Use direct substitution to verify that y(t) is a solution of the given differential equation in Exercise Group 1.1.9.15–20. Then use the initial conditions to determine the constants C or c1 and c2.

17. y′′+4y=0, y(0)=1, y′(0)=0, y(t)=c1cos2t+c2sin2t

18. y′′−5y′+4y=0,   y(0)=1 , y′(0)=0,   y(t)=c1et+c2e4t

19. y′′+4y′+13y=0, y(0)=1, y′(0)=0, y(t)=c1e−2tcos3t+c2e−3tsin3t

27. The growth of a population of rabbits with unlimited resources and space can be modeled by the exponential growth equation, dP/dt=kP.

Write a differential equation to model a population of rabbits with unlimited resources, where hunting is allowed at a constant rate α.

Write a differential equation to model a population of rabbits with unlimited resources, where hunting is allowed at a rate proportional to the population of rabbits.

Write a differential equation to model a population of rabbits with limited resources, where hunting is allowed at a constant rate α.

Write a differential equation to model a population of rabbits with limited resources, where hunting is allowed at a rate proportional to the population of rabbits.

30. Radiocarbon Dating.

Carbon 14 is a radioactive isotope of carbon, the most common isotope of carbon being carbon 12. Carbon 14 is created when cosmic ray bombardment changes nitrogen 14 to carbon 14 in the upper atmosphere. The resulting carbon 14 combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis. Animals acquire carbon 14 by eating plants. When an animal or plant dies, it ceases to take on carbon 14, and the amount of isotope in the organism begins to decay into the more common carbon 12. Carbon 14 has a very long half-life, about 5730 years. That is, given a sample of carbon 14, it will take 5730 years for half of the sample to decay to carbon 12. The long half-life is what makes carbon 14 dating very useful in dating objects from antiquity.

Consider a sample of material that contains A(t) atoms of carbon 14 at time t. During each unit of time a constant fraction of the radioactive atoms will spontaneously decay into another element or a different isotope of the same element. Thus, the sample behaves like a population with a constant death rate and a zero birth rate. Make use of the model of exponential growth to construct a differential equation that models radioactive decay for carbon 14.

Solve the equation that you proposed in (a) to find an explicit formula for A(t).

The Chauvet-Pont-d'Arc Cave in the Ardèche department of southern France contains some of the best preserved cave paintings in the world. Carbon samples from torch marks and from the paintings themselves, as well as from animal bones and charcoal found on the cave floor, have been used to estimate the age of the cave paintings. If a particular sample taken from the Cauvet Cave contains 2% of the expected cabon 14, what is the approximate age of the sample?

Give a direct proof for the 2nd Isormorphism Theorem of bi-modules over rings.

(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