How would you do the work to find which PDEs below can be separated into ODEs by assuming a product of unknown functions x,y and z?

1) Uxx+Uyy+Uzz=0

2) Uxx+Uyy+Ux+Uy=0

3) Uxx+3Uxy+Uyy=0

4) Uxx+3Uxy+7Uy=0

5) x^2Uxx+yUyy=0

6) aUxy+bu=0

Discuss a relationship between two quantities associated with your major (business finance). Include your major and describe the relationship 1) in words, 2) graphically, 3) and as an equation. Define the terms in the equation, and properly label the graph and all terms related to it.

Let a and b be integers. Recall that a pair of Bezout coefficients for a and b is a pair of integers m, n ∈ Z such that ma + nb = (a, b).

Prove that, for any fixed pair of integers a and b, there are infinitely many pairs of Bezout coefficients.

1.) Use the product rule to find the derivative of

(−10x⁶−7x⁹)(3e^x+3)

2.) If f(t)=(t²+5t+8)(3t²+2) find f'(t)
   
Find f'(4)

3.) Find the derivative of the function
g(x)=(4x²+x−5)e^x

g'(x)=

4.) If f(x)=(5−x²) / (8+x²) find:

f'(x)=

5.) If f(x)=(6x²+3x+4) / (√x) , .  then:

f'(x) =    

f'(1) =

6.) Find the derivative of the function g(x)=(e^x) / (3+4x)

g'(x)=

7.)

Differentiate: y=(ln(x)) /( x⁶)

(dy) / (dx) =

8.) Given that
f(x)=x⁷h(x)
h(−1)=2
h'(−1)=5

Calculate f'(−1)

9.) The dose-response for a specific drug is f(x)=(100x²) / (x²+0.14) where f(x) is the percent of relief obtained from a dose of x grams of a drug, where 0≤x≤1.5

Find f'(0.7)

f'(0.7) =

and select the appropriate units.

a)Grams per percent relief

b)Percent relief per gram

c)Percent relief

d)Grams

10.)Let f(x)= (x) / (x+6) . Find the values of x where f'(x)=5

Give exact answers (not decimal approximations).


The greater solution is x=   
The lesser solution is x=  

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