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.

The Last Stop Boutique has a special sale of five days. Each day, Starting Monday, the price will decrease 10% of the previous day. For Example, the original price of the product is $20.00, the Monday of the sale will cost $18.00, on Tuesday it will cost $16.20, On Wednesday it will cost $14.58, on Thursday it will cost $13.12 and on Friday the price will be $11.81.

The price of the product that will going to use will be $25.00

A. Develop the algorithm and flowchart that calculates the rate of Withdrawal deduction according to the following values:

1. R = 10% 2. P = 8% 3. G=5%

An application of Newtonian Cooling is calculating the time of death of a person. When healthy, a human body has a steady temperature of 37◦C. Once a person dies, the regulatory mechanisms stop working, and the body temperature rises or falls, depending on the ambient temperature of the environment they died in.

1. The temperature of the body will decrease at a rate proportional to the difference in the current temperature and the ambient temperature. Based on this statement, show how we can model body temperature with the ODE

dT/ dt + kT = kT∞

where T is the temperature of the body, T∞ is the ambient temperate and k is constant.

2. Solve the 1st order ODE for T(t) given the body is initially 37◦C and ambient temperature is constant at 24◦C. Leave k as an unknown.

3. The body takes 2 hours to drop to 32◦C. Use this information to calculate the cooling coefficient k.

4. Plot the temperature of the body for 12 hours after death in MATLAB. Make sure both axes are clearly labelled. Comment on whether the plot is correctly modelling the temperature of the body and provide reasoning as to why (Hint: there are three observations you should be able to comment on!).

5. You find another body of a person related to the first (i.e. same k value), but this was outside where the ambient temperature can be approximated as:

T∞(t) = 6sin( πt /12 ) + 24

Solve the ODE in 1. for T(t) using this function for T∞.

Define the following on R3:
〈(a, b, c), (a′, b′, c′)〉 = 2aa′ + bb′ + 3cc′.

(a) Prove that 〈 , 〉 is an inner product on R3.
(b) Let B = {(1,1,0),(1,0,1),(0,1,1)}. Is B an orthogonal basis for R3 under the inner product defined above. If not, use the Gram-Schmidt algorithm to transform B into an orthogonal basis.

Using the inner product on 〈p, q〉 = ∫(0 to1)  p(x)q(x)dx on P2, write v as the sum of a vector in U and a

vector in U⊥, where v=x^2, U =span{x+1,9x−5}.

List and describe the types of carrying costs and ordering costs?

What is the ROP? How is it determined?

The school population for a certain school is predicted to increase by 80 students per year for the next 14 years. If the current enrollment is 800 students, what will the enrollment be after 14 ​years?

Joe's annual income has been increasing each year by the same dollar amount. The first year his income was ​$24,700​, and the 12th year his income was $37,900. In which year was his income $ 43,900?

How many terms are there in each of the following sequences?

a. 39,40,41,42...,539

b. 1,2,2²,2³,...2⁶⁰

c. 100, 200,300,400,...3000

d. 1,2,4,8,16,32,...2048

PLEASE SHOW WORK

Set and solve a linear system find a polynomial pp of degree 4 such that

p(0)=1, p(1)=1, p(2)=11, p(3)=61, and p(4)=205.

Your answer will be an expression in x.

Modifying your calculation, and without starting from scratch, find a polynomial qq of degree 4 such that q(0)=2, q(1)=3, q(2)=34, q(3)=167, and q(4)=522.

q(x) = ?

find the n-th order Taylor polynomial and the remainder for f(x)=ln(1+x) at 0

There are (m − 1)n + 1 people in a room. Show that either there are m people who mutually do not know each other, or there is a person who knows at least n others.

(a) Let n be odd and ω a primitive nth root of 1 (means that its order is n). Show this implies that −ω is a primitive 2nth root of 1. Prove the converse: Let n be odd and ω a primitive 2nth root of 1. Show −ω is a primitive nth root of 1. (b) Recall that the nth cyclotomic polynomial is defined as Φn(x) = Y gcd(k,n)=1 (x−ωk) where k ranges over 1,...,n−1 and ωk = e2πik/n is a primitive nth root of 1. Compute Φ8(x) and Φ9(x), writing them out with Z coefficients. Show your steps.