1. Consider the system;

dx/dt=x(1-x/10)-1/2xy

dy/dt=2y(1-y/6)-xy

(a) Find all equilibrium solutions.

(b) Draw the phase plane.

(c) On the phase plane, mark
• all equilibrium solutions.

• the lines where dx/dt =0, dy/dt =0

• an arrow indicating the general direction of the flow of the system in each region on the picture.

(d) If you were given initial conditions x = 3 and y = 4, what do you think would happen to the solution?

Show many ways can you tile a 3xn checkboard with a 2x1 tile? Create a recursive relation and show how to derive  the number of ways.

An eagle is flying horizontally at the altitude of 400 ft with a velocity of 16 ft/s when it drops a dead prey. A frog jumps 2 seconds later at a site facing away from the eagle and 60 ft from the point beneath the original position of the bird. If the frog catches the falling prey at the height of 144 ft, determine the initial speed and angle of inclination of the frog. Please use an advanced math way using vector-valued functions to solve it (Calculus 3).
A note: h = 400 , b = 144 , a = 60

Let y =

U1 =

U2 =

Find the distance from y to the plane in R3 spanned by U1 and U2. Exact answer please.

Engineering system of type k-out-of-n is operational if at least k out of n components are operational. Otherwise, the system fails. Suppose that a k-out-of-n system consists of n identical and independent elements for which the lifetime has Weibull distribution with parameters r and λ. More precisely, if T is a lifetime of a component, P(T ≥ t) = e−λtr, t ≥ 0. Time t is in units of months, and consequently, rate parameter λ is in units (month)−1. Parameter r is dimensionless. Assume that n = 8,k = 4, r = 3/2 and λ = 1/10. (a) Find the probability that a k-out-of-n system is still operational when checked at time t = 3. (b) At the check up at time t = 3 the system was found operational. What is the probability that at that time exactly 5 components were operational? Hint: For each component the probability of the system working at time t is p = e−0.1 t3/2. The probability that a k-out-of-n system is operational corresponds to the tail probability of binomial distribution: IP(X ≥ k), where X is the number of components working. You can do exact binomial calculations or use binocdf in Octave/MATLAB (or dbinom in R, or scipy.stats.binom.cdf in Python when scipy is imported). Be careful with ≤ and <, because of the discrete nature of binomial distribution. Part (b) is straightforward Bayes formula.

Let ? be a ?−algebra in ? and ?:? ⟶[0,∞] a measure on ?. Show that ?(?∪?) = ?(?)+?(?)−?(?∩?) where ?,? ∈?.

Find a particular solution, Yp, of the non-homogenous DE y" + 3y' + 2y = 1/1+e^x

Prove using the short north-east diagonals or any other mathematical method of your preference, that if A is enumerable, then it is also countable with an enumeration that lists each of its members exactly three (3) times. Hint. Your proof will consist of constructing an enumeration with the stated requirement.

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y < y^(−1) then x < −1.

Given a group G with a subgroup H, define a binary relation on G by a ∼ b if and only if ba^(-1)∈ H.

(a) (5 points) Prove that ∼ is an equivalence relation.

(b) (5 points) For each a ∈ G denote by [a] the equivalence class of a and prove that [a] = Ha = {ha | h ∈ H}. A set of the form Ha, for some a ∈ G, is called a right coset of H in G.

(c) (5 points) Let a ∈ G. For all g ∈ G prove that Hg = Ha if and only if g ∈ Ha. Hint: two elements are equivalent if and only if their equivalence classes coincide.

(d) (5 points) Prove that the map ρa : H → Ha given by ρa(h) = ha, h ∈ H, is a bijection.

find the equation of the line that has slope - 2/3 and which passes through (-1,-6)

The SkyLight Company produces two light fixtures (products 1 and 2) that require both metal frame parts and electrical components. Management wants to determine how many units of each product to produce so as to maximize profit. For each unit of product 1, two unit of frame parts and two units of electrical components are required. For each unit of product 2, three units of frame parts and two unit of electrical components are required. The company has 240 units of frame parts and 200 units of electrical components. Each unit of product 1 gives a profit of $1, and each unit of product 2 gives a profit of $3. No more than 60 units of product 2 should be produced.

1. Formulate this linear programming model algebraically.
2. Use the graphical method (by drawing the objective function line) to solve this model, determine the optimal solution and maximum profit.
3. Write down the coordinate of all corner points.
4. How does the optimal solution change if the maximum unit of product 2 is increased to 80 units? Use the graphical method to solve the new model.
5. Formulate the model in (a) using Excel spreadsheet. Identify the data cells, the changing cells, and the objective cell.
6. Solve the model using Excel Solver. Write down the optimal solution, and value of slack variables of each constraint.

3. Consider the volume E as the solid tetrahedron with vertices (1, a, 0), (0, 0, 0), (1, 0, 0), and (1, 0, 1) where a > 0. (a) Write down the region E as a type I solid. (b) Find a such that RRR E x^2 yz dV = 1.