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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.
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