(Differential Equations) Consider the differential equation xy’-x⁴y³+y=0

1. Verify that the function y = (Cx²-x⁴)^-1/2 is a solution of the differential equation where C is an arbitrary constant.

2. Find the value of C such that y(-1) = 1. State the solution of the initial value problem.

3. State the interval of existence.

Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x, y, z, and w in terms of the parameters t and s.)

(x, y, z, w) = ( )

*Last person who solved this got it wrong

1. Ross White (see Problem 9) wants to reconsider his decision of buying the brackets and is considering making the brackets in-house. He has determined that setup costs would be $25 in machinist time and lost production time, and 50 brackets could be produced in a day once the machine has been set up. Ross estimates that the cost (including labor time and materials) of producing one bracket would be $14.80. The holding cost would be 10% of this cost.

(a) What is the daily demand rate?

(b) What is the optimal production quantity?

(c) How long will it take to produce the optimal quantity? How much inventory is sold during this time?

(d) If Ross uses the optimal production quantity, what would be the maximum inventory level? What would be the average inventory level? What is the annual holding cost?

(e) How many production runs would there be each year? What would be the annual setup cost?

(f) Given the optimal production run size, what is the total annual inventory cost?

(g) If the lead time is one-half day, what is the ROP?

One-Way ANOVA and Multiple Comparisons

The purpose of one-way analysis of variance is to determine if any experimental treatment, or population, means, are significantly different. Multiple comparisons are used to determine which of the treatment, or population, means are significantly different. We will study a statistical method for comparing more than two treatment, or population, means and investigate several multiple comparison methods to identify treatment differences.

-Search for a video, news item, or article (include the link in your discussion post) that gives you a better understanding of one-way analysis of variance and/or multiple comparison methods, or is an application in your field of study.

-Explain in your post why you chose this item and how your linked item corresponds to our One-Way ANOVA and Multiple Comparisons course objectives.

-Then describe how you could use any of these methods in your future career or a life situation.

Three frogs are placed on three vertices of a square. Every minute, one frog leaps over another frog, in such a way that the "leapee" is at the midpoint of the line segment whose endpoints are the starting and ending positions of the "leaper". Will a frog ever occupy the vertex of the square which was originally unoccupied?

The n- dimensional space is colored with n colors such that every point in the space is assigned a color. Show that there exist two points of the same color exactly a mile away from each other.

Find the vector and parametric equations for the plane. The plane that contains the lines r1(t) = <6, 8, 8,> + t<-2, 9, 6> and r2 = <6, 8, 8> + t<5, 1, 7>.

Given the parametrized curve r(u) = a cos u(1 − cos u)ˆi + a sin u(1 − cos u)ˆj, u ∈ [0, 2π [ , (with a being a constant)

i) Sketch the curve (e.g. by constructing a table of values or some other method)

ii) Find the tangent vector r 0 (u). What is the tangent vector at u = 0? And at u = 2π? Explain your result.

iii) Is the curve regularly parametrized? Motivate your answer using the definition.

iv) Compute the length of the arc corresponding to the interval [0, π/2].

A person borrows money at i^(12) = .12 from Bank A, requiring level payments starting one month later and continuing for a total of 15 years (180 payments). She is allowed to repay the entire balance outstanding at any time provided she also pays a penalty of k% of the outstanding balance at the time of repayment. At the end of 5 years (just after the 60th payment) the borrower decides to repay the remaining balance, and finances the repayment plus the penalty with a loan at i^(12) = .09 from Bank B. The loan from Bank B requires 10 years of level monthly payments beginning one month later. Find the largest value of k that makes her decision to refinance correct.

Let V be a vector space and let U and W be subspaces of V . Show that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V .

The linear transformation is such that for any v in R², T(v) = Av.

a) Use this relation to find the image of the vectors v₁ = [-3,2]^T and v₂ = [2,3]^T. For the following transformations take k = 0.5 first then k = 3,

T₁(x,y) = (kx,y)

T₂(x,y) = (x,ky)

T₃(x,y) = (x+ky,y)

T₄(x,y) = (x,kx+y)

For T₅ take theta = (pi/4) and then theta = (pi/2)

T₅(x,y) = (cos(theta)x - sin(theta)y, sin(theta)x + cos(theta)y)

b) Plot v₁ and v₂ and their images under the transformations. Write a short description saying what the transformations is doing to the vectors.

Use Newton-Raphson to find the real root to five significant figures 64x^3+6x^2+12-1=0. First graph this equation to estimate. Use the estimate for Newton-Raphson

“A real number is rational if and only if it has a periodic decimal expansion”

Define the present usage of the word periodic and prove the statement

1) Determine whether these statements are true or false. Please explain why so I can understand where the answers came from

a) ∅ ∈ {∅}

b) ∅ ∈ {∅, {∅}}

c) {∅} ∈ {∅}

d) {∅} ∈ {{∅}}

e) {∅} ⊂ {∅, {∅}}

f ) {{∅}} ⊂ {∅, {∅}}

g) {{∅}} ⊂ {{∅}, {∅}}

2) Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Find

a) A x B x C

d) B x B x B

3) Show that if A and B are sets, then

b) (A ⊕ B) ⊕ B = A. (prove every step! you may use the fact that symmetric difference of sets is associative)