Let (s_n) be a sequence that converges.

(a) Show that if s_n ≥ a for all but finitely many n, then lim s_n ≥ a.

(b) Show that if s_n ≤ b for all but finitely many n, then lim s_n ≤ b.

(c) Conclude that if all but finitely many sn belong to [a,b], then lim s_n belongs to [a, b].

Prove the Weierstrass M-test for uniform convergence of Series of Functions.

What real world applications exist for Euler Circuits?

Matlab project:

Solve using Matlab three problems:

One using the combination formula

One using the permutation of n objects formula

One using the permutation of r objects out of n objects

You can pick these problems from the textbook or you can make up your own questions.

A detailed answer will be appreciate.

6. To prove that for all x1, x2, ..., x9 ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, there exists a

value of x10 for the check digit in the code ISBN-10.

7. To prove that for every x1, x2, ..., x12 ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, there exists a

value of x13 for the check digit in the code ISBN-13.

Solve this Initial Value Problem using the Laplace transform.

x''(t) + 6 x'(t) + 25x(t) = cos(t),

x(0) = 0,

x'(0) = 1

Consider the equation y''− (sin x)y = 0.

Find the general solution as a power series centered at x = 0. Write the first six nonzero terms of the solution. Write the two linearly independent solutions that form the general solution.

Differential Equations

Q.1 Marks 2 (CLO 3)

From the following data, you are required to: -

(a) Fit the regression line Y on X and predict Y if x = 20

(b) Fit the regression line X on Y and predict X if y = 10

Y	X
14	4
4	12
2	8
2	6
4	4
6	4
4	16
12	8

Note : Answers should be in Word or Excel Format

Minimize ?(?,?)=3?−2?+6f(x,y)=3y−2x+6 subject to ?≥−3,    ?≤4,    4?+7?≤23,    8?+7?≥−31.

Let D be a division ring, and let M be a right D-module. Recall that a subset S ⊂ M is linearly independent (with respect to D) if for any finite subset T ⊂ S, and elements a_t ∈ D for t ∈ T, if sum of ta_t = 0, then all the a_t = 0.

(a) If S ⊂ M is linearly independent, show that there exists a maximal linearly independent subset U of M that contains S, and that U is a basis for M (that is,M is a free D-module).

(b) Suppose that S is a generating set (that is, for every element m ∈ M, there exists a finite subset T ⊂ S and a_t ∈ D such that m = sum of ta_t). Show that there exists a subset U ⊂ S that is a basis for M.

(c)* Bonus for proving that all bases of M have the same cardinality, if it has a finite basis. (It is also true for infinite bases.)

Assess the symmetry of your body measurements by computing the proportions of your body using phi. Do the same with your facial measurements. (you may upload camera image).

Consider the cities B,C,D,E,F,G
The costs of the possible roads between cities are given below:

c(B,F) = 11
c(B,G) = 10
c(C,G) = 8
c(D,E) = 12
c(D,F) = 13
c(E,F) = 9
c(E,G ) = 7

What is the minimum cost to build a road system that connects all the cities?

Solve each of the following equations by finding an integrating factor:

x dy + y dx + 3x^3y^4 dy = 0

Consider a formula of propositional logic consisting of a conjunction of clauses of the form (±p⊕±q), where p and q are propositional variables (not necessarily distinct) and ±p stands for either p or ¬p. Consider the graph in which the vertices include p and ¬p for all propositional variables p appearing in the formula, and in which there is an edge (1) connecting p and ¬p for each variable p, and (2) connecting two literals if their exclusive-or is a clause of the formula. Prove that the formula is satisfiable if and only if the graph is 2-colorable.

Please formulate and solve each of the following problems. For each problem, you should include the final SOLVER printout (either your final spreadsheet or an answer report), as well as (1) clear and precise definitions for all decision variable; (2) your objective function indicating whether it is to be maximized and minimized; (3) all constraints, including non-negativity and integrality (if necessary); and (4) what the optimal decision is (in words) and what outcome will be produced.

1. A manufacturer of stereos has plants in Atlanta, New Haven and Dallas, and distributions centers in San Francisco, Boston, Washington, D.C., and Cleveland. The tables show weekly production capacities, demand requirements, and unit transportation costs (in dollars).

ORIGIN	PRODUCTION	DESTINATION	REQUIREMENTS
Atlanta	65	San Francisco	50
New Haven	75	Boston	35
Dallas	45	Washington, D.C.	35

                                                                                Cleveland                                       65

UNIT TRANSPORTATION COSTS
	San Francisco	Boston	Washington, D.C.	Cleveland
Atlanta	13	9	6	5
New Haven	11	6	7	4
Dallas	7	8	15	10

The goal is to minimize total transportation costs.

3. A company has five jobs, each of which must be assigned to a single machine. The table shows the dollar costs for each possible job-machine assignment:

JOB                                                    MACHINE

                                    A                     B                     C                     D                     E

1                                  138                  127                  118                  121                  143

2                                  157                  138                  129                  132                  160

3                                  143                  129                  131                  130                  172

4                                  111                  119                  123                  107                  120

5                                  102                  120                  100                  119                  100     

Find the set of assignments with the lowest possible total cost.