Purchase of a new home

John and Jane had planned to save $60 000 dollars over the next five years as a down payment on a house. Jane assured John that if they contributed $1000 each month to a savings account that pays an annual rate of interest of 2.5% compounded monthly that they would have enough money to put a down payment of $60 000 on their new house. Wanting their daughter to have a house, Jane’s parents (The Henrys) have offered to lend John and Jane $65 000, which they have suggested (perhaps naively) John and Jane pay back by contributing to a savings account in the Henrys’ name as per Jane’s original savings plan. John’s worried this is not fair to his in-laws. Is he correct? If so, devise a fair repayment plan that would see the Henrys repaid at a rate of 2.5% compounded monthly over the 5 years.

The Does have qualified for a mortgage of $500,000 to be amortized over 25 years. Their mortgage broker has offered them the following options:

1. A 5 year fixed rate with monthly payments at an annual interest rate of prime+1%
2. A 10 year fixed rate with biweekly payments at an annual interest rate of prime+2%

Prime is currently at 1.5% and projected to increase by 0.25% every year for the next 10 years. Which Mortgage terms should they accept given that their goal is to pay as much principle as possible over the next 10 years?

Teacher's notes:

To begin you know the payment size, the number of payments and the interest rated and what you need to determine is the FV of those payments using the formula: FV=PMT[((1+i)^n - 1)/i].  

This will tell you how much money the Does will save (or pay back to the Henrys) under Janes original plan. You can use this same formula to ask how big the PMTs would have to be to ensure the Does pay back the Henrys exactly 65 000 dollars. Of course this means the Henrys earn no interest.

If you want to ensure the Henrys earn interest as per Jane’s original payment plan, then you’ll need to calculate the PMTs needed based on a present value of 65000 using the formula: PV=PMT((1-(1+i)^-n)/i). This will tell you how big the PMTs would have to be to ensure the Does pay back the Henrys 65 000 dollars with interest as per Jane’s original plan.

It is like a class discussion and we're supposed to write a discussion about Vector spaces, subspaces and bases

Overview (what to write on the discussion): So we're supposed to discuss and explain about these following points:

1. Vector Spaces, Definitions, Examples, Non Examples
2. Spanning Sets, Linear Independence
3. Basis, Dimension
4. Rank
5. Change of Basis, Coordinates

PUT EXAMPLES AND DEFINITIONS (EX: Showing examples to vector addition (closure under addition, etc) and scalar multiplication (distributive property, etc)

Instructions:

Add a new discussion topic. In your post, please include the following:

1. Details on proofs.
2. Illustrate examples of special interest.
3. Apply your mathematical knowledge of given structures in proving or disproving assertions regarding Vector Spaces.

The Tax Cuts & Jobs Act enhanced the deduction for charitable contributions by raising the limit that can be contributed in any one year. The limit is now 60% of adjusted gross income (AGI), up from 50%. Assume your client still has a charitable deduction limitation due to their AGI. The client might lose the charitable deduction because of achieving this limitation.

• Recommend at least two tax planning strategies to avoid losing the deductions. Provide support for your response

Solve by variation of parameters.

y''+4y =sin(2x)

y'''-16y' = 2

Let S be a set of n numbers. Let X be the set of all subsets of S of size k, and let Y be the set of all ordered k-tuples

(s₁, s₂,   , s_k)

such that

s₁ < s₂ <    < s_k.

That is,

(a) Define a one-to-one correspondence

f : X → Y.

Explain why f is one-to-one and onto.

(b) Determine |X| and |Y|.

An ice cream company collected data on their ice cream cones sales over a month in July in a Chicago suburb, along with daily temperature and the weather. The company is interested to develop a correlation between ice cream sales to the hot weather. Market research showed that more people come out in certain neighborhoods, to either enjoy the nice weather, or venture out if they do not have air conditioning in their apartments. The Chicago Police also tracked crime statistics during the same period. Crime statistics included murder, assault, robbery, battery, burglary, theft and motor vehicle theft. The data are shown below:

1.)Develop a linear regression model for ice cream sales over daily temperature. Show the linear equation in the form of y = ax + b, and the coefficient of determination.

What would be the projected forecast of ice cream sales in units, for daily temperature of 94 F?

2.) On July 15 & 16 there were heavy down pour of rain, which might have prevented some to venture out to purchase ice cream during the day. If you were to override those 2 data points, what would be the linear regression model be (by deleting July 15 & 16 data).

which would be considered a better forecast for ice cream sales

3.) Develop a linear regression on ice cream sales to crime statistics. Show the linear equation in the form of y = ax + b, and the r-square value.

Does this correlation demonstrate causation, that high ice cream sales cause crime statistics to go up?

Solve using the Laplace transform: y" + 4y = g(t) where y(0) = y'(0).

Hint: Use the convolution theorem to write your answer. You may leave your answer expressed in terms of an integral.

A model for the population

P(t)

in a suburb of a large city is given by the initial-value problem

= P(10⁻¹ − 10⁻⁷P),    P(0) = 3000,

where t is measured in months. What is the limiting value of the population?


At what time will the population be equal to one-half of this limiting value? (Round your answer to one decimal place.)
months

In order to apply Green’s theorem, the line integral of the boundary should be evaluated such that the integration region inside the boundary lies always on the left as one advances in the direction of integration. What happens if the region lies on the right? How can you apply the theorem then? Explain.

The centers of two circles are 4 cm apart, with one circle having a radius of 3 cm and the other a radius of 2 cm. Find the area of their intersection.

PART 2: Instructions: Write three paragraphs to answer this question:

Suppose a sociologist at John Jay College is given a grant to study “whether there is a binge drinking problem on the campus (and in the dorms and nearby neighborhood) of Small Town College (STC). Describe three different data collection methods that the John Jay sociologist could use to gather evidence to determine whether or not STC has a serious “binge drinking problem” on its campus.For each data collection method, suggest the questions or issues that the sociologist would focus upon.

Prove that (ZxZ, *) where (a,b)*(a',b') = (a+a',b+b') is a group

Information theory

Consider a random variable representing coin throws (Bernoulli Variable with Σ = {0,1} ). Let the true
probability distribution be p(0) = r, p(1) = 1-r.
Someone guesses a different distribution q(0) = s, q(1) = 1-s.
(a) Find expressions for the Kullback–Leibler distances D(p||q) and D(q||p) between the
two distributions in terms of r and s.
(b) Show that in general, D(p||q) ≠ D(q||p) and that equality occurs iff r = s.
(c) Compute D(p||q) and D(q||p) for the case r = 1/2 and s = 1/4.