How would I do this problem?

A mass of 1 kg is attached to a spring whose constant is 5 N/m. Initially the mass is released 1 m below the equilibrium position with a downward velocity of 5 m/s, and the subsequent motion takes place in a medium that offers a damping force numerically equal to two times the instantaneous velocity. Find the equation of motion if the mass is driven by an external force equal to f(t) = 12 cos (2t) + 3 sin (2t)

Let T be a connected graph and z ∈ℤ between the closed interval of 1 and the least degree of a vertex in T. Let a z - matching be a A ⊆ E s.t. there aren’t vertices with more than z edges in A. Let a z - cover be a X ⊆ E s.t. all vertices belong to at least z edges in X.

Let:

δ (T) = Max {|A| : A is a z - matching}

μ (T) = Min {|X| : X is a z - cover}

Show that δ (T) + μ (T) = zn

For the following questions, say you were given a line and a plane as below

                Line: r(t)= < x(t), y(t), z(t) > = < t-2, t+1, 3t > ,        Plane:   : a(x+2) + b(y-3) - 4z = 2

                a)            What relationship would have to exist between scalars a and b for the line not to intersect with the plane?

                Hint 1) Plug in x, y, z given in the line into the plane equation.

                Hint 2) Say you were solving this equation and got the following. What would that indicate?

                                i)             t = 2:      ________________________________________________________

                                ii)            6 = –5:   ________________________________________________________

                                iii)           4 = 4:     ________________________________________________________

                                So what must be true about the coefficient of t? What also can’t be true about the constant term?

                b)            What relationship would have to exist between scalars a and b for the plane to contain the line?

                c)            Assuming the line is not contained in the plane but does intersect it, give an expression for the time t that the line intersects the plane. Also give the point of intersection.
                                (Both answers will have scalars a and b in them)

Prove by induction that:

1) x^n - 1 is divisible by x-1

2)2n < 3^n for all natural numbers n

SOLVE IN MATLAB. SHOW CODE.

You have decided to enter the candy business. You are considering producing two types of candies: Slugger Candy and Easy Out Candy, both of which consist solely of sugar, nuts, and chocolate. At present, you have in stock 100 oz of sugar, 20 oz of nuts, and 30 oz of chocolate. The mixture used to make Easy Out Candy must contain at least 20% nuts. The mixture used to make Slugger Candy must contain at least 10% nuts and 10% chocolate. Each ounce of Easy Out Candy can be sold for 25¢ , and each ounce of Slugger Candy for 20¢. Formulate an LP that will enable you to maximize your revenue from candy sales.

Find the pure and mixed strategy Nash equilibriums for the following game.

Show computation how you got it

Let X, Y be Banach spaces. Show that T ∈ L(X, Y) is continuous if and only if T is bounded.

1- Let W1, W2 be two subspaces of a vector space V . Show that
both W1 ∩ W2 and W1 +W2 are subspaces.?and Show that W1 ∪ W2 is a subspace
only when W1 ⊂ W2 or W2 ⊂ W1.
(recall that W1 + W2 = {x + y | x ∈ W1, y ∈ W2}.)

Problem 5. Recall that Mn,n(R) is the vector space of all n by n real matrices.

(a) Show that W = {A | tr(A) = 0} is a subspace of Mn,n(R).?

(b) Determine the dimension of W and find a basis for it.?

(c) Show that the trace map tr : Mn,n(R) → R is a linear transformation.?

4)

You are measuring the length of catheters from your production line. The catheters have a

mean length of 30 cm and a standard deviation of 2 cm.

a. What is the probability that a catheter will be longer than 31.7 cm?

b.What is the probability that a catheter will be between 29.3 and 33.5 cm long?

c.What is the probability that a catheter will be less than 25.5 cm long?

5) In a survey of senior citizens, you find that there is an 60% chance that any given senior citizen will disapprove of legalized marijuana.

a.What is the chance you will find between 7 – 9 senior citizens out of 12 surveyed will

disapprove of legalized marijuana?

b. What is the chance you will find between 7 – 9 senior citizens out of 12 surveyed will

approve of legalized marijuana?

c. What is the chance that no more than 5 senior citizens out of 12 surveyed will approve of

legalized marijuana?

d.What is the chance that at least 8 senior citizens out of 12 surveyed will disapprove of legalized marijuana?

Sara Sanders purchased 50 shares of Apple stock at $ 190.35 per share using the prevailing minimum initial margin requirement of 56 %. She held the stock for exactly 6 months and sold it without any brokerage costs at the end of that period. During the 6​-month holding​ period, the stock paid $ 1.54 per share in cash dividends. Sara was charged 4.5 % annual interest on the margin loan. The minimum maintenance margin was 25 %.

a. Calculate the initial value of the​ transaction, the debit balance​, and the equity position on​ Sara's transaction.

b. For each of the following share​ prices, calculate the actual margin​ percentage, and indicate whether​ Sara's margin account would have excess​ equity, would be​ restricted, or would be subject to a margin​ call: ​(1) $ 174.66​, ​(2) $ 207.62​, and​ (3) $ 122.17.

c. Calculate the dollar amount of​ (1) dividends received and​(2) interest paid on the margin loan during the 6​-month holding period.

d. Use each of the following sale prices at the end of the 6​-month holding period to calculate​ Sara's annualized rate of return on the Apple stock​ transaction: ​(1) $ 184.47​, ​(2) $ 194.61​, and​ (3) $ 206.26.

Solve the linear system Az = b using the following methods:

I. The GE+PP algorithm for sparse (banded) linear systems, which is the default algorithm used by Matlab’s “\” operator when the matrix (call it Asparse) is of sparse type. You may find it easiest to set up the matrix using the spdiags command.

Let B ={A1, A2, ..., An}⊆Mmn, and write B′ ={A^T 1, A^T 2, ..., A^T n}⊆Mnm. Show that:

a. B is independent if and only if B′ is independent.

b. B spans Mmn if and only if B′ spans Mnm.