a) Prove that if X is Hausdorff, then X is T₁

b) Give an example of a space that is T₁ , but not Hausdorff. Prove that the space you give is T₁ and prove it is not Hausdorff.

For each of the following relations, determine if f is
• a function,
• surjective, or
• injective.
Conclude by stating if the relation represents a bijective function.
For each point, state your reasoning in proper sentences.
a) f = {(a, b) ∈ N
2 × N | a ∈ N
2
, a = (a1, a2), b, a1, a2 ∈ N, b = a1a2}
b) f = {(x, y) ∈ S
2
| y = x
2}, where S = {x ∈ R | x ≥ 0}

Solve the following differential equations.

a.) (2xy^2 +2x)dx−(4x^2 +1)dy=0

b.) (3ye^(3xy) +4xy)dx+(3xe^(3xy) +2x^2)dy=0

Find two linearly independent power series solutions for the following differential equation. Write the first four terms for each.

y′′ − xy = 0

1. An 8 pound weight stretches a spring 2 feet. The surrounding medium offers a damping force that is numerically equal to 2 times the instantaneous velocity. It is then released from rest from a point 3 feet below the equilibrium point.
a. Determine the equation of motion.

b. Is the system underdamped, overdamped, or critically damped?

Q11: Use the Lagrange interpolating polynomial of degree three or less and four-digit chopping arithmetic to approximate cos 0.750 using the following values. Find an error bound for the approximation.
cos 0.698 = 0.7661 ,cos 0.733 = 0.7432 cos 0.768 = 0.7193 cos 0.803 = 0.6946.

Use the Laplace transform to solve the given initial value problem.

y⁽⁴⁾ − 4y''' + 6y'' − 4y' + y = 0;

y(0) = 1,

y'(0) = 0,

y''(0) = 0,

y'''(0) = 1

For this discussion, you will reflect on the many applications and uses of statistics. Develop a main response in which you address the following:

• Identify two (or more) possible applications of statistics.
• Explain how statistics are beneficial and useful in these situations.
• Discuss what additional information statistics can provide.

d^2y/dx^2 − dy/dx − 3/4 y = 0,

y(0) = 1, dy/dx(0) = 0,

Convert the initial value problem into a set of two coupled first-order initial value problems

and find the exact solution to the differential equatiion

Use mathematical induction to prove that for each integer n ≥ 4, 5ⁿ ≥ 2²ⁿ⁺¹ + 100.

solve the following using Siri Solutions. Verify your solution using usual methods (if possible).

1. y" - 2y' + y = 0

2. y" - 2xy' + y = 0

In this project we explore how two populations develop when one preys on the other. Clearly if there are no predators, the prey population will keep growing, whereas if there are no prey, the predators will go extinct. Suppose x and y denote the populations of the prey and predators respectively.

If y = 0, we will assume that

dx/dt = ax, a > 0.

If y does not equal 0, it is natural to assume that the number of encounters between predators and prey is jointly proportional to x and y. If we further assume a proportion of these encounters leads to the prey being eaten, we have

dx/dt = dx − bxy, a, b > 0.

Similarly, we have

dy/dt = −cy + dxy, c, d > 0.

This system of equations is called Volterra’s predator-prey equations.

Part a) Solve this system of equations to find solutions in the form g(y) = f(x). You will see that we cannot explicitly find y in terms of x, so our solutions are implicit. We can still, however, study these solutions.

Part b) Suppose g(y) = C1, where C1 is a constant. Determine how many solutions there are to this equation by using calculus techniques. Note this may well depend on the value of C1. Do the same thing for f(x) = C2, where C2 is a constant.

Part c) Hence determine the shape of the trajectories in the x, y-plane (do a sketch!), and their directions.

Part d) Clearly the system has a rest point at x = c/d and y = a/b. By making the substitutions x = c/d + X and y = a/b + Y , assuming X and Y are small enough that we can neglect any second order terms in X and Y , show that near the rest point, trajectories are approximately ellipses.

Part e) Finally, sketch graphs of x(t) and y(t) against t on the same axes. To help, show that d^2 * y / dt^2 > 0 whenever dx/dt > 0 are think about what this means in terms of the shapes of the graphs.

Hello,

In your own words, please if you were to teach geometry such as triangles and quadrilaterals, algebra to calculate perimeter and area. and understanding of similarity by problem-solving.

what kind of difficulties you might face and the implications in classroom practice

how would you teach it,

why do you think it would be hard for students to learn.

I want this to be about 500 words essay.

Please answer this in essay-based format

Topic: Math - Linear Algebra

Focus: Matrices, Linear Independence and Linear Dependence

Consider four vectors v1 = [1,1,1,1], v2 = [-1,0,1,2], v3 = [a,1,0,b], and v4 = [3,2,a+b,0], where a and b are parameters. Find all conditions on the values of a and b (if any) for which:

1. The number of linearly independent vectors in this collection is 1.

2. The number of linearly independent vectors in this collection is 2.

3. The number of linearly independent vectors in this collection is 3.

4. The number of linearly independent vectors in this collection is 4.

Explain the difference between supervised and unsupervised learning. Provide examples.