Question

Let f(x,y)=e^(−5x)sin(3y).

(a) Using difference quotients with Δx=0.1 and Δy=0.1, we
estimate

fx(3,2)≈

fy(3,2)≈

(b) Using difference quotients with Δx=0.01 and Δy=0.01, we find
better estimates:

fx(3,2)≈

fy(3,2)≈

Answer #1

Let f(x, y) = 2x^3y^2 + 3xy^3 4x^3 y. Find
(a) fx
(c) fxx
(b) fy
(d) fyy
(e) fxy
(f) fyx

Given the function f(x, y) = e^(x)*sin(y) + e^(y)*sin(x)
approximate f(0.1, -0.2) using P(0, 0).

Let fx,y (x,y) = 3 e^-(x+y) for 0 < x <1/2y and y>0. a)
Find f x(x) and f y( y) . b) Write out the integral
necessary to find , Fx,y ( u v) . DO NOT EVALUATE THE INTEGRAL.

(1 point)
Find all the first and second order partial derivatives of
f(x,y)=7sin(2x+y)−2cos(x−y)
A. ∂f∂x=fx=∂f∂x=fx=
B. ∂f∂y=fy=∂f∂y=fy=
C. ∂2f∂x2=fxx=∂2f∂x2=fxx=
D. ∂2f∂y2=fyy=∂2f∂y2=fyy=
E. ∂2f∂x∂y=fyx=∂2f∂x∂y=fyx=
F. ∂2f∂y∂x=fxy=∂2f∂y∂x=fxy=

Question about using the convolution of distribution:
1. we have the formula: integral fx(x)fy(z-x)dx=integral
fx(z-x)fy(x)dx
I know this are equivalent. However, how do I decide which side
I should use ?
For example,X~Exp(1) and Y~Unif [0,1] X and Y independnt and the
textbook use fx(z-x)fy(x)dx.
However, can I use the left hand side fx(x)fy(z-x)dx???is there
any constraint for using left or right or actually both can lead me
to the right answer???
2. For X and Y are independent and...

Let F ( x , y , z ) =< e^z sin( y ) + 3x , e^x cos( z ) + 4y
, cos( x y ) + 5z >, and let S1 be the sphere x^2 + y^2 + z^2 =
4 oriented outwards Find the flux integral ∬ S1 (F) * dS. You may
with to use the Divergence Theorem.

Let f(x, y) = sin x √y.
Find the Taylor polynomial of degree two of f(x, y) at (x, y) =
(0, 9).
Give an reasonable approximation of sin (0.1)√ 9.1 from the
Taylor polynomial of degree one of f(x, y) at (0, 9).

Let X and Y have the joint probability density function f(x, y)
= ⎧⎪⎪ ⎨ ⎪⎪⎩ ke−y , if 0 ≤ x ≤ y < ∞, 0, otherwise. (a) (6pts)
Find k so that f(x, y) is a valid joint p.d.f. (b) (6pts) Find the
marginal p.d.f. fX(x) and fY (y). Are X and Y independent?

4. Let X and Y be random variables having joint probability
density function (pdf) f(x, y) = 4/7 (xy − y), 4 < x < 5 and
0 < y < 1
(a) Find the marginal density fY (y).
(b) Show that the marginal density, fY (y), integrates to 1
(i.e., it is a density.)
(c) Find fX|Y (x|y), the conditional density of X given Y =
y.
(d) Show that fX|Y (x|y) is actually a pdf (i.e., it integrates...

Let
f(x,y,z)=exy4z5+xy2+y3z
Calculate.
fx fy fz fxx fxy fyy
fxz fyz fzz fzyy fxxy fxxyz
1

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