4) Find the volume of the solid formed by the region bounded by the graphs of y= x3 , y=x for x=0 and x=1

-Sketch the region bounded by the graphs of the functions and find the area of the region bounded by the graphs of y=x-1 and y= (x − 1)3

-calculate the arc length of the graph y= x=1 to x=2 14x7 + 101x5 from

-Use the washer method to find the volume of the solid formed by revolving the region bounded by the graphs by y=-2x+1 and y= x2 +1

Problem for submission: For which positive integers k can a simple graph G = (V, E) be constructed such that: G has k vertexes, that is, |V | = k, G is bipartite, and its complement G is bipartite? Prove your answer is correct

Please show and explain your full proof.

A binary string is a “word” in which each “letter” can only be 0 or 1

Prove that there are 2^n different binary strings of length n.

Note:

• Your goal is to produce a properly constructed proof by induction, but this does not mean you have to follow
• Mathematical induction, step-by-step..
• Write the statement with n replaced by k
• Write the statement with n replaced by k+1.
• Identify the connection between the kth statement and the (k+1)th statement.
• Complete the induction step by assuming that the n=k version of the statement is true, and using this assumption to prove that the n=k+1 version of the statement is true.
• Complete the induction proof by proving the base case.
• point-by-point.
• ANOTHER IMPORTANT NOTE:
• Make sure you are addressing the given statement: "there are 2ⁿ different binary strings of length n” !!! So your solution should primarily be discussing binary strings. Yes, the fact that 2^k+1 = 2^k∙2 will be useful in your solution, but it should not be the sole content of your solution, as by itself that equality is a fact about exponents, not a fact about binary strings.

The commanding officer (CO) plans to move a part of his battalion to
another location. There are two types of vehicle available to him, a Ford
vehicle that can carry 25 m3 plus 5 personnel, and a Holden vehicle that
can carry 15 m3 plus 10 personnel. For security reason, all vehicles must
move together and they will be used for a single trip. The required
materials to be taken to the new location are organized in unit pallet
load of 2 m3. The CO requires transporting of a total of 60 pallet loads
and 40 personnel. There are a maximum of 30 Ford and 40 Holden
vehicles available. Each Ford vehicle is estimated to use 50 L of fuel per
trip, whereas the Holden vehicle will only use 30 L. If the CO wants to
move all the required materials and personnel at minimum fuel use,
what mix of Ford and Holden vehicles should the CO choose? Formulate
the problem as an LP model.

Consider the IVP t²y''−(t² + 2)y' + (t + 2)y = t³ with y(1) = 0 and y'(1) = 0.

• One function in the fundamental set of solutions is y₁(t) = t. Find the second function y₂(t) by setting y₂(t) = w(t)y₁(t) for w(t) to be determined.

• Find the solution of the IVP

Consider the IVP t²y''−(t² + 2)y' + (t + 2)y = t³ with y(1) = 0 and y'(1) = 0.

• One function in the fundamental set of solutions is y₁(t) = t. Find the second function y₂(t) by setting y₂(t) = w(t)y₁(t) for w(t) to be determined.

• Find the solution of the IVP

A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 900, 1080, and 840, respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows. (For example, to make 1 unit of product A requires 2 hours of work from Dept. I, 3 hours of work from Dept. II, and 2 hours of work from Dept. III.)

How many units of each product should the company produce in order to maximize its profit?

What is the largest profit the company can realize?
$  

Are there any resources left over? (If so, enter the amount remaining. If not, enter 0.)

15. Let r be a positive real number. The equation for a circle of radius r whose center is the origin is (x^2)+(y^2)= r^2 .

(a) Use implicit differentiation to determine dy/dx .

(b) Let (a,b) be a point on the circle with a does not equal 0 and b does not equal 0. Determine the slope of the line tangent to the circle at the point (a,b).

(c) Prove that the radius of the circle to the point (a,b) is perpendicular to the line tangent to the circle at the point (a,b).

Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to

The integers satisfy a property known as mathematical induction. This is a familiar topic in high school textbooks.

(a) The First Principle of Mathematical Induction is stated as follows. Suppose S is a subset of N with the following properties: (i) The number 1 is in S. (ii) If n is in S, then n + 1 is in S. Using well-ordering, prove S = N.

(b) The Second Principle of Mathematical Induction is stated as follows. Suppose S is a subset of N with the following properties: (i) The number 1 is in S. (ii) If n is in S and if every natural number k, where k ≤ n, is in S, then n + 1 is in S. Using well-ordering, prove S = N.

Show, using mathematical induction, that 2k−1 ≤ k! for all natural numbers k, by doing parts (a) and (b).

(a) Let S be the set of natural numbers for which the inequality is true. Show that 1 is in S.

(b) Now conclude, using the induction hypothesis and the theorems about inequalities, that 2 n = 2 · 2 n−1 ≤ 2 · n! ≤ (n + 1) · n! = (n + 1)! That is, if n is in S, then n + 1 is in S.

Define the following order on the set Z × Z: (a, b) < (c, d) if either a < c or a = c and b < d. This is referred to as the dictionary order on Z × Z.

(a) Show that there are infinitely many elements (x, y) in Z × Z satisfying the inequalities (0, 0) < (x, y) < (1, 1).

(b) Show that Axioms O1–O3 ( Trichotomy, Transitivity, Addition for inequalities) are satisfied for this ordering.

(c) Give an example that shows that Axiom O4 (Multiplication for inequalities) is not satisfied for this ordering.

(d) Is the well-ordering axiom satisfied for Z × Z with the dictionary order?

Let G be an abelian group and n a fixed positive integer. Prove that the following sets are subgroups of G.

(a) P(G, n) = {gⁿ | g ∈ G}.

(b) T(G, n) = {g ∈ G | gⁿ = 1}.

(c) Compute P(G, 2) and T(G, 2) if G = C₈ × C₂.

(d) Prove that T(G, 2) is not a subgroup of G = D_n for n ≥ 3 (i.e the statement above is false when G is not abelian).

1) Find the solution of the given initial value problem and describe the behavior of the solution as t → +∞

y" + 4y' + 3y = 0, y(0) = 2, y'(0) = −1.

2) Find a differential equation whose general solution is Y=c₁e^2t + c₂e^-3t

3) Determine the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution. Do not attempt to find the solution t(t − 4)y" + 3ty' + 4y = 2 = 0, y(3) = 0, y'(3) = −1.

4) Consider the ODE: y" + y' − 2y = 0. Find the fundamental set of solutions y1, y2 satisfying y1(0) = 1, y'1 (0) = 0, y2(0) = 0, y'2 (0) = 1.

Determine the number of permutations of {1,2,3,...,n-1,n} where n is any positive integer and no even integer is in its natural position.