At what point do the curves r1(t) = t, 3 − t, 35 + t2 and r2(s) = 7 − s, s − 4, s2 intersect? (x, y, z) = Find their angle of intersection, θ, correct to the nearest degree. θ = °

Consider G = (Z₁₂, +). Let H = {0, 3, 6, 9}.

a. Show that H is a subgroup of G.

b. Find all the cosets of H in G and denote this set by G/H. [Note: If x ∈ G then H +₁₂ [x]₁₂ = {[h + x]₁₂?? | [h]₁₂ ∈ H} is the coset generated by x.]

c. For H +₁₂ [x]₁₂, H +₁₂ [y]₁₂ ∈ G/H define (H+_12[x]₁₂)⊕(H+₁₂[y]₁₂) by(H+₁₂ [x]₁₂)⊕(H+₁₂ [y]₁₂)=H+₁₂ [x+y]₁₂.

d. Show that ⊕ is well defined and construct the addition table for G/H with the operation ⊕.

find the projection vector of the vector v = (2,3,5) onto the plane z = 2x + 3y -1

Second order Differential equation:

Find the general solution to [ y'' + 6y' +8y = 3e^(-2x) + 2x ] using annihilators method and undetermined coeficients.

*PLEASE SHOW ALL WORK*

Consider a damped, forced mass/spring system. Let t denote time (in seconds) and let x(t) denote the position (in meters) of the mass at time t, with x = 0 corresponding to the equilibrium position. Suppose the mass m = 1 kg, the damping constant c = 3 N·s/m, the spring constant k = 2 N/m, the external force is F (t) = 20 cos(2t), the initial position

x(0) = 1 m, and the initial velocity x′(0) = 2 m/s. a. Find the position function x(t).

b. What part of the solution is the transient part and what part is the steady-state part?

1. For an m x n matrix A, the Column Space of A is a subspace of what vector space?

2. For an m x n matrix A, the Null Space of A is a subspace of what vector space?

Use the method of undetermined coefficients to find the complete solutions of the following differential equations.

d²y/dx² − 3 dy/dx + 2y = 2x² + e^x + 2xe^x + 4e^3x .

Given the curve −→r (t) = <sin³ (t), cos³ (t),sin² (t)> for 0 ≤ t ≤ π/2 find the unit tangent vector, unit normal vector, and the curvature.

Another model for a growth function for a limited population is given by the Gompertz function, which is a solution to the differential equation

dP/dt=cln(K/P)P

where c is a constant and K is the carrying capacity. Answer the following questions

1. Solve the differential equation with a constant c=0.15, carrying capacity K=3000, and initial population P0=1000

Answer: P(t)=?

2. With c=0.15, K=3000, and Po=1000, find limt→∞P(t).

Limit:?

Find the periodic payment R required to amortize a loan of P dollars over t years with interest charged at the rate of r%/year compounded m times a year. (Round your answer to the nearest cent.)

a. P = 50,000, r = 4, t = 15, m = 4

b. P = 90,000, r = 3.5, t = 17, m = 12'

c. P = 120,000, r = 5.5, t = 29, m = 4

A cup of hot coffee has a temperature of 201˚F when freshly poured, and is left in a room at 72˚F. One minute later the coffee has cooled to 191˚F.

a. Assume that Newton's Law of cooling applies. Write down an initial value problem that models the temperature of the coffee.

u'= -k(_____)

u(o)= _____

b. Determine when the coffee reaches a temperature of 183˚F. Give answer in minutes, and round the answer to two decimal places.

Solve the following linear equations with constant coefficients, using characteristic equations and undetermined coefficients as needed.

y''+4y'-12y=x+e^2x y(0)=1,y'(0)=2

Use the truth-tree decision procedure (relying on Proof Tools or pen/pencil and paper) to determine whether P→Q,Q⊨P is deductively valid.