Decide which special case applies. case 1) no feasible region 2) unboundedness for maximizing the objective function 3)Redundant constraint 4)More than one optimal solution

Problem - A candidate for mayor is planning an advertising campaign. Radio Ads cost $200 and reach 3,000 voters and TV ads cost $500 and reach 7,000 voters.The campaign manager

has stipulated that at least 10 ads of each type must be used. Also, the manager insists that the number of radio ads be at least as great as the number of TV ads. How many ads of each type should be purchased in order to reach the maximum number of voters?

Theorem 1 (Mean Value Theorem). Suppose ? ⊂ R is open and ? : ? → R. If ? is differentiable on the open interval (?, ?) ∈ ? then there exists ? ∈ (?, ?) such that ?(?) − ?(?) ? − ? = ? ′ (?).

4. An alternative interpretation of the Mean Value Theorem is that if ? is differentiable on (?, ? + ℎ), then there exists ? ≤ ? ≤ ? + ℎ such that ?(? + ℎ) = ?(?) + ??(?)(ℎ) (1)

a. The Mean Value Theorem fails if ? : R^? → R^? even with ? = 1 and ? = 2. Let ? : [0, 1] → R^2 be given by ?(?) = (? − ?^2 , ? − ?^3 ). Show that Equation (1) fails for ? when ? = 0 and ℎ = 1.

b. On the other hand, let ?(?, ?, ?) = ?? + ?^2 , a = (0, 0, 0) and b = (2, 1, 2). Let ? : [0, 1] → R^3 be the parameterization of the line segment going from a to b. Find (? ∘ ?)(?) and (? ∘ ?) ′ (?).

c. Find a value ?0 ∈ (0, 1) such that ?(b) − ?(a) = (? ∘ ?) ′ (?0).

WeHaul Trucking is planning its truck purchases for the coming year. It allocated $600,000 for the purchase of additional trucks, of which three sizes are available. A large truck costs $150,000 and will return the equivalent of $15,000 per year to profit. A medium-sized truck costs $90,000 and will return the equivalent of $12,000 per year. A small truck costs $50,000 and will return the equivalent of $9,000 per year. WeHaul has maintenance capacity to service either four large trucks, five medium-sized trucks, or eight small trucks, or some equivalent combination. WeHaul believes that it will be able to hire a maximum of seven new drivers for these added trucks. The company cannot spend more than one/half of the total funds it actually spends to purchase medium-sized trucks. (Hint: this is not necessarily one half of the total funds it has allocated for the purchase of additional trucks).   

You must submit your linear programming formulations and show the linear programming software solution to this problem to receive credit. If you solve this using another linear programming approach, you may submit that instead of the software solution.   

a) Formulate a linear programming model to be used for determining how many of each size of truck to purchase if the company wants to maximize its profit. Ignore the time value of money. Provide the linear programming variables, the objective function, and the constraints for the problem.

b) At optimality, how much profit will result and what is the optimal combination of trucks? You must submit your linear programming formulations and show the linear programming software solution to this problem to receive credit. If your answer is in fractional units of trucks that is acceptable – do not round to whole number of trucks.

c) Using your sensitivity analysis output, provide two sensitivity analysis interpretations. One must be for the objective function and one must be for one of the constraints. You must provide the source of your answers from the sensitivity analysis output.

d) Now suppose that there is a requirement that WeHaul must purchase at least two small trucks for each medium size truck. Also, the number of larger trucks cannot be more than the total number of medium and small trucks. Write the constraint(s) for this requirement. However, you do not need to resolve the problem.

Let ? : R^2 → R be given by ?(?, ?) = √︀ |??|.

a. Show ? is continuous at (0, 0).

b. Show ? does not have a directional derivative at (0, 0) along (1, 1).

c. Is ? differentiable at (0, 0)?

Find the maximum point between x = -1 to 0 Find the minimum point between x = 0 to 1

Use Golden Search section method

Use 10 iterations

Determine the truth value of each statement in the case when the universe comprises all nonzero integers, and in the case when the universe consists of all nonzero real numbers.

(a) ∃x∀y (xy = 1);

(b) ∀x∃y ((2x + y = 5) ∧ (x − 3y = −8)).

What is the answer for a) and )b? Please thoroughly explain, I've only taken Logic and Quantifiers so far.
Please make sure it's clear and readable, 100% I'll thumbs ups if it's good.

Using linear regression, model the following data. The table contains world internet users in millions (Source: International Telecommunication Union; ICT database). Model the data as a linear function. Note the x and y are tabulated. Your answer needs to be an equation in the form y = mx + b

LabView: Using MathScript create a VI that computes and plots the second order polynomial y = Ax² + Bx + C. The VI should use controls on the front panel to input the coefficients A, B, and C, and also should use front panel controls to enter the number of points N to evaluate the polynomial over the interval x₀ to x_N-1. Plot y versus x on a waveform graph indicator.

