Prove

1. For each u ∈ R n there is a v ∈ R n such that u + v= 0

2. For all u, v ∈ R n and a ∈ R, a(u + v) = au + av

3. For all u ∈ R n and a, b ∈ R, (a + b)u = au + bu

4.  For all u ∈ R n , 1u=u

[Note that, in this example, the mesh sizes in x and y are identical (h); strictly speaking, this need not be true. In some applications, we may need more resolution along the x- or y-axis; we could then use separate mesh sizes h_x and h_y.]

By definition, the partial derivative of a function f ( x , y ) with respect to x is

∂ f ∂ x = L i m h ⟶ 0 f ( x i + h , y j ) − f ( x i , y j ) h

and the partial derivative with respect to y is

∂ f /∂ y = L i m h ⟶ 0 f ( x i , y j + h ) − f ( x i , y j )/ h

If we applied these formulas to our grid values, we would get the finite difference expressions

∂ f /d x ( x i , y j ) ≅ f ( x i + 1 , y j ) − f ( x i , y j )/ h

Note: To avoid round-off error, retain at least six decimal places in all of your calculations.

• Assume the function f is defined as f(x, y) = 3 tan x cos y
• Use differentiation rules to find the exact partial derivatives ∂ f /∂ x and ∂ f /∂ y , and evaluate those exact partial derivatives at (1.56, -2.1).  
• Use the finite difference formulas to estimate ∂ f /∂ x and ∂ f/ ∂ y at (1.56, -2.1) for three different values of the mesh size
  • h = 0.01
  • h = 0.001
  • h = 0.0001

• Use your calculated values to fill in this table:

Answer the following questions:

• For which partial derivative is the finite difference approximation more accurate?
• Why is the finite difference approximation for the other partial derivative less accurate?  Under what real-world conditions might we see such poor approximations?

1. Prove that there exist infinitely many positive real numbers r such that the equation 2^x + 3^y + 5^z = r has no solution (x,y,z) ∈ Q × Q × Q.

(Hint: Is the set S = {2^x + 3^y + 5^z : (x,y,z) ∈ Q × Q × Q} countable?)

Show that the number of solution to x² - y² ≡ a (p) is p-1 if p not divides a and 2p -1 if p divides a, Use u = x+y and v= x-y

Then show that summation y=0 to p-1 ((y^2 +a)/p) = -1 if p not divides a and

= p-1 if p divides a

Step (D) of the divide-and-conquer strategy (i.e. combine the solutions to smaller instances of the problem to obtain the solution of the original instance) is not a necessary step for this design strategy. Mergesort is an example of such cases.

Select one:

True

False

Example 10.5: Verify the divergence theorem for the vector field F = 2xzi + yzj +z2k and V is the volume enclosed by the upper hemisphere x2 + y2 + z2 = a2, z ≥ 0

Compute the quartic interpolating polynomial for the Hermite interpolation problem

p(0) = 2, p'(0) = -9

p(1) = -4, p'(1) = 4

p(2) = 44

with respect to the Newton basis. Compute the divided differences. Find a quintic interpolating polynomial that additionally satisfies p(3) = 2.

We have already derived the integral formulae for the mass m, the moment My about the y-axis, and the moment Mx about the x-axis, of the region R where a lamina with density ρ(x) resides in the xy-plane. The method we used was to:

-Slice R into n rectangles, where y = f(x) bounded R above and y = g(x) bounded R below, on [a, b].

-Compute the area, mass, and moments (about both the y-axis and the x-axis), of the i th rectangle Ri .

-Take the Riemann sum limit to derive the integral formulae for m, My, and Mx.

There are analogous integral formulae for m, My, and Mx, of R in terms of y (in class we did it in terms of x). Indeed now assume the region R is bounded to the right by x = f(y) and to the left by x = g(y) on [c, d] with density ρ(y).

Adapt the method we did in class to derive the formulae for m, My, and Mx, as y-integrals.

You must label or define relevant variables and quantities, and at the end take the Riemann sum limit.

Note: Only by replacing x with y in the x-integral formulae does not yield the correct y-integral formulae.

please please focus on "note" and it is also for "y integral"

I posted the question earlier but the answer was not the professor is looking for

An individual possesses 5 umbrellas which he employs in going from his home to the office, and vice versa. If he is at home at the beginning of a day and it is raining, then he will take an umbrella with him to the office provided there is one to be taken. Similarly, if he is at the office and at the end of a day it is raining, he will take one to go home (provided there is one to be taken at the office). If it is not raining, then he never takes an umbrella. Assume that, independent of the past, it rains at the beginning or at the end of a day with probability 0.35.

(a)Define a Markov chain for this system by the construction of the one-step transition matrix (Hint: Define the states of the chain as the number of umbrellas the individual has in the place he is at (home or office). Assume that there is a transition each time he changes places (from home to the office or vice versa)

(b)Find the steady state probabilities, by the formulation of the steady state equations.

(c) What fraction of time does the man get wet? Justify your answer.

Suppose ?⃗ (?,?)=−??⃗ +??⃗ and ? is the line segment from point ?=(2,0) to ?=(0,3).

(a) Find a vector parametric equation ?⃗ (?) for the line segment ? so that points ? and ? correspond to ?=0 and ?=1, respectively. ?⃗ (?)=

(b) Using the parametrization in part (a), the line integral of ?⃗ along ? is ∫??⃗ ⋅??⃗ =∫???⃗ (?⃗ (?))⋅?⃗ ′(?)??=∫?? ?? with limits of integration ?= and ?=

(c) Evaluate the line integral in part (b).

(d) What is the line integral of ?⃗ around the clockwise-oriented triangle with corners at the origin, ?, and ?? Hint: Sketch the vector field and the triangle.

Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions (if any exist).

i.     x₁+ x₂ +x₄ = 3

             x₂ + x₃ = 4

       x₁ + 2x₂ + x₃ + x₄ = 8

ii.    x₁ + 2x₂ + x₃ = 4

       x₁ + 2x₂ = 6

iii.   x₁ + x₂ =1

     2x₁ + x₂=3

     3x₁ + 2x= 4

let R = Z x Z. P be the prime ideal {0} x Z and S = R - P. Prove that S^-1R is isomorphic to Q.