Analyticity of trigonometric functions (a) Directly from the definition, construct the Taylor Series centered at x = 0 for the function f(x) = cos(x). (b) Show that this series converges for all x ∈ R. (c) Show that this series converges to cos(x) for all x ∈ R.  

DIRECTIONS: Show all the work in the space provided. Box the final answers, and follow the indicated directions.

1. Solve the DE:

y"-3y'+2y=e^xsinx

The graph of a function

y = g(x)

on the domain

−8 ≤ x ≤ 8

consists of line segments and semicircles of radius 2 connecting the points

(−8, 0), (−4, 4), (0, 4), (4, 4), (8, 0).

(a) What is the range of g?

0 < y < 60 ≤ y ≤ 4    0 ≤ y ≤ 20 < y < 40 ≤ y ≤ 6

(b) Where is the function increasing? (Select all that apply.)

−8 ≤ x ≤ −2−2 ≤ x ≤ 22 ≤ x ≤ 44 ≤ x ≤ 8−8 ≤ x ≤ 8

Where is the function decreasing? (Select all that apply.)

−8 ≤ x ≤ −2−2 ≤ x ≤ 22 ≤ x ≤ 44 ≤ x ≤ 8−8 ≤ x ≤ 8

(c) Find the multipart formula for y = g(x) if

(d) If we restrict the function to the smaller domain

−6 ≤ x ≤ 0,

what is the range?

0 ≤ y ≤ 62 ≤ y ≤ 6    0 ≤ y ≤ 42 ≤ y ≤ 40 ≤ y ≤ 2

(e) If we restrict the function to the smaller domain

0 ≤ x ≤ 4,

what is the range?

2 ≤ y ≤ 40 ≤ y ≤ 2    0 ≤ y ≤ 64 ≤ y ≤ 60 ≤ y ≤ 4

Find an optimal parenthesization of matrices whose sequence of dimensions is: <5, 10, 12, 5, 50>. Please write out both the m[·, ·] and s[·, ·] tables.

8) Solve x^2y’’ – 2y = ln(x)

2)   Solve the system of equations below
          dx/dt – 3x – 6y = t^2
          dx/dt + dy/dt – 3y = e^t

Use the method of variation of parameters to find the complete solution of the differential equation d²y/ dx² + 4 dy /dx + 4y = e ^−2x ln(x), x > 0.

Q: Asking for assistance in understanding and solving this example on Modern Algebra II with the steps of the solution to better understand, thanks.

**Please give the step by steps with details to completely see how the solution came about.

1) Determine all elements in an integral domain that are their own inverses under multiplication.

2) Let F be a finite field with n elements. Prove that x^(n-1 )= 1 for all nonzero x in F. Hint: Use the fact that the nonzero elements of F form a group under the multiplication operation.

Please show picture over a rectangle.

a) Let f(x,y) = 2sin(πx)−3cos(πy). Calculate Uf(P) and Lf(P) over the partition P = {1,1.5,2}×{2,2.5,3}.

b) Explain what techniques would be necessary to calculate Uf(P) and Lf(P) for f(x,y) = 2sin(x)−3cos(y) over the partition P = {1,1.5,3}×{2,2.5,3}.

what is the tenth letter of the standard english alphabet?

Let f : Z × Z → Z be defined by f(n, m) = n − m

a. Is this function one to one? Prove your result.

b. Is this function onto Z? Prove your result

Give a clear simple argument that each of the following sets is uncountable.

a. R × Q

b. N ∪ P(N)

c. P(R)

d. (0, π) ∪ {4, 5, 6, 7}

Solve for x,y,z using the inverse if possible.

x+2y+5z=2

2x+3y+8z=3

-x+y+2z=3

describe a project list two task that may be performed in parallel and two tasks that need to be performed sequeooontially