<Vector Calculus>

Please show all step by step work and formulas. Thanks

A current in a river is pushing a ship in a direction of 25 degrees north of east with a speed of 12 knots (nautical miles per hour). There is a 7 knots (kn) wind pushing the ship in a direction 80 degrees south of east. Find the velocity vector of the ship vector of the ship relative to the water if the ship is moving due east at a speed of 40 knots (kn) relative to the ground.

A recombinant protein production in a productionscale
process typically has a level of host-cell protein (HCP) by ELISA of 30 ppm. The
most recent batch showed a shift in HCP to 80 ppm.What steps can you take to confirm
this result?

PROBLEM # 1: Plot the values of (P/A, i, 15) for i= 1%, 5%, 7%, 10%, 15%

PROBLEM # 2: Plot the values of (P/F, i, 15) for i= 1%, 5%, 7%, 10%, 15%

PROBLEM # 3: Plot the values of (P/G, i, 15) for i= 1%, 5%, 7%, 10%, 15%

find the general solution (2x+3)dy/dx=(2x+3)/(x^2 +3x+1) -y

Let P be a partial order on a finite set X. Prove that there exists a linear order L on X such that P ⊆ L. (Hint: Use the proof of the Hasse Diagram Theorem.) Do not use induction

1.Describe the effects of the FCS length on error detection performance

2.Describe the effect of frame length on the CRC performance

3.Summarize the burst error detection performance for a given FCS length in terms

of burst size and probabilities.

1. Let W be the collection of polynomials, p(x), for which p'(5)=0 (the derivative of the polynomial when x = 5 is 0).

1. State two polynomials that would belong to this set, W.
2. State one polynomial that does not belong to this set.
3. Is W a subspace of the vector space containing all polynomial functions? That is, is this subset of polynomials closed under addition and scalar multiplication, and does it contain a zero vector? Justify your answer. (You do not have to give a detailed proof, but you should thoroughly explain your thought process using the theorems in section 4.1.)

2.  If the definition of the subset W is changed to the collection of polynomials, p(x), for which p'(5)=1, does that change your verdict as to whether W is a subspace? Explain your answer

Solve y ' ' + 10 y ' + 26 y = 0 , y ( 0 ) = 1 , y ' ( 0 ) = − 3

A country's census lists the population of the country as 246 million in 1990, 268 million in 2000, and 286 million in 2010. Fit a second-degree polynomial passing through these three points. (Let the year 2000 be x = 0 and let p(x) represent the population in millions.) p(x) = million

Use this polynomial to predict the population in 2020 and in 2030.

2020 million

2030 million

Let G be a group, show that: |a^{-1}ba| = |aba^{-1}| = a for all a,b in G.

Number Theory Course

1. Let ? and ? be integers. Show that if ? + ? is odd, then ? and ? are of the opposite parity:
a. By giving a proof by contradiction

2. Let n be a natural number. Prove that (?+ 1)2 −1 is even if and only if n is even. You may use any suitable proof technique.

3. Using a proof by induction, show that 5 | 6? − 1, for all natural numbers ? .

Create a portfolio using the four stocks and information below:

(Do not round intermediate calculations. Record your answers in decimal form and round your answers to 4 decimal places. Ex. X.XXXX)

What is the variance of A?

What is the variance of B?

What is the variance of C?

What is the variance of D?

What is the Correlation (A,A)?

What is the Correlation (B,B)?

What is the Correlation (C,C)?

What is the Correlation (D,D)?

What is the Covariance (A,A)?

What is the Covariance (A,B)?

What is the Covariance (A,C)?

What is the Covariance (A,D)?

What is the Covariance (B,A)?

What is the Covariance (B,B)?

What is the Covariance (B,C)?

What is the Covariance (B,D)?

What is the Covariance (C,A)?

What is the Covariance (C,B)?

What is the Covariance (C,C)?

What is the Covariance (C,D)?

What is the Covariance (D,A)?

What is the Covariance (D,B)?

What is the Covariance (D,C)?

What is the Covariance (D,D)?

What is the expected return on the portfolio above?

What is the variance on the portfolio above?

What is the standard deviation on the portfolio above?

(a) Solve the linear system Az = b using the following three methods:

I. The GE+PP algorithm for sparse (banded) linear systems, which is the default algorithm used by Matlab’s “\” operator when the matrix (call it Asparse) is of sparse type. You may find it easiest to set up the matrix using the spdiags command.

II. The GE+PP algorithm for dense linear systems, again using “\”. Here, you need a dense version of A which you can obtain either with the diag command or more simply by typing Adense = full(Asparse)

III. The Gauss-Seidel iterative algorithm, also using the matrix Asparse. You may make use of the gs2 code from lectures, setting the parameters tol = 1e-8 and maxiter = 1e5, and taking initial guess z0 = (1, 1, . . . , 1)T .

For each method, compute the solution with n = 100 and n = 1000 points. How do the three methods compare in terms of cost? (use elapsed time from Matlab’s tic / toc as a proxy for cost). How well do your three answers agree with each other? (use the vector 2-norm to compare the differences). Which result do you think is most accurate?

A survey of 77 customers was taken at a bookstore regarding the types of books purchased. The survey found that 46 customers purchased MYSTERIES, 38 purchased SCIENCE FICTION, 32 purchased ROMANCE NOVELS, 20 purchased MYSTERIES AND SCIENCE FICTION, 17 purchased MYSTERIES AND ROMANCE NOVELS, 13 purchased SCIENCE FICTION AND ROMANCE NOVELS, and 6 purchased all three types of books.

a). how many of the customers surveyed purchased only mysteries?

b). how many purchased mysteries and science fiction, but not romance novels?

c). how many people purchased mysteries or science fiction?

d). how many people purchased mysteries or science fiction, but not romance novels?

e). how many purchased exactly two types of books?