3. Find the Fourier integral representation of each of the following functions.

f(x) = sin(x)/x

1. Performance metrics

   Buffalo BBQ Restaurant is trying to become more efficient in training its chefs. It is experimenting with two training programs aimed at this objective. Both programs have basic and advanced training modules. The restaurant has provided the following data regarding the two programs after two weeks of implementation:

   	Training Program A						Training Program B
   New chef #	1	2	3	4	5		6	7	8	9	10
   Hours of basic training	23	26	27	20	21		25	23	27	31	22
   Hours of advanced training	8	6	7	9	12		6	4	0	2	4
   Number of chef mistakes	11	13	17	15	15		8	6	7	7	7

   a. Compute the following performance metrics for each program:

   (1) Average hours of employee training per chef, rounded to one decimal place.

   Program A: hrs. per chef

   Program B: hrs. per chef

   (2) Average number of mistakes per chef, rounded to one decimal place.

   Program A: mistakes per chef

   Program B: mistakes per chef

Let T : P3(R) → P4(R) be defined by T(f(x)) = 5f′(x)-∫ f(t)dt (integral from 0 to x)

1. Show that T is a linear transformation.

2.Find dim (P3(R)) and dim (P4(R)).

3.Find rank(T). Find nullity(T)
4. Is T one-to-one? Is T onto? Justify your answers.

Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker P is W^perp.

1. Find all solutions to the following linear congruences using Fermat’s Little Theorem or Euler’s Theorem to help you. Show all your steps.

(a) 3462x ≡ 6 173 (mod 59)

(b) 27145x ≡ 1 (mod 42)

D^2 (D + 1)y(t)= (D^2 +2)f(t)

a.) Find the characteristic polynomial, characteristic equation, characteristic roots, and characteristic modes of the system.

b.) Find y_o(t), the zero-input component of response y(t) for t>=0, if the the initial conditions are   y_0 (0) = 4, y_0' (0) = 3, and y_0'' (0) = -1

Let f : [0, 1] → R and suppose that, for all finite subsets of [0, 1], 0 ≤ x1 < x2 < · · · < xn ≤ 1,

we have |f(x1) + f(x2) + · · · + f(xn)| ≤ 1. Let S := {x ∈ [0, 1] : f(x) ̸= 0}. Show that S is countable

Number Systems Choose any system of numeration (Egyptian, Roman, Mayan, Chinese, Hindu-Arabic, Greek, Babylonian, etc.) and answer the following questions: Is it an additive, multiplicative, or ciphered system? Why? Is there a radix or base? What is it? Is there a schema or rule for combining the numerals to represent numbers? Briefly describe the rule(s).

explain interest rate risk or maturity price risk faced by short term and long term investors in bonds using an example

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove

(a). (a^m)(a^n)=a^(m+n)

(b). (a^m)^n=a^(mn)

y"-3y'+2y=4t-8 , y(0)=2 , y'(0)=7
y(t)=?