In: Advanced Math

Q. Show that the ellipse x^2/4+ y^2/9= 1 in R2 is
connected, and show that the function f(x, y) = 3 x^3− 5y^3 − 4 has
a zero on the above ellipse.

On R2, consider the function f(x, y) = ( .5y,
.5sinx). Show that f is a strict contraction on R2. Is
the Banach contraction principle applicable here? If so, how many
fixed points are there? Can you guess the fixed point?

5. Given the function y = q(x) =
(x^2)/(x-1)
a. What is the domain of q(x)?
b. What are the vertical asymptotes?
c. What are the horizontal asymptotes?
d. Where is q(x) increasing/decreasing (draw a line and
specify by intervals – be sure to include points where q isn’t
defined)?
e. Where is q(x) concave up/down ((draw a line and
specify by intervals – be sure to include points where q isn’t
defined)?
f. Find rel max/min.
g. Find inflection...

a.maximize and minimize −2xy on the ellipse x^2+4y^2=4
b.Determine whether or not the vector function is the gradient
∇f (x, y) of a function everywhere defined. If so, find all the
functions with that gradient. (x^2+3y^2)i+(2xy+e^x)j

Show that the cylinder x 2 + y 2 = 4 and the sphere x 2 + y 2 +
z 2 − 8y − 6z + 21 = 0 are tangent at the point (0, 2, 3). That is,
show that the cylinder and sphere intersect at that point, and that
they share the same tangent plane at that point.

(6) Consider the function f(x, y) = 9 − x^2 − y^2 restricted to
the domain x^2/9 + y^2 ≤ 1. This function has a single critical
point at (0, 0)
(a) Using an appropriate parameterization of the boundary of the
domain, find the critical points of f(x, y) restricted to the
boundary.
(b) Using the method of Lagrange Multipliers, find the critical
points of f(x, y) restricted to the boundary.
(c) Assuming that the critical points you found were...

4. The joint density function of (X, Y ) is
f(x,y)=2(x+y), 0≤y≤x≤1
. Find the correlation coefficient ρX,Y
.
5. The height of female students in KU follows a normal
distribution with mean 165.3 cm and s.d. 7.3cm. The height of male
students in KU follows a normal distribution with mean 175.2 cm and
s.d. 9.2cm. What is the probability that a random female student is
taller than a male student in KU?

a) Calculate the maximum MaxDisk value of the function f (x,y) =
9 ln * (x^2+(1+1)y^2+(0.5+0.5)) on the circle disk with the center
of origin and radius 4.
b) Also calculate the maximum the value MaxBorder that the
function assumes on the border.

f(x,y)=30(1-y)^2*x*e^(-x/y). x>0. 0<y<1.
a). show that f(y) the marginal density function of Y is a Beta
random variable with parameters alfa=3 and Beta=3.
b). show that f(x|y) the conditional density function of X given
Y=y is a Gamma random variable with parameters alfa=2 and
beta=y.
c). set up how would you find P(1<X<3|Y=.5). you do not
have to do any calculations

Find y as a function of x if
y''''−4y'''+4y''=−128e^{-2x}
y(0)=2, y′(0)=9, y″(0)=−4, y‴(0)=16.
y(x)=?

x
y
fxy(x,y)
-1
-2
1/8
-0.5
-1
1/4
0.5
1
1/2
1
2
1/8
Show that the function above satisfies the probabilities of a
joint probability mass function.
Find E(X), E(Y), V(X), V(Y)
Find marginal probability distribution of X
Find the covariance and correlation.

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