Question

1. A manufacturer produces its products at two locations. Let x
and y be the number of units produced at these two locations. The
cost function is modeled
by
.

C(x,y)=0.25x^2-25x+0.05y^2-12y+3000

Find the number of units that should be produced at each location
to minimize the cost.

2. Given a surface f(x,y)=x^2+y^2-xy and a point (2,1,3) on the surface, find the slopes of the surface at the point in the x-direction and the y-direction.

Please show work

Answer #1

A manufacturer has two factories at which wooden benches can be
produced. Let x and y be the number of units (in thousands)
produced at the two locations. The cost function is ? =
0.25?2 − 10? + 0.35?2 − 12? + 0.5?? + 500.
Find the number that should be produced at each location to
minimize cost.

A firm produces two products A and B. The total cost function of
producing “x” items of product A and “y” items of product B is TC =
2x2 + y2 – xy – 49x + 425. Find the number of each type that should
be produced to minimize the total cost if the firm is committed the
producing 21 items in total.

1. Let f(x, y) = 2x + xy^2 , x, y ∈ R.
(a) Find the directional derivative Duf of f at the point (1, 2)
in the direction of the vector →v = 3→i + 4→j .
(b) Find the maximum directional derivative of f and a unit
vector corresponding to the maximum directional derivative at the
point (1, 2).
(c) Find the minimum directional derivative and a unit vector in
the direction of maximal decrease at the point...

A corporation manufactures candles at two locations. The cost of
producing x1 units at location 1 is
C1 =
0.02x12 +
2x1 + 400
and the cost of producing x2 units at
location 2 is
C2 =
0.05x22 +
2x2 + 225.
The candles sell for $14 per unit. Find the quantity that should
be produced at each location to maximize the profit
P = 14(x1 +
x2) − C1
− C2.

The joint cost (in dollars) for two products is given by
C(x, y) =
25 + x2 + 3y +
2xy
where x represents the quantity of product X
produced and y represents the quantity of product
Y produced.
(a) Find the marginal cost with respect to x if 7 units
of product X and 11 units of product Y are
produced.
$ ???
(b) Find the marginal cost with respect to y if 7 units of
product X and...

Let f(x, y) = sqrt( x^2 − y − 4) ln(xy).
• Plot the domain of f(x, y) on the xy-plane.
• Find the equation for the tangent plane to the surface at the
point (4, 1/4 , 0).
Give full explanation of your work

(a) Find an equation of the plane tangent to the surface xy ln x
− y^2 + z^2 + 5 = 0 at the point (1, −3, 2)
(b) Find the directional derivative of f(x, y, z) = xy ln x −
y^2 + z^2 + 5 at the point (1, −3, 2) in the direction of the
vector < 1, 0, −1 >. (Hint: Use the results of partial
derivatives from part(a))

16.
a. Find the directional derivative of f (x, y) = xy at P0 = (1,
2) in the direction of v = 〈3, 4〉.
b. Find the equation of the tangent plane to the level surface
xy2 + y3z4 = 2 at the point (1, 1, 1).
c. Determine all critical points of the function f(x,y)=y3
+3x2y−6x2 −6y2 +2.

Suppose we toss a fair coin twice. Let X = the number of heads,
and Y = the number of tails. X and Y are clearly not
independent.
a. Show that X and Y are not independent. (Hint: Consider the
events “X=2” and “Y=2”)
b. Show that E(XY) is not equal to E(X)E(Y). (You’ll need to
derive the pmf for XY in order to calculate E(XY). Write down the
sample space! Think about what the support of XY is and...

K 2. Point Source Interference Pattern
Consider a two-dimensional x-y space in which there are two
identical and synchronized point sources located at y=±D/2 emitting
waves of wavelength λ. The sources are said to constructively
interfere at any location if the phase difference between waves
received from the two sources at that location differ by an
integral multiple of 2π. Equivalently, the distances from the
location to the two sources differ by an integral number of
wavelengths. Derive the complete...

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