. Determine whether K4 (the complete graph on 4 vertices contains the following: i) A walk that is not a trail. ii) A trail that is not closed and is not a path. iii) A closed trail that is not a cycle.

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E.
b) Prove that every line segment between two points in R^k
is connected, that is Ep,q = {tp + (1 − t)q |
t ∈ [0, 1]} for any p not equal to q in R^k.
C). Prove that every convex subset of R^k is connected.

Real Mathematical Analysis, Pugh, 5.29 : Prove Corollary18 that r^th-order differentiability implies symmetry of D^rf, r ≥ 3. Use induction to show that (D^rf)_p (v₁,.....,v_r) is symmetric with respect to permutations of v₁,...,v_r−1 and of v₂,...,v_r. Then take advantage of the fact that r is strictly greater than 2. (Please provide a formal proof. Thanks)

Corollary 18: The r^th derivative, if it exists, is symmetric: Permutation of the vectors v₁,...,v_r does not affect the value of (D^rf)_p(v₁,...,v_r). Corresponding mixed higher-order partials are equal.

Problem 1.      Smoke Sensors, Inc. (SSI), is experiencing a tremendous growth in demand for its household smoke detectors. SSI produces both an AC model and a battery-operated model. It has an opportunity to be the exclusive supplier for a major department store chain, The Seers Company. Seers wishes to receive at least 20,000 AC models and 10,000 battery-operated models each week.

SSI's unanticipated prosperity has left it short of sufficient capacity to satisfy the Seer's contract over the short run. However, there is a subcontractor who can assist SSI by supplying the same types of smoke detectors. SSI must decide how many units it will make of each detector and how many units it will buy from the subcontractor. Data below summarize the production, price, and cost parameters.

Model

(hours per unit)

The subcontractor can supply any combination of battery or AC models up to 20,000 units total each week. The cost per unit to SSI is $21.50 and $20.00, respectively, for the AC and battery models. The contract with Seers calls for SSI to receive $25.00 for each AC model and $29.50 for each battery model.

Hint: Table below shows the possibilities.

1. Formulate the LP model which would allow SSI to determine the number of units of each type to produce and to buy to maximize total profit.
2. Use solver to solve the problem. Give the values of the decision variables, slacks, and the value of the objective function.

Solve the following initial value problem, showing all work. Verify the solution you obtain. y^''-2y^'+y=0; y(0)=1,y^' (0)=-2.

Consider the definite integral ∫₀⁵ ((3x − 1)/(x + 2)) dx

a. How large an n do we need to use to approximate the value of the integral to within 0.001 using the Midpoint Rule?

b. How large an n do we need to use to approximate the value of the integral to within 0.001 using Simpson’s Rule?

E.C. 2. (10 pts.) Suppose that (sn) is a sequence of real numbers such that sn ≥ 0 for all n ∈ N. (a) Show that the set of subsequential limits of S satisfies S ⊆ [0,∞) ∪ {+∞}. (b) Is it possible for S = [0,∞) ? (Hint: apply Theorem 11.9.)

Legible handwriting is a must

Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to chemical C. Initially there are 15 grams of A and 32 grams of B, and for each gram of B, 1.7 grams of A is used. It has been observed that 11.75 grams of C is formed in 15 minutes. How much is formed in 40 minutes? What is the limiting amount of C after a long time ?

Let A ∈ R ^ (nxn) with autovalue λ associated with the vector V_λ ∈ R ^ (n), determine the shape of the eigenvalues and eigenvectors of

a) 8A + I

b) A^(2) + λA

c) -2A + 5I

d) 2A^(2) + 3λA

Let G be a nontrivial nilpotent group. Prove that G has nontrivial center.

What is a vector space? Provide an example of a finite-dimensional vectors space and an infinite- dimensional vector space.

What is an orbit of a group action? Give an example of a group action on a set with 10 elements with exactly two orbits.

Consider a vibrating system described by the initial value problem. (A computer algebra system is recommended.) u'' + 1/4u' + 2u = 2 cos ωt, u(0) = 0, u'(0) = 6

(a) Determine the steady state part of the solution of this problem.
u(t) =

(b) Find the amplitude A of the steady state solution in terms of ω.
A =

(d) Find the maximum value of A and the frequency ω for which it occurs.