Use Lagrange multipliers to find the minimum and maximum values for the following functions subject to the given constaints.

a) f(x,y) = 8x²+y² ; x⁴+y^{4 = 4}

b) f(x,y,z) = 2z-8x² ; 4x²+y²+z² = 1

c) f(x,y,z) = xyz ; x²+4y²+3z² = 36

Question 1

Problem 2.64 Use MATLAB to solve for and plot the response of the following models for 0≤t ≤1.5, where the input is f (t) =5t and the initial conditions are zero: 3¨ x +21˙ x +30x = f (t)

A)  Use math (analytically) to find the response of the models given in Problem 2.64 of your book. Show all assumptions and all steps of your derivation. (Turn in the calculations.)

B) Use MATLAB to solve the same problem 2.64

C) (Turn in the MATLAB script and answers from MATLAB,.m file, screen shots if needed)

E) Plot the response for Problem 2.64 . (Turn in the MATLAB plot with t being time in SI units)

F) Comment on the response the analytical solution compared with the MATLAB and the plots. (Do the calculations and MATLAB agree ?

Why and Why not.? Do the plots make sense?)

Which of the following are subrings of the field R of real numbers:

(d) D = {a + b (³√3) + c (³√9) | a, b, c ∈ Q}.

(e) E = {m + n(1 + √ 3)/2 | m, n ∈ Z}.

(f) F = {m + n(1 + √ 5)/2 | m, n ∈ Z}.

Discuss the ratio of circumference to diameter for circles on the sphere.

the Euclidean radius is the distance to the (Euclidean) center of the circle -- which
is not a point on the sphere. But the circle also has a center on the
sphere -- this would be the North (or South) Pole for lines of
latitude. So the spherical radius of such a circle is different, namely
the spherical distance from the (spherical) center.

Derive formulas relating these various notions of radius and circumference.

Let A be a subset of all Real Numbers. Prove that A is closed and bounded (I.e. compact) if and only if every sequence of numbers from A has a subsequence that converges to a point in A.

Given it is an if and only if I know we need to do a forward and backwards proof. For the backwards proof I was thinking of approaching it via contrapositive, but I am having a hard time writing the proof in a way that is understandable to the readers.

Prove that "congruent modulo 3" is an equivalence relation on Z. What are the equivalence classes?

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 is not to exceed 209,000 hr; and the amount of labor for on-site work is to be less than or equal to 231,000 labor-hours. Determine how many houses of each type Boise should produce to maximize its profit from this new venture.

Explain why our method of contradiction showing "there are infinitely many primes in the form of 4k-1" DOESN'T WORK for the primes in the form of 4k+1?

I asked this question before and the answer was incorrect, so I am asking it again. It DOESN'T want to use contradiction method to prove , It wants to explain why we cannot use contradiction method. PLEASE DON'T ANSWER IF YOU DON'T KNOW THE CORRECT ANSWER.

Write the converse of each statement of the following.

1. Healthy plant growth follows from sufficient water.

2. Increased availability of information is a necessary condition for further technological advances.

3. Errors were introduced only if there was a modification of the program

4. Fuel savings implies good insulation or storm windows throughout,

Let (E,d) be a metric space and K, K' disjoint compact subsets of E. Prove the existence of disjoint open sets U and U' containing K and K' respectively.

please provide good answer.

1). Let S = { 1,2,3, …., n} for some positive integer n. Define the operations + and . on S as x + y = max{ x, y }, and x.y = min{ x, y }. Is it possible to make S into a Boolean algebra with these two operations? Explain your reasoning.

[ Note: max{ x, y } returns the maximum of the values x and y, and min{ x, y } returns the minimum of the values x and y. For example: max{3, 8} = 8, and min{ 3, 8 } = 3, and max{ 3, 3 } = min{ 3, 3 } = 3.]

give an example of an error with a proof by case

Solve the following initial value problems and graph the solutions at times t = 1, 2, and 3:
(a) u_t − 3u_x = 0, u(0, x) = e^−x^{2}
(b) u_t + 2u_x = 0, u(−1, x) = x/(1 + x^2)
(c) u_t + u_x + 1/2u = 0, u(0, x) = tan^−1 x;
(d) u_t− 4u_x + u = 0, u(0, x) = 1/(1 + x^2)\

What is an example of a situation that you might be able to use an equation with a single unknown to help understand? What is an example of a situation that you might not be able to use an equation with a single unknown to understand? What makes an equation with a single unknown helpful in one of your examples but not the other? What patterns exist in your two examples that might be helpful in determining when to use a simple equation?

A golf ball is hit and lands 53 m away. The path of the ball took it just over a 9 m tree. Determine a function that models the path of the golf ball