Prove that the polynomial x^3 + x^2 – x + 1 has no integer roots

1. Here is a model:

Domain: {Wikipedia, 2, Isaac Newton}

Referents: ​n​: Isaac Newton

Extensions: P: Ø Q: {2} R: {Wikipedia, Isaac Newton}

Is the following proposition true or false on this model? Explain your answer.

∀​x​(​Qx ​∨​Rx​) → ​Pn

2. Is the following argument valid or invalid? If it is valid, explain why there cannot be a countermodel. If it is invalid, provide a countermodel.

ᄀ∀​xHx

∃​xMx

∴ᄀ​∃​x​(​Mx ​∧​Hx​)

Bunce’s farm has a 10 L water tank fed by a stream running from Boggis’ Chicken Ranch. Water from the stream flows into and out of Bunce’s tank at a rate of 10 L/minute which keeps it well-mixed. One day Bunce finds out that Boggis has been dumping chicken manure into the river which adds Salmonella to the stream at an unknown concentration of a mcfu/L (Note: mcfu is a unit of measurement for bacteria standing for “millions of colony forming units”). At the time of his discovery, Bunce measures the concentration of Salmonella in his tank to be 1 mcfu/L and one minute later measures it again to be 2 mcfu/L. Use these measurements to determine the concentration of Salmonella in the river and the limiting concentration of resulting Salmonella in Bunce’s tank.

Joan borrowed ​$17 comma 000.00 to buy a car. She repaid ​$4100.00 four months later and ​$7900.00 six months later. After twelve​ months, she borrowed an additional ​$6200.00​, and repaid ​$5100.00 after 20 months. She paid the entire​ balance, including the​ interest, after 24 months. Interest was 9​% compounded monthly for the first year and 9.6​% compounded monthly for the remaining time. What was the size of the final​ payment?

4. Compute the minimum square error for
f(x) =x/π (−π < x < π).

Suppose a 1 kg mass is attached to the end of a string that is stretched 0.2 m by a force of 500 N (newtons). An exterior force F0 cos 50t acts on the mass. Find the position function x(t) if the initial conditions are given by x(0) = 0, x′(0) = 50.

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Consider an mass-spring system with the following IVP for its disagreement y (t) at time t greater than or equal to 0. You may assume it is underdamped.

y" + y' + 5y = 0 , y (0) = -2 , y '(0) = -1

(a) Convert this to a DE system IVP in displacement y and velocity v.

(b) Without using technology or solving the second order DE, make a rough sketch of the system solution on a phase plane. It does not need to be precise, but briefly show and explain the direction you give at the initial value, and the long term behavior as t heads towards infinity

Discuss the applications of PDE's in Engineering, Science and Economics with a detailed example for each. In at least one example, discuss the solution to the PDE and how it has practical implications.

Let L = {aⁱb^j | i ≠ j; i, j ≥ 0}.

Design a CFG and a PDA for this language. Provide a direct design for both CFG and PDA (no conversions from one form to another allowed).

Question 1.​ Answer the following questions by making a truth table. Be sure to explain what feature of the truth table you’ve drawn justifies your answer. (That is, indicate which part, or parts, of the table show what the answer to the question is and why)

a) Is ¬P, P ↔ Q, P → Q a logically consistent set of sentences?

b) Is A ∧ (B ∨ C), ¬((A ∨ B) ∧ C) a logically consistent set of sentences?

c) Does ? → ¬? entail ¬(?∨?)?

d) Does the set ¬(L ∧ M), ¬(M ∧ N) entail ¬M?

If an undamped spring-mass system with a mass that weighs 24 lb and a spring constant 9 lbin is suddenly set in motion at t=0 by an external force of 288cos(4t) lb, determine the position of the mass at any time. Assume that g=32 fts2. Solve for u in feet.

Enclose arguments of functions in parentheses. For example, sin(2x).