CRT and solving linear congruences (No credit will be given for simply guessing and proving that your answer is correct): (a) Solve the system of linear congruences: x ≡ [1, 2, 3] (mod [3, 5, 7]). (b) If 3x ≡ 2 (mod 7) and 5x ≡ 3 (mod 13), find a solution for x modulo 91 if it exists. (c) Solve for x if 3x ≡ 15 (mod 18) and 4x ≡ 5 (mod 15), if it exists. (d) Solve for x if 3x ≡ 15 (mod 18) and 4x ≡ 6 (mod 15), if it exists.

Show that every sequence contains a monotone subsequence and explain how this furnished a new proof of the Bolzano-Weierstrass Theorem.

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x +2y +z, 2x +2y +3z +w, x +4y +2w)

a) Find the dimension and basis for Im T (the image of T)

b) Find the dimension and basis for Ker ( the Kernel of T)

c) Does the vector v= (2,3,5) belong to Im T? Justify the answer.

d) Does the vector v= (12,-3,-6,0) belong to Ker? Justify the answer.

Use the simplex method to solve the following linear programming problems. Clearly indicate all the steps, the entering and departing rows and columns and rows, the pivot and the row operations used. An investor has up to N$450,000 to invest in three types of investments. Type A pays 6% annually and has a risk factor of 0. Type B pays 10% annually and has a risk factor of 0.06. Type C pays 12% annually and has a risk factor of 0.08. To have a well-balanced portfolio, the investor imposes the following conditions. The average risk factor should be no greater than 0.05. Moreover, at least one-half of the total portfolio is to be allocated to Type A investments and at least one-fourth of the portfolio is to be allocated to Type B investments. How much should be allocated to each type of investment to obtain a maximum return?

solve using MATLAB.
Title: Numerical differentiations.

Use Richardson extrapolation to estimate the first derivative of y = cos x at x = π/4 using step sizes of h₁ = π/3 and h₂ = π/6. Employ centered differences of O(h²) for the initial estimates.

solve using MATLAB.

What are the eigenvalues and eigenvectors of a diagonal matrix? Justify your answer (explain)

These questions follow the exercises in Chapter 4 of the book, Scientic Computing: An Introduction, by Michael Heath. They study some of the properties of eigenvalues, eigenvectors, and some ways to compute them.

x^2*y''+x*y'+(x^2-1)y=0 what is the solution by using Frobenius method

Let (X, d) be a metric space.

Prove that every metric space (X, d) is homeomorphic to a metric space (Y, dY ) of diameter at most 1.

Consider the following initial value problem: ?? − 2?? = √? − 2? + 3 ?? ?(0) = 6

1. Write the equation in the form ?? ?? = ?(?? + ?? + ? ), where ?, ?, ??? ? are constants and ? is a function.

2. Use the substitution ? = ?? + ?? + ? to transfer the equation into the variables ? and ? only.

3. Solve the equation in (2).

4. Re-substitute ? = ?? + ?? + ? to write your solution in terms of ? and ?.

5. Use the initial condition to write the solution for this initial value problem.

6. Discussion and conclusion.

please share the solution with explanation as it mentioned in part 6.

(PDE)

Find the series soln to Ut=Uxx on -2<x<2, T>0

with Dirichlet boundary { U(t,-2)=0=U(t, 2)

initial condition { U(0,x) = { x, IxI <1

Use the method of steepest ascent to approximate the optimal solution to the following problem: max⁡ z=-(x1-2)^2-x1-(x2)^2 . Begin at the point(2.5,1.5)

(p.s. The answer already exists on the Chegg.Study website is incorrect)

You have agreed to pay off an $8,000 loan in 30 monthly payments of $298.79 per month. The annual interest rate is 9% on the unpaid balance.

(a) How much of the first month’s payment will apply towards reducing the principal of $8,000?

(b) What is the unpaid balance (on the principal) after 12 monthly payments have been made?

x' = -6x - 3y + te^2t

y' = 4x + y

Find the general solution using undetermined coeffiecients

x' = -6x - 3y + te^2t

y' = 4x + y

Find the general solution using undetermined coeffiecients

Suppose that over a certain region of space the electrical potential V is given by the following equation.

V(x, y, z) = 4x² − 4xy + xyz

(a) Find the rate of change of the potential at P(3, 6, 6) in the direction of the vector v = i + j − k.
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(b) In which direction does V change most rapidly at P?

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(c) What is the maximum rate of change at P?

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