Apply the Laplace Transform to solve the initial value problems

1. y' + 2y = 2cos(3t) , y(0) = 1

2. y'' - 3y' + 2y = 2 - 10e^-3t , y(0) = -1 , y'(0)= 1

• What is Clustering?
• What is the purpose of clustering?
• What assumptions are needed for clustering?
• Why do you need to transform qualitative variables to be presented by numbers in clustering?
• What is the purpose of normalizing quantitative variables?

Question H:

Let’s keep using the AM-radio style signal from Question G: f(x) = sin(12x) * sin(x) on [0,2*pi].

It’s okay to use Wolfram Alpha or Desmos to do the following integrals.

i)

ii) What do you get if you graph 0.5*cos(11x)+-0.5*cos(13x) ?

. During droughts, water for irrigation is pumped from the ground. When ground water is pumped excessively, the water table lowers. In California, lowering water tables have been linked to reduced water quality and sinkholes. A particular well in California’s Inland Empire has been monitored over many years. In 1994, the water level was 250 feet below the land surface. In 2000, the water level was 261 feet below the surface. In 2006, the water level was 268 feet below the surface. In 2009, the water level was 271 feet below the surface. In 2012, it was 274 feet below the surface. And in 2015, it was 276 feet below the surface. (a) Find a cubic (degree 3) polynomial model for this data on water level. First, define what x and y mean here, and write the data points you use. Then, find a cubic which is a best fit for this data, in the least squares sense. (b) Use your model to predict the water level of the well, in feet below the surface, in 2020. You may assume that the trends of 1994 to 2015 continue to 2020.

Let p be an element in N, and define d _p to be the set of all pairs (l,m) in N×N such that p divides m−l. Show that d_p is an equivalence relation

Good afternoon interns,

As you all know, you are all vying for the 15 openings to be offered at the end of the internship. Another chance has come up for you to show us that you deserve one of those positions here at Firmament Financials Inc. New clients, Mr. and Mrs. Perez are planning on starting a family and would like to start saving for their child’s college education. The want to know what they can afford when their child is ready for college. They would like to utilize our Yearly College Savings Plan which grows a yearly deposit at an APR of 15% with continuous compounding. Your job is to analyze their information and come up with feasible recommendations. Their information can be found in the email they sent. Of course, you will be graded on your analysis. A grading rubric has been provided.

Good luck interns!

Dear Ms. Robles,

As satisfied customers of Firmament Financial, we of course look to you to help us save for our upcoming child’s college education. We feel that we can afford $100 per month for the 18 years until college in your College Savings Plan. We have a list of universities that we have researched with the approximate cost per year for each. Which if any of the university options will we be able to afford for our child? We would prefer to send them to an elite private university if we can afford it.

Anthony & Veronica Perez

Let b be a vector that has the same height as a square matrix A. How many solutions can Ax = b have?

Suppose your instructor asked you to look at the influence of age (X) on automobile insurance premium rates (Y). List three variables you might control for in your regression analysis and why you would control for them (that is, what effect would these control variables have on insurance rates?).

7. Find the solution of the following PDEs:
ut−16uxx =0
u(0,t) = u(2π,t) = 0

u(x, 0) = π/2 − |x − π/2|

Write a funtion and apply multi-var. Newton-Raphson method.

QUESTION 1

Vector Space Axioms

Let V be a set on which two operations, called vector addition and vector scalar multiplication, have been defined. If u and v are in V , the sum of u and v is denoted by u + v , and if k is a scalar, the scalar multiple of u is denoted by ku . If the following axioms satisfied for all u , v and w in V and for all scalars k and l , then V is called a vector space and its elements are called vectors.

1) u + v is in V

2) u + v = v + u

3) (u + v) + w = u + (v + w)

4) 0 + v = v

5) v + (−v) = 0

6) ku is in V

7) k(u + v) = ku + kv

8) (k + l)u = ku + lu

9) k(lu) = (kl)(u)

10) 1v = v

Task: Show that the set V of all 3×3 matrices with distinct entries and also combination of positive and negative numbers is a vector space if vector addition is defined to be matrix addition and vector scalar multiplication is defined to be matrix scalar multiplication.

QUESTION 2

Suppose u, v, and w are all vectors in a vector space V and c is any scalar. An inner product on the vector space V is a function that associates with each pair of vectors in V, say u and v, a real number denoted by u, v that satisfies the following axioms.

(a) < u, v > = < v, u > (Symmetry axiom)

(b) < u + v, w > = < u, w + v, w > (Additive axiom)

(c) < cu, v > = < c u, v > (Homogeneity axiom)

(d) < u, u > ≥ 0 and < u, u > = 0 if and only if u = 0 (Positivity axiom)

A vector space along with an inner product is called an inner product space.

Task: Show that the set V of all 3×3 matrices with distinct entries and also combination of positive and negative numbers is a inner product space if vector addition is defined to be standard matrix addition and vector scalar multiplication is defined to be matrix scalar multiplication.