Draw a graph having the given properties stated below, or explain why no such graph exists: In each case assume simple graphs (no self loops and no parallel edges) a. Six vertices each with degree 3 b. Five vertices each with degree 3. c. Four vertices each with degree 1. d. Six vertices and four edges. e. Four edges; four vertices having degrees 1, 2, 3, and 4.

Which study design is best to minimize confounders--cohort or case-control?

Big-M (describe process up to initial tableau and how to
recognize if infeasible) careful to distinguish between Max LP and
a Min LP

Use method of undetermined coefficients to find a particular solution of the differential equation ?′′ + 9? = cos3? + 2. Check that the obtained particular solution satisfies the differential equation.

s QUESTION 10 Car payments are determined using simple interest. If you don’t remember this formula, go back and look it up! To finance your vehicle purchase you have two choices. Write two equations, one for each bank, that models the banks' loan options using x for the price of the vehicle and y to represent the total cost (price, interest and origination fees) paid on each loan. Be sure to label which equation goes with which bank. Bank A: Finance the full price of the vehicle at 5% with an origination fee of $300 paid over 5 years. Bank B: Finance the full price of the vehicle at 3.5% paid over 6 years (no additional fees).

QUESTION 12 Use the equations you came up with for bank financing options to calculate the following: Total cost of your vehicle using bank A Monthly payment of your vehicle using bank A Total cost of your vehicle using bank B Monthly payment of your vehicle using bank B Show all of your work using the equation editor and label each solution. my car costs $31,690.00

Use the Laplace transform to solve the given system of differential equations. d2x dt2 + 3 dy dt + 3y = 0 d2x dt2 + 3y = te−t x(0) = 0, x'(0) = 4, y(0) = 0

Using the constructive proof of the Chinese Remainder Theorem, find the unique x(mod 100) satisfying the congruences

x ≡ 1(mod 25),

x ≡ 0(mod 4).

1. A local developer is selling homes for $125,000 with a required down payment of 6%. Find the amount of the required down payment and the mortgage.

2. A couple obtained a loan for $130,000 to purchase their home. How much would the loan origination fee be if the couple had to pay 3 points on the loan. (Loan origination fee = Mortgage * Points).

3. New houses in a neighborhood are selling for $175,000. A down payment of $18,000 is required and a 25-year mortgage at an annual interest rate of 8% is available. Find the monthly mortgage payment.

4. The monthly mortgage payment on a house is $824.36, and the homeowner must pay an annual property tax of $930. Find the total monthly payment for the mortgage and the property tax.

Let x be a set and let R be a relation on x such x is simultaneously reflexive, symmetric, and antisymmetric. Prove equivalence relation.

. For this problem, you may find Proposition 3.1.5 useful which in turn implies that tan x is continuous whenever x is not an odd multiple of π 2 . Moreover, you can assume that sin x and cos x are positive and negative on appropriate intervals. Let I := (− π 2 , π 2 ). (a) Show that tan x is strictly increasing and, hence, injective on I. (b) Prove that lim x→− π 2 + tan x = −∞ and lim x→π 2 − tan x = ∞ Use this to conclude that f(I) = R. You may find Exercise 6 in Section 3.7 useful here. (c) Show that arctan x = tan−1 x maps R to I and is differentiable everywhere with d dx tan−1 x = 1 1 + x 2 for all x ∈ R. (d) Prove that limx→∞ tan−1 x = π 2 and lim x→−∞ tan−1 x = − π 2

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is continuous on [a, b]

(62). (Transient Behavior of a Mixing tank): A tank initially holds 80 gal of a brine solution containing 0.125 lb of salt per gallon. At t=0, another brine solution containing 1 lb of salt per gallon is poured into the tank at a rate of 4 gal/min, while the well-stirred mixture leaves the tank at a rate of 8 gal/min. (a) Formulate a model for describing the transient behavior of the tank (b) Find the amount of salt in the tank when the tank contains exactly 40 gal of the solution.

Solve by solving the dual problem.

Minimize

z = 30x₁ + 15x₂ + 28x₃,

subject to