Calculus w/ analytical geometry:

please be concise

- Fubini's Theorem: if a function is
continuous on the domain R, then the triple
integral can be evaluated in any order that
describes R.
(a) Explain the significance of this theorem.
(b) Provide an example to illustrate this
theorem.

- multi-integration
(a) Explain the purpose of changing variables
when double or triple integrating.
(b) Post an example illustrating such a change of variables.

Suppose you draw a card from a well- shuffled deck of 52 cards. Determine the following probabilities .

a. drawing a 9

b.drawing a 3 or king

c.drawing a spade

d.drawing a black card

e. DRAWING A RED KING

3) We have not forgotten Halmos Pal. In class I asked you what’s wrong with this “proof” of Halmos’ that all horses are the same color? It’s time to tell me what you found. (Try the web.)

consider the solid S bounded by the two cylinders x^2+y^2=3 and y^2+z^2=3 in R^3

a.Find the volume of S by setting up and evaluating a double integral.

b.Find the surface area of the solid S. You may use symmetry to simplify the computation.

NEW NUMBERS: A chemical plant stores spare parts for maintenance in a large warehouse. Throughout the working day, maintenance personnel go to the warehouse to pick up supplies needed for their jobs. The warehouse receives a request for supplies, on average, every three minutes. The average request requires 2.75 minutes to fill. Maintenance employees are paid $21.50 per hour and warehouse employees are paid $16 per hour. The warehouse operates 8 hours per day.

a) Based on the number of maintenance employees in the system, an 8 hour work day, and the given arrival and service rates. What is the system cost per day (to the nearest $) if there is only 1 warehouse employees working?

b) Based on the number of maintenance employees in the system, an 8 hour work day, and the given arrival and service rates. What is the system cost per day (to the nearest $) if there are 2 warehouse employees working?

c) Based on the number of maintenance employees in the system, an 8 hour work day, and the given arrival and service rates. What is the system cost per day (to the nearest $) if there are 3 warehouse employees working?

d) What is the optimal number of warehouse employees to staff the warehouse?

f(x)=〖2x〗^3-cosx/5+2e^(-x) given of f(x) function,
   a-Fill the table f(x) column using calculator f(x) for given x values.
   b-After all calculation of table find f(1,3) Neville’s Method approximation x0=1,2 and x1=1,4
   c-Find f(1,3) Neville’s Method approximation x0=1,2 x1=1,4 and x3=1,5
Tell which result is more reliable and precise in case b, and c. Why?

Solve the question

Suppose C and D are two matrices where CD is defined

a) fill in the blank: Each column vector of CD is a linear combination of ___________ and each column vector of CD is in the span of_________ .

b) Using part A, what is the relation of the column space of CD, the column space of C and the column space of D

How do you solve this?

Find the general solution of the given differential equation

Show all steps

y''-5y'-6y=10tsin(3t)

u(t−c) =uc(t) ={0, 0≤t<c,1, t≥c.}

USE Laplace Transform to solve

y′′+ 2y′+ 2y=δ(t−5)e^tcost, y(0) = 1, y′(0) = 2, whereδ(t)is the Dirac delta. Does the solution show a

resonance?

1. Prove that there is only one possible multiplication table for G if G has exactly 1, 2, or 3 elements. Analyze the possible multiplication tables for groups with exactly 4 elements, and show that there are two distinct tables, up to reordering the elements of G. Use these tables to prove that all groups with < 4 elements are commutative.

(You are welcome to analyze groups with 5 elements using the same technique, but you will soon know enough about groups to be able to avoid such brute-force approaches.)

The Company manufactures paring knives and pocket knives. Each paring knife requires 3​ labor-hours, 7 units of​ steel, and 4 units of wood. Each pocket knife requires 6​ labor-hours, 5 units of​ steel, and 3 units of wood. The profit on each paring knife is​ $3, and the profit on each pocket knife is​ $5. Each day the company has available 96 ​labor-hours, 134 units of​ steel, and 120 units of wood. Suppose that the number of​ labor-hours that are available each day is increased by 27. Use sensitivity analysis to determine the effect on the optimal number of knives produced and on the profit.

The Company should produce___ paring knives and ___ pocket knives each day for a profit of

​$_____.

Determine whether the given differential equation is exact. If it is exact, solve it.

i) (x 3 + y 3 )dx + 3xy2 dy = 0, Ans. x 4 + 4y 3x = C

ii) (y ln y − e −xy) + (1 y + x ln y) dy dx = 0, Ans. not exact

iii) (e −x sin y − 3)dx − (3x 2 − e x sin(2y))dy = 0, Ans. not exact

iv) (xy − 1)dx + (x 2 − xy)dy = 0, Ans. not exact.

4. Show that the equation (y ln y)dx + (x − ln y)dy = 0 is not exact. Find an integration factor to find a one parameter family of solutions. Ans: 2x ln y − (ln y) 2 = C.

Find the solution of the given initial value problem:

y′′′+y′=sec(t), y(0)=6, y′(0)=7, y′′(0)=−3