Suppose a function f : R → R is continuous with f(0) = 1. Show that if there is a positive number x₀ for which f(x₀) = 0, then there is a smallest positive number p for which f(p) = 0. (Hint: Consider the set {x | x > 0, f(x) = 0}.)

C. Prove the following claim, using proof by induction. Show your work.

Let d be the day you were born plus 7 (e.g., if you were born on March 24, d = 24 + 7). If a = 2d + 1 and b = d + 1, then aⁿ – b is divisible by d for all natural numbers n.

Prove for the system of ordinary differential equations x'=-x, and y'=-5y the origin is lyapunov stable, attracting and asymptotically stable using the EPSILON DELTA definition of each. The epsilon and delta that make the definitions hold must be found.

Determine if a real-world system could be represented as a particle

Please give a exemple

7) (a) (2 pts) How many permutations are there of the word QUARANTINE?

(b) (4 pts) How many of the permutations of QUARANTINE do NOT the same letter appearing consecutively? (Thus, any permutation with the substring "AA" should not be counted and any permutation with the substring "NN" should not be counted.)

(c) (6 pts) A mountain permutation is defined as one where the first portion of the permutation has letters in strictly increasing alphabetical order, and the second portion with the letters in strictly decreasing alphabetical order. One mountain permutations of the letters in QUARANTINE is AINTURQNEA. Notice that only a set of letters with a unique "maximum" alphabetic letter have mountain permutations. For this permutation, A < I < N < T < U, and U > R > Q > N > E > A. How many mountain permutations are there of the letters in the word QUARANTINE?

Match each sequence to a good candidate for a closed form. Note that for each of the given sequences, the initial value of the index n is given.

1. A. 1, 3, 7, 15, 31, 63, ... (n>=0)
2. B. 3, 8, 13, 18, 23, 28,... (n>=1)
3. C. 3, 8, 15, 24, 35, ... (n>=1)
4. D. 1, 1, 2, 3, 5, 8, 13, 21, ... (n>=1)
5. E. Corresponding sequence not listed.

p(x) = a₀ + a₁x + a₂x² + · · · + a_kx^k is a polynomial of degree k. What is the Taylor series of p(x), and its radius and interval of convergence.

1. Suppose that A is a 6 x 6 matrix that can be written as a product of matrices A = BC where B is 6 x 4 and C is 4 x 6. Prove that A is not invertible.

2. An economist builds a Leontief input-output model for the interaction between the mining and energy sectors of a local economy using the following assumptions:

• In order to produce 1 million dollars of output, the mining sector requires 0.1 million dollars of input from the mining sector and 0.5 million dollars of input from the energy sector.

• In order to produce 1 million dollars of output, the energy sector requires 0.6 million dollars of input from the mining sector and 0.2 million dollars of input from the energy sector.

(a) Construct the consumption matrix C for this model.

(b) Compute the matrix (I – C) ¹.

(c) Find the equilibrium production level when the final demand is d = (10, 40).

   (d) Also compute the equilibrium production levels for final demands (1, 0) and (11, 40).

(f) In light of your answers to parts (c), (d), and (e) above, interpret the entries in

the matrix (I – C) ¹.

(g) Suppose that due to the growth of green energy companies, the energy sector requires only 0.3 million dollars of input from the mining sector. Compute the new consumption matrix C* and then new (I – C*) ¹. Interpret the entries of the inverse matrix and compare to your answer to part (f) to explain how the change in the energy sector will affect this economy. .

3. Let L be a line in R² defined by y = mx + b. That is, L has y-intercept (0, b) and slope m. In this problem, you will consider different cases for the line L and and how to reflect points in that line. You do not need to multiply out the products to a single matrix; you can simply leave your answer as a few matrices multiplied together

1. Suppose that L is the x-axis.

1. What is m? What is b?

2. Find a 3x3 matrix that when multiplied with a point (x, y) in homogeneous coordinates will give its image under a reflection in the line L.

2. Suppose that L does not intersect the x-axis.

1. What is m?

2. Find a 3 x 3 matrix that will translate L to the x-axis. Since we don’t know what b is (other than b 6 ≠ 0), the matrix will have to include the unknown b.

3. Find another 3 x 3 matrix that will translate the x-axis to L. Again, this matrix will have to include b.

4. Find a product of 3 x 3 matrices that when multiplied with a point (x, y) in homogeneous coordinates will give its image under a reflection in the line L.

3. Now suppose that L does intersect the x-axis, does so at the origin, and does so at an angle of θ (measured from the positive direction).

1. What is b? By trigonometry, m = tan(θ).
2. Find a 3 x 3 matrix that will rotate the line L to the x-axis.

3. Find another 3 x 3 matrix that will rotate the x-axis to the line L.

4. Find a product of 3 x 3 matrices that when multiplied with a point (x, y) in homogeneous coordinates will give its image under a reflection in the line L.

4. Finally, suppose that L does intersect the x-axis, but not at the origin, and does so at an angle of θ (measured from the positive direction). Find a product of 3x3 matrices that when multiplied with a point (x, y) in homogeneous coordinates will give its image under a reflection in the line L.

4. Let A be an n x n invertible matrix. Prove that

det(A^-1 ) = 1/ det(A)

Prove

1. Let f : A→ B and g : B → C . If g 。 f is one-to-one, then f is one-to-one.

2. Equivalence of sets is an equivalence relation (you may use other theorems without stating them for this one).

x"(t)- 4x'(t)+4x(t)=4e^2t ; x(0)= -1, x'(0)= -4

Graph theory

In a certain school, there are 39 clubs and 194 students. Some students do not belong to any clubs but every club has at least one member and no two clubs have the same number of members. Prove that there is a way to choose one student from each club ending up with 39 different students.

find the general solution of the given differential equation.

1. y''+2y'−3y=0

2.  6y''−y'−y=0

3.  y''+5y' =0

4.  y''−9y'+9y=0

Given that a particular solution of 2y′′ +3y′ +y = x^2 +7x+8 is yp1=x^2+x+1 and that a particular solution of 2y′′ + 3y′ + y = 2sinx+4cosx is yp2=sinx-cosx, find a particular solution for 2y" +3y' +y =3x^2 + 21x + 24 -sinx -2cosx

can a vector with dimensions R^N and a vector with dimensions R^N+1 be a matrix? and if so what would it dimension size be?