An air conditioning manufacturer produces room air conditioners at plants in Houston, Phoenix, and Memphis. These are sent to regional distributors in Dallas, Atlanta, and Denver. The shipping costs vary, and the company would like to find the least-cost way to meet the demands at each of the distribution centers. Dallas needs to receive 800 air conditioners per month, Atlanta needs 600, and Denver needs 200. Houston has 850 air conditioners available each month, Phoenix has 650, and Memphis has 300. The shipping cost per unit from Houston to Dallas is $8, to Atlanta is $12, and to Denver is $10. The cost per unit from Phoenix to Dallas is $10, to Atlanta is $14, and to Denver is $9. The cost per unit from Memphis to Dallas is $11, to Atlanta is $8, and to Denver is $12. How many units should be shipped from each plant to each regional distribution center? What is the total cost for this? I need the linear programming model for this problem and written variables, constraints, and minimize cost.

If you prove by strong induction a statement of the form ∀ n ≥ 1P(n), the inductive step proves the following implications (multiple correct answers are possible):

a) (P(1) ∧ P(2)) => P(3)

b) (P(1) ∧ P(2) ∧ P(3)) => P(4)

c) P(1) => P(2)

To make a phrase/words, solve the 18 application of derivatives below.  Then replace each numbered blank with the letter corresponding to the answer for that problem. Show all solutions on the answers given below.

"   __ __ __ __ __ __ __       __ __ __ __      __ __ __ __ __ __ __      __ __ __ __ __      __ __ __ __ __"       

"__ __ __ __ __ __ __      __ __      __      __ __ __ __ __ __ __      __ __ __ __ __      __ __ __ ."     

Derivative Application Problems:      

1.  Find the equation of the line normal to the curve  f(x) = x³ – 3x²  at the point (1, -2).          

2.  Find the equation of the line tangent to the curve  x² y – x = y³ – 8  at the point where x = 0.          

3.  Determine the point(s) of inflection of  f(x) = x³ – 5x² + 3x + 6.         

4.  Determine the relative minimum point(s) of  f(x) = x⁴ – 4x³.        

5.  A particle moves along a line according to the law s = 2t³ – 9t² + 12t – 4, where t≥0 .  Determine the total distance traveled between t = 0 and t = 4.             

6.  A particle moves along a line according to the law s = t⁴ – 4t³, where t≥0.    Determine the total distance traveled between t = 0 and t = 4.        

7.  If one leg, AB, of a right triangle increases at the rate of  2 inches per second, while the other   leg, AC,  decreases at 3 inches per second, determine how fast the hypotenuse is   changing (in feet per second) when AB = 6 feet and AC = 8 feet.     

8.  The diameter and height of a paper cup in the shape of a cone are both 4 inches, and water is leaking   out at the rate of ½ cubic inch per second.  Determine the rate (in inches per second) at which  the water level is dropping when the diameter of the surface is 2 inches.     

9.  For what value of y is the tangent to the curve  y² – xy + 9 = 0 vertical?        

10.  For what value of  k  is the line  y = 3x + k tangent to the curve  y = x³ ?        

11.  Determine the slopes of the two tangents that can be drawn from the point (3, 5) to the parabola    y = x² .              

12.  Determine the area of the largest rectangle that can be drawn with one side along the x-axis and  

two vertices on the curve  y = e^-x2

13.  A tangent drawn to the parabola  y = 4 – x²  at the point (1, 3)  forms a right triangle with the   coordinate axes.  What is the area of this triangle?         

14.  If the cylinder of largest possible volume is inscribed in a given sphere, determine the ratio of the   volume of the sphere to that of the cylinder.

15.  Determine the first quadrant point on the curve  y²x = 18 which is closest to the point  (2, 0).     

16.  Two cars are traveling along perpendicular roads, car A at 40 mph, car B at 60 mph.  At noon when   car A reaches the intersection, car B is 90 miles away, and moving toward it.  At 1PM, what is   the rate, in miles per hour, at which the distance between the cars is changing?

Prove by Induction.

Prop: if the (greatest common factor) gcf(a,m) = f then there is k, l with ka + lm = f

Steps that we have already established is,

m = a(q1) + (r1)

a = (r1)(q2) + (r2)

(r1) = (r2)(q3) + (r3)

...

r(n-2) = r(n+1)q(n) + r(n)

r(n-1) = r(n)q(n+1) + 0 is the very last step to this sequence

Let F(x, y, z) = z tan^−1(y^2)i + z^3 ln(x^2 + 7)j + zk. Find the flux of F across S, the part of the paraboloid x2 + y2 + z = 29 that lies above the plane z = 4 and is oriented upward.

1. Let A be an inductive subset of R. Prove that {1} ∪ {x + 1 | x ∈ A} is inductive.

2. (a) Let n ∈ N(Natural number) and suppose that k 2 < n < (k + 1)2 for some k ∈ N. Prove that n does not have a square root in N.

(b) Let c ∈ R \ {0}. Prove that if c has a square root in Z, then c has a square root in N.

(c) Prove that 2 does not have a square root in Z.

# Can you someone please help me with these questions.

#

(a) Show that the length of the broken line satisfies Length(L) ≥ |AB|.

(b) Show that L achieves the lower bound
Length(L) = |AB|

if and only if the vertices V1,...,Vk−1 all lie on the segment AB and appear in that orderonAB,i.e.,theysatisfyVi ∈Vi−1Vi+1 forall1≤i≤k−1.

Sketch the graph, shade the region and set up the integral and DO NOT EVALUATE.

8.Find the area A of the region bounded by the line y = x^2 and y= x is revolved

a) About the x axis using disks or washers

b) About the line x = 2 (any method you like)

find all solutions of the congruence x^2=60(mod77)

Prove rigid motions are:

a. continuous

b. one-to-one

c. closed under composition

There are five distinct white and seven distinct blue shirts in a wardrobe. Find the number of ways of taking four shirts from the wardrobe such that a) they could be either white or blue, b) they are all white, c) they are all blue, d) they are all the same color and e) 2 are white and 2 are blue.

a. Use mathematical induction to prove that for any positive integer ?, 3 divide ?^3 + 2?
(leaving no remainder).
Hint: you may want to use the formula: (? + ?)^3= ?^3 + 3?^2 * b + 3??^2 + ?^3.
b. Use strong induction to prove that any positive integer ? (? ≥ 2) can be written as a
product of primes.

Is it true that every symmetric positive definite matrix is necessarily nonsingular? (Need to show some form of proof, not just yes or no answer)

Please add some commentary for better understanding.

The Fellowship of the Ring traveling through the mountains of Moria (which are described by the functionf(x,y)=12x2y2e(-x2-y2)) and they find themselves at the point A=(1.5, -1, 1.05). They need to get to the top of the mountain they are climbing so they decide to travel along the plane y=-1. How steep is their climb when they start at point A? A diagram: https://www.geogebra.org/3d/jbkkyugx

1. What if they decided to go in a different direction, say they started walking towards the point (0,0,0) on a plane that is “vertical”. How steep would be the mountain going in that direction?

2. What is the general equation for planes that are vertical, like the one that you used to travel from A to (0,0,0)?

3. What would change in your calculations if we change where point A is? Can you generalize how to find the steepness in different directions from a given point A?

Let p be an integer other than 0, ±1.

(a) Prove that p is prime if and only if it has the property that whenever r and s are integers such that p = rs, then either r = ±1 or s = ±1.

(b) Prove that p is prime if and only if it has the property that whenever b and c are integers such that p | bc, then either p | b or p | c.