Give an example of a family f₁ of subsets of a set X which has finite intersection property. Justify your choice of example.
See Answer → Find in
where d is the metric given by
Suppose that is given by f(t) = (t,t²) and
is given by g(x, y) = (x2, xy, y2-x2). Compute the derivative of gof.
Find the directional derivative of the function defined by
at the point (1,2,-1,-2) in the direction v = (1,0,-2,2).
See Answer →Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.
See Answer →Show that the function f defined by
is not differentiable at (0,0). Does the partial derivatives of f exists at (0,0)? or any at any other point in R²? Justify your answer.
See Answer →Let F be the function from to
defined by
F(x, y) = (x2 + y2 , xy)
Show that F is differentiable at (2,1) . Find the differential matrix of F.
See Answer →Find the interior and closure of the set of rationals in
with standard metric.
Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.
See Answer →If E is a subset of with standard metric, then show that
Which of the following subsets of R are compact w.r.t. the metric given against them. Justify your answer.
i) A = (1,0) in of −
with standard metric.
ii) A = [4,3] − with discrete metric.
iii) {(x, y) ∈ y > 0} −
with standard metric
Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set D U E where
Find the Fourier series of f (t) = t2 on [−π,π].
See Answer →For the equation , x2 + y3 + z3 = at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form z = f (x, y) .
See Answer →Find and classify the extreme values of (x, y) = xy Subject to the constraint
Does Cantor's intersection theorem hold for the metric space X = (0,1] with the standard metric? Justify your answer.
See Answer →