Identify (basically, state the name of) each conic and sketch a
graph. Label important point(s).
a)...
Identify (basically, state the name of) each conic and sketch a
graph. Label important point(s).
a) x = (y-2)(y+2)
b) (x+y)(x-y) = 9
c) 25x2 + 4y2 = 100
d) For part c), give a set of parametric equations for the
graph; show work.
make a contour map with x=0, y=0, z=0, z=1, z=2, and z=4 for
z=x^2+y^2 then sketch...
make a contour map with x=0, y=0, z=0, z=1, z=2, and z=4 for
z=x^2+y^2 then sketch the graph
1. Consider the relations R = {(x,y),(y,z),(z,x)} and S =
{(y,x),(z,y),(x,z)} on {x, y, z}. a)...
1. Consider the relations R = {(x,y),(y,z),(z,x)} and S =
{(y,x),(z,y),(x,z)} on {x, y, z}. a) Explain why R is not an
equivalence relation. b) Explain why S is not an equivalence
relation. c) Find S ◦ R. d) Show that S ◦ R is an equivalence
relation. e) What are the equivalence classes of S ◦ R?
Prove that as rings, Z[x]/〈2, x〉 ∼= Z/2.
Explain why this proves that 〈2, x〉 is...
Prove that as rings, Z[x]/〈2, x〉 ∼= Z/2.
Explain why this proves that 〈2, x〉 is a maximal ideal in
Z[x].
Consider function f(x, y) = x 2 + y 2 − 2xy and the 3D graph...
Consider function f(x, y) = x 2 + y 2 − 2xy and the 3D graph z =
x 2 + y 2 − 2xy. (a) Sketch the level sets f(x, y) = c for c = 0,
1, 2, 3 on the same axes. (b) Sketch the section of this graph for
y = 0 (i.e., the slice in the xz-plane). (c) Sketch the 3D
graph.
Prove that the rings, Z[x]/<2,x> is isomorphic to Z/2 and
explain why this means that <2,x>...
Prove that the rings, Z[x]/<2,x> is isomorphic to Z/2 and
explain why this means that <2,x> is a maximal in Z[x]
For the 3-CNF
f = (x’ +y’+z)& (x+y’+z’)&(x+y+z’)&
(x’+y+z)&(x’+y+z’) &(x+y+z)
where “+” is or, “&” is...
For the 3-CNF
f = (x’ +y’+z)& (x+y’+z’)&(x+y+z’)&
(x’+y+z)&(x’+y+z’) &(x+y+z)
where “+” is or, “&” is and operations, “
’ ” is negation.
a)give 0-1 assignment to variables such that
f=1 x= ______ y= ______ z= ____
f=0 x= ______ y= ______ z= ____
-
b) Draw the corresponding graph and mark the
maximum independent
set.
(you can draw on paper, scan and insert here)
(Lagrange Multipliers with Three Variables) Find the global
minimum value of f(x,y,z)=(x^2/4)+y^2 +(z^2/9) subject to x...
(Lagrange Multipliers with Three Variables) Find the global
minimum value of f(x,y,z)=(x^2/4)+y^2 +(z^2/9) subject to x - y + z
= 8. Now sketch level surfaces f(x,y,z) = k for k = 0; 1; 4 and the
plane x-y +z = 8 on the same set of axes to help you explain why
the point you found corresponds to a minimum value and why there
will be no maximum value.