Question

1. Let f (x, y) =

xy((x^2-y^2)/(x^2+y^2)) if, (x, y) 6= (0, 0),

0, if (x, y) = (0, 0) (it's written as a piecewise function)

(a) Compute ∂f/∂y (0, 0) and ∂f/∂x (0, 0).

(b) Compute ∂f/∂y (x, 0) for all x, and ∂f/∂x (0, y) for all y

(c) Use part (a) and (b) to compute ∂^2f/∂y∂x (0, 0) and ∂^2f/ ∂x∂y (0, 0), then verify that:

∂^2f/∂y∂x (0, 0) does not equal ∂^f/ ∂x∂y (0, 0)

Answer #1

1. for 0<= x <=3 0<=x<=1 f(x,y) = k(x^2y+ xy^2)
a. Find K joint probablity density function.
b. Find marginal distribution respect to x
c. Find the marginal distribution respect to y
d. compute E(x) and E(y) e. compute E(xy)
f. Find the covariance and interpret the result.

Let X and Y have joint density f(x, y) = 6/7(x + y)^2 if 0 ≤ x ≤
1, 0 ≤ y ≤ 1, 0 otherwise, where c is a positive constant.
Compute the marginal densities of X and of Y (be explicit about
all cases!).
Compute P(Y + 2X < 1).
Determine whether X and Y are independent. Justify your
answer.

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

Let f(x, y) = 1/2 , y < x < 2, 0 ≤ y ≤ 2 , be the joint
pdf of X and Y .
(a) Find P(0 ≤ X ≤ 1/3) .
(b) Find E(X) .
(c) Find E(X|Y = 1) .

Consider F and C below.
F(x, y, z) = yz i + xz j + (xy + 12z) k
C is the line segment from (2, 0, −3) to (4, 6, 3)
(a) Find a function f such that F =
∇f.
f(x, y, z) =
(b) Use part (a) to evaluate
C
∇f · dr along the given curve C.

Let F be the defined by the function F(x, y) = 3 + xy - x - 2y,
with (x, y) in the segment L of vertices A (5,0) and B (1,4). Find
the absolute maximums and minimums.

Let the random variable X and Y have the joint pmf f(x, y) =
xy^2/c where x = 1, 2, 3; y = 1, 2, x + y ≤ 4 , that is, (x, y) are
{(1, 1),(1, 2),(2, 1),(2, 2),(3, 1)} .
(a) Find c > 0 .
(b) Find μX
(c) Find μY
(d) Find σ^2 X
(e) Find σ^2 Y
(f) Find Cov (X, Y )
(g) Find ρ , Corr (X, Y )
(h) Are X...

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...

Please write down the details of reasons for your solutions.
Thanks.
Questions 2
a) Let f(x,y) = sin (y^2 + e^(2x)). Find f(xy) and f(yx) and
verify their equality.
b) Find the equation of the tangent plane to the surface
z=e^(2x)*cos(3y) at P (0, pi/3, -1)
c) Find the direction in which the function f(x,y) =
((x-2y)/(2x+y))^(1/3) increases most rapidly at (1,0).

Let f(x,y) = xe^sin(x^2y+xy^2) /(x^2 + x^2y^2 + y^4)^3 . Compute
∂f ∂x (√2,0) pointwise.

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