Let G be connected, and let e be an edge of G. Prove that e is a bridge if and only if it is in every spanning tree of G.

Show that if a function f(z) is analytic in a domain D then it has derivatives of all orders in D.

(a) Prove the following claim: in every simple graph G on at least two vertices, we can always find two distinct vertices v,w such that deg(v) = deg(w).

(b) Prove the following claim: if G is a simple connected graph in which the degree of every vertex is even, then we can delete any edge from G and it will still be connected.

Sketch the direction field of the equation dy/dx=y-4y^3. Sketch the phase portrait. Find the equilibrium solutions and classify each equilibrium as stable, unstable or semi-stable. Sketch typical solutions of the equation.

3) Laplace Transform and Solving first order Linear Differential Equations with Applications The Laplace transform of a function, transform of a derivative, transform of the second derivative, transform of an integral, table of Laplace transform for simple functions, the inverse Laplace transform, solving first order linear differential equations by the Laplace transform Applications: a)))))) Series RL circuit with ac source [electronics]

Suppose f : X → S and F ⊆ P(S). Show, f −1 (∪A∈F A) = ∪A∈F f −1 (A) f −1 (∩A∈F A) = ∩A∈F f −1 (A)

Show, if A, B ⊆ X, then f(A ∩ B) ⊆ f(A) ∩ f(B). Give an example, if possible, where strict inclusion holds.

Show, if C ⊆ X, then f −1 (f(C)) ⊇ C. Give an example, if possible, where strict inclusion holds.

For any r, s ∈ N, show how to order the numbers 1, 2, . . . , rs so that the resulting sequence has no increasing subsequence of length > r and no decreasing subsequence of length > s.

Many elderly people have purchased medigap insurance policies to cover a growing Medicare copayment. These polices cover some or all of the medical costs not covered by Medicare. use economic theory to explain how these policies likely influence the demand for health care by elderly people.

u=αx + αy

find
slope, graph
convexty
continuity

Let f : X → Y and g : Y → Z be functions. We can define the composition of f and g to be the function g◦ f : X → Z for which the image of each x ∈ X is g(f(x)). That is, plug x into f, then plug theresultinto g (justlikecompositioninalgebraandcalculus). (a) If f and g arebothinjective,must g◦ f beinjective? Explain. (b) If f and g arebothsurjective,must g◦ f besurjective? Explain. (c) Suppose g◦ f isinjective. What,ifanything,canyousayabout f and g? Explain. (d) Suppose g◦f issurjective. What,ifanything,canyousayabout f and g? Explain.

please write clearly

Consider the equation: ?̇ +2? = ?(?) with initial condition x(0) = 2

(a) If u(t) = 0, find the solution ?(?). What is ?(?) as t -> ∞?

(b) If u(t) = 4+t, find the solution ?(?). What is ?(?) as t -> ∞?

(c) If u(t) = ?^3?, find the solution ?(?). What is ?(?) as t -> ∞?

(d) If u(t) = δ(t), find the solution ?(?). What is ?(?) as t -> ∞?

For this Variation of parameter problem, consider the following method of solving the general linear equation of first order

y' + p(t)y= g(t)

(a) If g(t) = 0 for all t, show that the solution is

y= Aexp[ - the integral of p(t)dt] , where A is a constant

Please use good handwriting and show as many steps as possible. Thank you.