Find the real Fourier series of the piece-wise continuous periodic function

f(x)=1+x+x^2 -pi<x<pi

Find the fourier series of the following

a. ?(?) = 1 + ? ; −? ≤ ? < ?

b. ?(?) = { 1 0 ≤ ? < 2 −1 2 ≤ ? < 4

c. ?(?) = ? 2 + 1 ; −1 ≤ ? < 1

help guys struggling student here thankies

A store has 5 years remaining on its lease in a mall. Rent is $2,100 per month, 60 payments remain, and the next payment is due in 1 month. The mall's owner plans to sell the property in a year and wants rent at that time to be high so that the property will appear more valuable. Therefore, the store has been offered a "great deal" (owner's words) on a new 5-year lease. The new lease calls for no rent for 9 months, then payments of $2,600 per month for the next 51 months. The lease cannot be broken, and the store's WACC is 12% (or 1% per month).

1. Should the new lease be accepted? (Hint: Be sure to use 1% per month.)

   -Select- Yes or No

2. If the store owner decided to bargain with the mall's owner over the new lease payment, what new lease payment would make the store owner indifferent between the new and old leases? (Hint: Find FV of the old lease's original cost at t = 9; then treat this as the PV of a 51-period annuity whose payments represent the rent during months 10 to 60.) Do not round intermediate calculations. Round your answer to the nearest cent.

   $____________  

3. The store owner is not sure of the 12% WACC—it could be higher or lower. At what nominal WACC would the store owner be indifferent between the two leases? (Hint: Calculate the differences between the two payment streams; then find its IRR.) Do not round intermediate calculations. Round your answer to two decimal places.

     __________%

Let f(x) = −3+(3x² −x+1) ln(2√x−5)

(a) Find the derivative of f.

(b) Using the derivative and linear approximation, estimate f(9.1).

Exercise 2.4.1: Proofs by contradiction. About Give a proof for each statement.

(c)The average of three real numbers is greater than or equal to at least one of the numbers.

(e)There is no smallest integer.

Discrete Math / Proofs

Directions: Show all work/steps. State all assumptions as well as the goal of the proof.

Define A = { all binary sequences of length 4 }

So < 1, 1, 0 1 > ε A, <0, 0, 0, 0 > ε A, <1, 0, 0, 1> ε A etc.

i.) What is | A | ?

Define a relation R on A as follows:

For 1, a₂, a₃, a₄ > R 1, b₂, b₃, b₄> ε A

( 1, a₂, a₃, a₄> R 1, b₂, b₃, b₄> if and only if a₁ - a₂ + a₃ - a₄ = b₁ - b₂ + b₃ - b₄ )

e.g. <0, 0, 1, 1> R <1, 1, 0, 0> since 0 - 1 + 1 - 1 = 0 = 1 - 1 + 0 - 1 and <1, 0, 1, 1> R <0,0,1,0> since 1 - 0 + 1 - 1 = 1 = 0 - 0 + 1 - 0   etc.

ii.) Prove R is an equivalence relation on A. Please be clear in your exposition.

iii.) List the elements (binary sequences of length 4) in each equivalence class and name each equivalence class with one of its names.

iv.) Suppose an operation on the set of equivalence classes is intended to be defined as follows:

[<a₁, a₂, a₃, a₄>] * [<b₁, b₂, b₃, b₄>] = [a₁b₁, a₂b₂, a₃b₃, a₄b₄>]

Show by specific example that this operation is not well-defined.

Find the charge on the capacitor in an LRC-series circuit at

t = 0.04 s

when

L = 0.05 h,

R = 1 Ω,

C = 0.04 f,

E(t) = 0 V,

q(0) = 5 C,

and

i(0) = 0 A.

(Round your answer to four decimal places.)
___________C

Determine the first time at which the charge on the capacitor is equal to zero. (Round your answer to four decimal places.)
_____________s

Use an appropriate infinite series method about

x = 0

to find two solutions of the given differential equation. (Enter the first four nonzero terms for each linearly independent solution, if there are fewer than four nonzero terms then enter all terms. Some beginning terms have been provided for you.)

y'' − xy' − 3y = 0

using MATLAB :

x = [0.3036 0.6168 0.7128 0.7120 0.9377 0.7120 0.3989 0.3028 0.3036 0.5293];

y = [0.1960 0.2977 0.4169 0.1960 0.2620 0.5680 0.6697 0.7889 0.5680 0.5020];

reflect the model about the horizontal line y = 0.6180 and translate the image .0484 units along the x axis.

and then reflect the model about the vertical line x = 0.5078 and translate the image -.3720 units along the y axis

Provide an example of a study when extrapolation would definitely lead to incorrect results. Explain your reasoning.

[Discrete math]

Show that it is possible to arrange the numbers 1, 2, . . . , n

in a row so that the average of any two of these numbers

never appears between them. [Hint: Show that it suffices

to prove this fact when n is a power of 2. Then use mathematical

induction to prove the result when n is a power

of 2.]

I saw the solution but I don't understand why permutation pi is using here.. please explain it with more detail.

1. Solve the initial value problem below using the method of Laplace transforms.

56w′′−4w′+4w=16t+56​, w(−3)=2​, w'(-3)=2

2. Solve for​ Y(s), the Laplace transform of the solution​ y(t) to the initial value problem below.

ty′′−6y′+9y=cos5t−sin5t​, y(0)=4​, y'(0)=4

1. Solve the​ third-order initial value problem below using the method of Laplace transforms.

60y′′′+8y′′+11y′−20y=−60​, y(0)=11​, y′(0)=−10​, y"(0)= 50

2. Use the method of Laplace transforms to find a general solution to the differential equation below by assuming that a and b are arbitrary constants.

5y′′+6y′+25y=5​, y(0)=a​, y'(0)=b

The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system.

x'₁ = 6x₁ + 6x₂ + 2x₃

x'₂ = -8x₁ - 8x₂ - 6x₃

x'₃ = 8x₁ + 8x₂ + 6x₃

What is the general solution in matrix form?
x(t) =

Feature selection is the process of reducing the dimensionality of the feature space to increase performance and decrease running time (since there are fewer features). Outline a feature selection method for the unigram words feature representation using word relations Please explain the methods, not just theory. This question is from the book:Text Data Management and Analysis: A Practical Introduction to Information Retrieval and Text Mining