Give a vector parametric equation for the line that passes through the point (2,0,5), parallel to the line parametrized by 〈t−3,t−3,−3−3t〉:

Find the simplest vector parametric expression r⃗ (t) for the line that passes through the points P=(0,0,−1) at time t = 1 and Q=(0,−4,−1) at time t = 5

Need help with Calculus III

1. Let X and Y be non-linear spaces and T : X -->Y. Prove that if    T is One-to-one then T^-1 exist on R(T) and T^-1 : R(T) à X is also a linear map.

2. Let X, Y and Z be linear spaces over the scalar field F, and let T₁ ϵ B (X, Y) and T₂ ϵ B (Y, Z). let T₁T₂(x) = T₂(T₁x) ∀ x ϵ X.

(i) Prove that T₁T₂ ϵ B (X,Y) is also a bounded linear mapping.

(ii) Prove that ǀǀT₂T₁ǀǀ ≤ ǀǀT₂ǀǀ ǀǀT₁ǀǀ

3. If X is an inner product space, then for arbitrary x, y ϵ X, ǀ< x, y>ǀ ≤ ǀǀxǀǀ ǀǀyǀǀ, prove that the inner product < ∙ > is a continuous function on X by X (Cartesian product) domain.

A construction firm is considering buying a backhoe, since it pays $50/h to rent one, and it needs to use a backhoe for 150 hours per month. Assuming a backhoe lasts 5 years, and has monthly maintenance of $2,000 and a salvage value of $15,000, what’s the most the company should pay for this backhoe? (Use a 6% discount rate)

Bob​ Carlton's golf camp estimates the following workforce requirements for its services over the next two​ years: Quarter 1 2 3 4 5 6 7 8 Demand​ (hrs) 4,200 6,400 3,100 5,000 4,400 6,240 3,800 5,000

Each certified instructor puts in 480 hours per quarter regular time and can work an additional 120 hours overtime.​ Regular-time wages and benefits cost Carlton $7,200 per employee per quarter for regular time worked up to 480 hours, with an overtime cost of $20 per hour.

Unused regular time for certified instructors is paid at $15 per hour

. There is no cost for unused overtime capacity. The cost of​ hiring, training, and certifying a new employee is $10,000. Layoff costs are $4,000 per employee.

Currently 8 employees work in this capacity.

(a) Find a workforce plan using the level strategy that allows for no delay in service. It should rely only on overtime and the minimum amount of undertime necessary. What is the total cost of the​ plan? 701000

(b) Use a chase strategy that varies the workforce level with minimal undertime and without using overtime. What is the total cost of this plan? 809600

(c) Propose a low-cost, mixed strategy and calculate its total cost. (Any strategy that improves on both the chase and level strategies is acceptable; no need to find the optimal schedule.)

Find the general solution

1.(1+x²) (d²y/dx²) + x (dy/dx) + ax = 0

2. ρ(dθ/dρ) –2/ρ (dρ/dθ) = 0

3.(dy/dx)² -4x (dy/dx) +6y = 0

4.y(d²y/dx²) + (dy/dx)² = (dy/dx)

5.Solve simultaneously:

(dx/dt) + (dy/dt) + y –x = e^2t

(d²x/dt²) + (dy/dt) = 3 e^2t

6.Using method of variation of parameter, solve: y'' – 8 y' +16 y = 6x e^4x

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is the direct product of U and V (where V a subgroup of G) if only if there is a homomorphism f : G → U with    f|U = IdU

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows.

For the first year's production, a sum of $8,200,000 is budgeted for the building material; the number of labor-hours available for work in the factory (for prefabrication and partial assembly) is not to exceed 215,000 hr; and the amount of labor for on-site work is to be less than or equal to 234,000 labor-hours. Determine how many houses of each type Boise should produce to maximize its profit from this new venture. (Market research has confirmed that there should be no problems with sales.)

Let R be a UFD and let F be a field of fractions for R. If f(α) = 0, where f ∈ R [x] is monic and α ∈ F, show that α ∈ R

NOTE: A corollary is the fact that m ∈ Z and m is not an n^th power in Z, then ⁿ√m is irrational.

The​ input-output matrix for a simplified economy with just three sectors​ (agriculture, manufacturing, and​ households) is given below.

          Agriculture Manufacturing Households

A.  How many units from each sector does the agriculture sector require to produce 1​ unit?

The agriculture sector requires _____units from​ agriculture, ____units from​ manufacturing, and ___units from households.

B.  What production levels are needed to meet a demand of 32 units of​ agriculture, 34 units of​ manufacturing, and 34 units of​ households?

Production levels of ____units of​ agriculture,___units of​ manufacturing, and _____units of households are needed.  ​(Round to the nearest whole number as​ needed.)

C.  How many units of manufacturing are used up in the​ economy's production​ process?

The​ economy's production process uses up_____ units of manufacturing.

​(Round to the nearest whole number as​ needed.)

Previously, we listed all 29 topologies on the set X={a,b,c}. However, some of the resulting topological spaces are homeomorphic. Which are homeomorphic? Divide the set of 29 topological spaces into homeomorphism classes, and be sure to justify your choices. There are 9 homeomorphism classes in total. (To justify your choices, explain why the spaces within each class are homeomorphic to each other. Your explanations can be somewhat loose).

Suppose that a decision-maker’s preferences over the set A={a, b, c} are represented by the payoff function u for which u(a) = 0, u(b) = 1, and u(c) = 4.

(a) Are they also represented by the function v for which v(a) =−1, v(b) = 0, and v(c) = 2?

(b) How about the function w for which w(a) =w(b) = 0 and w(c) = 8?

(c) Give another example of a function f:A→R that represents the decision-maker’s preferences.

(d) Is there a function that represents the decision-maker’s preferences and assigns negative numbers to all elements of A?

Let u and v be orthogonal vectors in R3 and let w = 3u + 6v. Suppose that ||u|| = 5 and ||v|| = 4. Find the cosine of the angle between w and v.