Find all local extreme values of the given function and identify each as a local maximum, local minimum, or saddle point of the next functions.

1) f(x, y) = e-(x2 + y2 - 16x)

2) f(x, y) = x² + 100 - 20x cos y; -π < y < π

*(4) (a) Prove that if p=(x,y) is in the set where y<x and if r=distance from p to the line y=x then the ball about p of radius r does not intersect with the line y=x.

(b) Prove that the set where y<c is an open set.

Justify your answer

A fluid has velocity components of u=(8t2)m/s and v=(8y+3x)m/s, where x and y are in meters and t is in seconds.

Part A

Determine the magnitude of the velocity of a particle passing through point (1 m, 1 m) when t = 2 s.

V=

Part B

Determine the direction of the velocity of a particle passing through point (1 m, 1 m) when t = 2 s.

θv=

Part C

Determine the magnitude of the acceleration of a particle passing through point (1 m, 1 m) when t = 2 s.

a=

Part D

Determine the direction of the acceleration of a particle passing through point (1 m, 1 m) when t = 2 s.

θa=

Dr. Poh-Shen Loh advanced an alternative approach to solving quadratic equations.

1. Do a YouTube search of the method and state what does his approach consist of. (Provide Links to the YouTube Video used).
2. Do you think it is superior to other existing approaches? Why or why not?
3. What other concepts in Mathematics do you need to understand the Poh-Shen Loh’s Approach to solving quadratic equation

a)The demand function for a product is modeled by

p = 12,000

1 −

.

Find the price p (in dollars) of the product when the quantity demanded is x = 1000 units and x = 1500 units. What is the limit of the price as x increases without bound?

  x = 1000 units (Round your answer to two decimal places.)____$
  x = 1500 units (Round your answer to two decimal places.)___$
What is the limit of the price as x increases without bound?___$

B)The population P (in thousands) of Charlotte, North Carolina from 1980 through 2013 can be modeled by

P = 321e^0.0275t where t = 0 corresponds to 1980.† (Round your answers to the nearest whole number.)What was the population of Charlotte in 2013? people
In what year will the population of Charlotte reach 1,900,000?

Let G, H, K be groups. Prove that if G ≅ H and H ≅ K then G ≅ K.

I want to estimate the interpolation error in [a,b] where f is interpolated by polynomials at x_j, h=(b-a)/n, x_j=a+jh, j=0,1,...,n.

I want the answers in detail especially for n=2,3.

This is a problem from Jeff ’s notes - reproduced here for ease. The d-dimensional hypercube is the graph defined as follows. There are 2d vertices, each labeled with a different string of d bits. Two vertices are joined by an edge if and only if their labels differ in exactly one bit. See figures in Jeff ’s notes if you need to - but it would be more instructive to draw them yourself and recognize these objects. Recall that a Hamiltonian cycle is a closed walk that visits each vertex in a graph exactly once. Prove that for every integer d ≥ 2, the d-dimensional hypercube has a Hamiltonian cycle.

The Polynomial f(x) = X^3 - X^2 - X -1 has one real root a, which happens to be positive. This real number a satisfies the following properties:

- for i = 1,2,3,4,5,6,7,8,9,10, one has {a^i} not equal to zero

- one has

[a] = 1, [a^2] = 3, [a^3] = 6, [a^4] = 11, [a^5] = 21, [a^6] = 7, [a^7] = 71, [a^8] = 130

(for a real number x, [x] denotes the floor of x and {x} denotes the fractional part of x.)

find this real root a

Of the following intervals, which includes the most prime numbers?
A. 20 and 30
B. 30 and 40
C. 40 and 50
D. 50 and 60

Please explain clearly. How do we even find prime numbers between these intervals.

Thank you.

Thank You

Define the gcd of three integers a, b, c as the largest common divisor of a, b, c, and denote it by (a, b, c). Show that (a, b, c) = ((a, b), c) and that (a, b, c) can be expressed as a linear combination of a, b, c.

a, The vectors v1 = < 0, 2, 1 >, v2 = < 1, 1, 1 > , v3 = < 1, 2, 3 > , v4 = < -2, -4, 2 > and v5 = < 3, -2, 2 > generate R^3 (you can assume this). Find a subset of {v1, v2, v3, v4, v5} that forms a basis for R^3.

b. v1 = < 1, 0, 0 > , v2 = < 1, 1, 0 > and v3 = < 1, 1, 1 > is a basis for R^3 (you can assume this.) Given an arbitrary vector w = < a, b, c > write w as a linear combination of v1, v2, v3.

c. Find the dimension of the space spanned by x, x-1, x^2 - 1 in P2 (R).

In parts a, b, and c, determine if the vectors form a basis for the given vector space. Show all algebraic steps to explain your answer.

a. < 1, 2, 3 > , < -2, 1, 4 > for R^3

b. < 1, 0, 1 > , < 0, 1, 1> , < 2, 0, 1 > for R^3

c. x + 1, x^2 + 1, x^2 + x + 1 for P2 (R).

Part a (worth 60 pts): Formulate a linear programming model (identify and define decision variables, objective function and constraints) that can be used to determine the amount (in pounds) of Brazilian Natural and Colombian Mild that will maximize the total contribution to profit. For “Part a” you do NOT need to solve this problem using Excel, you just need to do the LP formulation in the standard mathematical format.

Part b (bonus worth 20 pts): Solve the LP problem that you formulated in “Part a” using Excel. Give the values of each decision variable and the objective function. You MUST attach a copy of the solution report.