1: (a_n) and (b_n) are bounded sequences:

(a) prove that limsup(-a_n) = -liminf(a_n)

(b) for any c>0, prove that

limsup(ca_n) = climsup(a_n)

and

liminf(ca_n) = climinf(a_n)

(c) prove that

limsup(a_n+b_n) ≤ (limsup(a_n)) + (limsup(b_n))

and

liminf(a_n+b_n) ≥ (liminf(a_n)) + (liminf(b_n))

(d) If a_n and b_n are made of nonnegative terms, prove that

limsup(a_nb_n) ≤ (limsup(a_n)) x (limsup(b_n))

and

liminf(a_nb_n) ≥ (liminnf(a_n)) x (liminf(b_n))

(e) prove that

limsup(a_n+1) = limsup(a_n)

and

liminf(a_n+1) = liminf(a_n)

True or False? If true, give a brief reason why; if false, give a counterexample. (Assume in all that V and W are vector spaces.)

a. If T : V → W is a linear transformation, then T(0) = 0.

b. Let V be a vector space with basis α = {v1, v2, . . . , vn}. Let S : V → W and T : V → W be linear transformations, and suppose for all vi ∈ α, S(vi) = T(vi). Then S = T, that is, for all v ∈ V , S(v) = T(v).

c. Every linear transformation from R3 to R3 has an inverse. That is, if T : R3 → R3 is a linear transformation, then there exists a linear transformation S : R3 → R3 such that S(T(v)) = T(S(v)) = v for all v ∈ R3 .

d. If T : Rn → Rm is a linear transformation and n > m, then Ker(T) 6= {0}.

e. If T : V → W is a linear transformation, and {{v1, v2, . . . , vn} is a basis of V , then {{T(v1), T(v2), . . . , T(vn)} is a basis of W.

Differential Geometry (Mixed Use of Vector Calculus & Linear Algebra)

1A. 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.

1B. Prove that the set where y<c is an open set.

Let X be Z or Q and define a logical formula p by ∀x ∈ X, ∃y ∈ X, (x < y ∧ [∀z ∈ X, ¬(x < z ∧ z < y)]).

Describe what p asserts about the set X. Find the maximally negated logical formula equivalent to ¬p. Prove that p is true when X = Z and false when X = Q

discrete math

Each different prime number represents a different colored block. For example, 2 represents a yellow block, 3 represents a red block, 5 represents a green block , and so on...

a. How many blocks would we need to build the smallest number that is divisible by all natural numbers from 1 to 30? From 1 to 50? From 1 to 100?

b. I’m thinking of a number n . It takes exactly 50 blocks to build the number that is the smallest number divisible by all natural numbers from 1 to n. What is n? Hint: there are actually several numbers that n could be. Can you find all of them?

Can anyone please explain step by step how to solve this by excel solver cause the solver won't accept the binary word

A group of college students is planning a camping trip during the upcoming break. The group must hike several miles through the woods to get to the campsite, and anything that is needed on this trip must be packed in a knapsack and carried to the campsite. On particular student, Tina Shawl, has identified eight items that she would like to take on the trip, but the combined weight is too great to take all of them. She has decided to rate the utility of each item on a scale of 1 to 100, with 100 being the most beneficial. The item weights in pounds and their utility values are given below.

Item    1    2 3 4    5    6 7 8

Weight 8    1 7    6 3 12 5 14

Utility 80    20 50    55    50 75 30    70

Recognizing that the hike to the campsite is a long one, a limit of 35 pounds has been set as the maximum total weight of the items to be carried.

a) Formulate this as a 0-1 programming problem to maximize the total utility of the items carried.Solve this knapsack problem using a computer.

b) Suppose item number 3 is an extra battery pack, which may be used with several of the other items.Tina has decided that she will only take item number 5, a CD player, if she also takes item number 3.On the other hand, if she takes item number 3, she may or may not take item number 5.Modify this problem to reflect this and solve the new problem.

Problem 3-12 (Algorithmic)

Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $57, $93, and $141, respectively. The production requirements per unit are as follows:

For the coming production period, the company has 320 fan motors, 380 cooling coils, and 3200 hours of manufacturing time available. How many economy models (E), standard models (S), and deluxe models (D) should the company produce in order to maximize profit? The linear programming model for the problem is as follows:

The computer solution is shown in the figure below.

1. What is the optimal solution, and what is the value of the objective function? If required, round your answers to the nearest whole number. If your answer is zero, enter "0".
   

   	Optimal Solution
   Economy models (E)
   Standard models (S)
   Deluxe models (D)
   Value of the objective function	$

2. Which constraints are binding?
   

   Fan motors:
   Cooling coils:
   Manufacturing time:

3. Which constraint shows extra capacity? How much? If constraint shows no extra capacity, enter 0 as number of units. If required, round your answers to the nearest whole number.
   

   Constraints	Extra capacity	Number of units
   Fan motors
   Cooling coils
   Manufacturing time

4. If the profit for the deluxe model were increased to $156 per unit, would the optimal solution change?
   
   The optimal solution   change because the profit of the deluxe model can vary from   to  . $156 is   this range without the optimal solution changing.