?′ (?) = −100?(?), ?(0) = y₀

1) Find the condition for the step size dt such that the explicit Euler scheme converges to the exact solution to the given differential equation.

2) Find the condition for the step size dt such that the implicit Euler scheme converges to the exact solution to the given differential equation.

Verify Stokes theorem for F =(y^2 + x^2 - x^2)i + (z^2 + x^2 - y^2)j + (x^2 + y^2 - z^2)k over the portion of the surface x^2 + y^2 -2ax + az = 0

Solve the linear second-order ODE for each case of b. Find constants using the given initial conditions.

y(0)=1, y'(0)=0

y''+by'+16y=0

b=0

b=2

b=8

b=10

say b represents damping constant. What is the effect of damping on the motion of a mass?

1.Describe all of the homomorphisms from Z20 to Z40.

2.Describe all of the homomorphisms from Z to Z12.

A poker company assembles three different poker sets. Each Royal Flush poker set contains 1000 poker​ chips, 4 decks of​ cards, 10 ​dice, and 2 dealer buttons. Each Deluxe Diamond poker set contains 600 poker​ chips, 2 decks of​ cards, 5 ​dice, and one dealer button. The Full House poker set contains 300 poker​ chips, 2 decks of​ cards, 5 ​dice, and one dealer button. The company has 2 comma 900,000 poker​ chips, 10,000 decks of​ cards, 25,000 ​dice, and 6000 dealer buttons in stock. They earn a profit of ​$38 for each Royal Flush poker​ set, ​$22 for each Deluxe Diamond poker​ set, and ​$12 for each Full House poker set. Complete parts​ (a) and​ (b) below.

​(a) How many of each type of poker set should they assemble to maximize​ profit? What is the maximum​ profit?

Begin by finding the objective function. Let x 1x1 be the number of Royal Flush poker​ sets, let x 2x2 be the number of Deluxe Diamond poker​ sets, and let x 3x3 be the number of Full House poker sets. What is the objective​ function?

Prove that the product of a finite number of compact spaces is compact.

Let f(x) = x + 2/x

a) Use quadratic Lagrange interpolation based on the nodes x₀=1, x₁=2, and x₂=2.5 to approximate f(1.5) and f(1.2)

b) Use cubic Lagrange interpolation based on the nodes x₀=0.5, x₁=1, and x₂=2 to approximate f(1.5) and f(1.2)

Find the lagrange polynomials that approximate f(x) = x³

a ) Find the linear interpolation polynomial P₁(x) using the nodes x₀= -1 and x₁ = 0

_b) Find the quadratic interpolation polynomial P2(x) using the nodes x₀= -1 and x₁ = 0 and x₂ = 1

c) Find the cubic interpolation polynomial P₃(x) using the nodes x₀= -1 and x₁ = 0 and x₂ = 1 and x₃=2

d) Find the linear interpolation polynomial P₁(x) using the nodes x₀= 1 and x₁ = 2

e) Find the quadratic interpolation polynomial P₂(x) using the nodes x₀= 0 and x₁ = 1 and x₂​​​​​​​ = 2

Calculate integlar c F(r)dr when F =[xlny, ye^x] using Green's theorem. R is a rectangle whose vertices are (0,1), (3,1), (3,2), (0,2).

Add in the indicated base

56 base 8 + 75 base 8 =

Subtract in the indicated base

230 base 5 + 32 base 5

Multiply in the indicated base:

42 base 5 x 43 base 5

65 base 7 x 43 base 7

Let S1 and S2 be any two equivalence relations on some set A, where A ≠ ∅. Recall that S1 and S2 are each a subset of A×A.

Prove or disprove (all three):

The relation S defined by S=S1∪S2 is

(a) reflexive

(b) symmetric

(c) transitive

Consider the matrix of Hibert H₁₀₀ and the system HX = b, where b is a column vector from 1 to 100. Solve the system in the following three ways -->

1) X = H^-1 b

2) Using the factorization H = LU

3) Using the factorization H = QR

Premultiplica H for each of the three solution vectors and make the comparison with vector b.

What method gives a better solution?

How many zeros are there at the end of 2019! in octal notation ?