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A Guest needs material to finish a room in a basement. The room is square and one wall measures 15'. The height of the room is 8'.
In Rn with the usual topology, let A be the set of points x = (x1 , x2 .....xn) such that x12 + x22 + .....+xn2 = 1.
Denote by Intx(A) the interior of A in the topological space X and by Inty(A) the interior of A in the topological space Y.
The intersection of two open sets is compact iff it is empty. Can the intersection of an infinite collection of open sets be a non-empty compact set?
Every discrete space is first-countable; it is moreover second-countable if and only if it is countable.
Y = R, T is the discrete topology, A is the set consisting of all the numbers that do not have 5, 3, or 2 in any of their decimal expansions.
Find the interior, the closure, the accumulation points, the isolated points and the boundary points of the set A = { x = (x_1, x_2 ...) e l^2 : -1
Let p be an odd prime, and n = 2p. Show that a(n-1) is congruent to a (mod n) for any integer a.
Prove that X is compact if and only if for each family {F_a} with a in I of closed subsets of X that has the finite intersection property, the intersection.
Show that each subset of S is compact and that therefore there are compact subsets of S that are not closed.
Show that, if Y has the discrete topology and if p: X x Y --> X is the projection onto the first factor, then p is a covering map.
Define a new metric d on X = (0, 1/2)2 by d((a,b), (r,s)) = 1 if a is not equal to r Or |b - s| if a = r.
Find the interior, the closure, the accumulation points, the isolated point and the boundary points of the following sets.
A particle of mass 2 moves along the x-axis and is attracted towards the origin O by a force F=-8xi.
Suppose a twisted curve is defined in terms of the arclength s by where ? is a constant parameter.
Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and lines about the y-axis.
Suppose that a small brick is put into an aquarium which is the shape of a rectangular solid 1 foot wide, 3 feet long, and 2 feet high.
A gallon of paint covers about 350 square feet. How many gallons would be required to paint the room? Round up to the nearest gallon.
Water is poured into a funnel at the constant rate of 1 in^3/sec and flows out at a rate of 1/2 in^3/sec.
Systems of equations can be solved by graphing, using substitution, or elimination. What are the pros and cons of each method?
Use the QR factors in 1) to determine the lease squares solution to Ax=b. Ax=b is the solution related to Rx=Q^Tb
Determine whether it is possible to find a circumscribable, isosceles "true" trapezoid (in which the parallel sides are not congruent).
It is a common experience to hear the sound of a low flying airplane, and look at the wrong place in the sky to see the plane.
Find the volume of the pyramid with a rectangular base 11 in height, 12 in length, and 10 in width.