

2

2

68

2

L x y

x

y

x

y

xy

where x is the number o

f hours required by a skilled

craftsperson and y is the number of hours required by a semi

-skilled

person. Find the values of x and

y that minimize the labor charge. Find the minimum labor charge.