3.2.1. Find the Fourier series of the following functions:

(g) | sin x |

(h) x cos x.

Let V = {P(x) ∈ P10(R) : P'(−4) = 0 and P''(2) = 0}. If V= M_3×n(R), find n.

Recall P2(t) is the set of polynomials of order less than or equal to 2. Consider the the set of vectors in P2(t).

B={t^2,(t−1)^2,(t+1)^2}

(a) Show B is a basis for P2(t).

(b) If E={1,t,t^2}is the standard basis, calculate the change of basis matrices PE→B and PB→E

(c) Given v= 2t^2−5t+ 3, find its components in B

Given a sequence of closed intervals arranged so that each interval is a subinterval of the one preceding it and so that the lengths of the intervals shrink to zero, then there is exactly one point that belongs to every interval of the sequence. (This is known as the nested interval property)

3. The California Emmisons cap was set at 322 millon metric tons of carbon dioxide equivilent in 2016 and was expected to drop to 303 million metric tons of arbon dioxide equivilent in 2024 if we assume that the decreases in the emmisons cap is linearly each year, then do the followig thre parts:

a. Determine the linear quation for the amount of emmisions, e (in millions of metric tons), in terms of the number of years after 2000, t. Show the steps that you performed to arrive at your answer. Your final answer should be in the form of a short summary sentence. Place a box around your final answer.

B. Interpret the slope of the equation in the context of the problem using the idea of the steepnes property.

C. If this trend continues, then when will there be approximately 290 million metric tons of carbon dioxide emmited. Show, in detail how you arrived at your answer. Write a short summary as your final answer.

most important part c)

Determining whether a quantified logical statement is true and translating into English, part 2.

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In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates:

• P(x): x was given the placebo
• D(x): x was given the medication
• A(x): x had fainting spells
• M(x): x had migraines

Suppose that there are five patients who participated in the study. The table below shows the names of the patients and the truth value for each patient and each predicate:

For each of the following quantified statements, indicate whether the statement is a proposition. If the statement is a proposition, give its truth value and translate the expression into English.

(c) ∃x M(x) ∧ D(x)

(f) ∀x ((M(x) ∧ A(x)) → ¬D(x))

(g) ∃x (D(x) ∧ ¬A(x) ∧ ¬M(x))

(h) ∀x (D(x) → (A(x) ∨ M(x)))

LU Decomposition

(i). Prove that for n equal to 2 or 3 there is a non-singular square (n by n) matrix which has no LU decomposition with L unit lower triangular and U upper triangular. (In fact, this is true for any integer ≥ 2.)

(ii). We will see that all non-singular square matrices do have an LUP decomposition (some time soon in class). Here P is a permutation matrix, also defined in Appendix D and used in Chapter 28.

Show that the inverse of a permutation matrix P is also a permutation matrix. (You can do this by explicitly defining what P −1 when P is given.)

An urn always contains two balls, where each ball is either red or blue.

At each stage a ball is randomly chosen from the urn. A drawn red
ball is always replaced with a blue ball. A drawn blue ball is equally
likely to be replaced by either a red or a blue ball. Suppose that the
urn initially has one red and one blue ball.
(a) Define a Markov chain that should be useful for the above model.
Define its states and give the transition probabilities.
(b) Find the probability that the second ball selected is red.
(c) Find the probability that the third ball selected is red.
(d) Find the long run proportion of time that both balls are red.
(e) Find the long run proportion of drawn balls that are red.

initially a large tank with a capacity of 300 gallons contains 150 gallons of clean water. A solution of salt with a concentration of 2 lb / gal flows into the tank at a rate of 50 gal / min. The solution is perfectly well mixed while the solution is extracted at a rate of 25 gal / min. Find:

a) the amount of salt in the tank at the time it is filled (in pounds)

b) the speed at which salt is coming out at this time (in pounds per minute)

c) The amount of salt that has left the tank since the beginning and until that moment (in pounds)

The given table shows the preference schedule for an election with 5 candidates. Find the complete ranking using the method of pairwise comparisons.​ (Assuming that ties are broken using the results of the pairwise comparisons between the tying​ candidates.)

Choose the answer below

( ) E,D,B,C,A

( ) A,B,C,D,E

( ) E,A,B,C,D

( ) D,C,E,A,B

We say that a set of system specifications is consistent if there is a way for all the specifications to be simultaneously true. Otherwise, the set of specifications is inconsistent. Consider the following set of system specifications:

If the file system is not locked, then new messages will be queued.

The file system is not locked if and only if the system is functioning normally.

If new messages are not queued, then they will be sent to the message buffer.

If the file system is not locked, then new messages will be sent to the message buffer.

New messages will not be sent to the message buffer.

Use propositional logic to determine if the set of system specifications consistent or inconsistent by doing the following:

Part a (2 pts) : First, define the appropriate propositional variables for representing the system specifications.

Part b (2.5 pts) : Then, translate each of the system specifications into propositional logic using the propositional variables you defined in part a.

Part c (2.5 pts) : Finally, is the set of system specifications consistent or inconsistent? Justify your answer.

4) Write a brief reflection of Task 2 and 3 which may include description on data type used to solve the given task, variable used, and objects created. Reflection should also include justification on logic used to solve the given task along with proper references

Use the Laplace transform to solve the given initial-value problem.

y'' − 7y' = 12e^6t − 6e^−t,    y(0) = 1, y'(0) = −1

Find the following Taylor Series with given centers. Use the algebraic methods rather than the derivatives.

1. f(x) = x^3 + x^2 -2x +3 centered at a= -1

2. sin(x) centered at a= pi/2

3. f(x) = (e^x - e^-x)/2 centered at a= 0