Question

Let h(x,y)=k*x^2+6xy+14*y^2+4y+10.

(a) Find the minimal value of the function h for k = 2.

(b) Using envelope theorem find approximate minimal value of h for k = 1.98.

Answer #1

1. Find the differential of f(x,y)=\sqrt{x + e^{4y}}f(x,y)=
x+e^4y and use it to find the approximate change in the function
f(x,y)f(x,y) as (x,y)(x,y) changes from (3,0)(3,0) to
(2.6,0.1)(2.6,0.1).

Consider the function given by f(x,y) = 3x2 −6xy + 2y3 +
23.
(a) Find all critical points of f(x,y) and determine their
nature.
(b) What are the minimum and maximum values of f(x,y) on the
straight line segment given by 0 ≤ x ≤ 3, y = 2?

Let X and Y have joint pdf f(x,y)=k(x+y), for 0<=x<=1 and
0<=y<=1.
a) Find k.
b) Find the joint cumulative density function of (X,Y)
c) Find the marginal pdf of X and Y.
d) Find Pr[Y<X2] and Pr[X+Y>0.5]

Let y=2−3x+∑n=2∞an x power n be the power series solution of the differential equation:
y″+6xy′+6y=0 about x=0. Find a4.

find the general solution to x^2*y''+6xy'+6y=0

4. Consider the function z = f(x, y) = x^(2) + 4y^(2)
(a) Describe the contour corresponding to z = 1.
(b) Write down the equation of the curve obtained as the
intersection of the graph of z and the plane x = 1.
(c) Write down the equation of the curve obtained as the
intersection of the graph of z and the plane y = 1.
(d) Write down the point of intersection of the curves in (b)
and...

Let f(x,y) = sqrt(22−2x^2−y^2). Find the linearization of the
function f at (1,2) and use it to approximate f(1.1,2.1).

1. Find the absolute maximum of f(x,y)=x^2+2y^2+4y-2 on the disk
x^2+y^2<=4
2. Let E be the solid bounded by x^2+y^2=4, y+z=2, and z=0. What
is the volume of the solid E?

Let X be a random variable such that P(X=k) = k/10, for k =
1,2,3,4. Let Y be a random variable with the same distribution as
X. Suppose X and Y are independent. Find P(X+Y = k), for k =
2,...,8.

The function f(x,y)=4x-4y has an absolute maximum value and
absolute minimum value subject to the constraint x^2-xy+y^2=9. Use
Lagrange multipliers to find these values.

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