Without using a calculator, find the cube root of 2, correct to 1 decimal place.

Show that a monotone sequence converges if and only if it is bounded.

a)

Select all solutions of (d^2/dx^2)y(x)+64y(x)=0.

b)

Select all solutions of (d^2/dx^2)y(x)+36y(x)=0.

3. The Bay City Parks and Recreation Department has received a federal grant of $600,000 to expand its public recreation facilities. City council representatives have demanded four different types of facilities—gymnasiums, athletic fields, tennis courts, and swimming pools. In fact, the demand by various communities in the city has been for 7 gyms, 10 athletic fields, 8 tennis courts, and 12 swimming pools. Each facility costs a certain amount, requires a certain number of acres, and is expected to be used a certain amount, as follows:

The Parks and Recreation Department has located 50 acres of land for construction (although more land could be located, if necessary). The department has established the following goals, listed in order of their priority:

(1) The department wants to spend the total grant because any amount not spent must be returned to the government.

(2) The department wants the facilities to be used by a total of at least 20,000 people each week.

(3) The department wants to avoid having to secure more than the 50 acres of land already located.

Give a counterexample:

a) Xn + Yn converges if and only if both Xn and Yn converge.

b) Xn Yn converges if and only if both Xn and Yn converge.

1)Find the power series solution for the equation y'' − y = x

2)Find the power series solution for the equation y'' + (sinx)y = x; y(0) = 0; y'(0) = 1

Provide the recurrence relation for the coefficients and derive at least 3 non-zero terms of the solution.

Find a particular solution to the following non homogenous equations

1) y''' + y = t^3 + sin (t) + 11e^t

2) y'' + y = 2tsin(t)

3) y''''' - 4 y''' = e^2t + t^2 +5t + 4

Solve the initial value problem

11. xdx−y2dy=0, y(0)=1

12. dydx=yx, y(1)=−2

16. dydx=sinxy, y(0)=2

17. xy′=√1−y2, y(1)=0

23. Mr. Ratchett, an elderly American, was found murdered in his train compartment on the Orient Express at 7 AM. When his body was discovered, the famous detective Hercule Poirot noted that Ratchett had a body temperature of 28 degrees. The body had cooled to a temperature of 27 degrees one hour later. If the normal temperature of a human being is 37 degrees and the air temperature in the train is 22 degrees, estimate the time of Ratchett's death using Newton's Law of Cooling.

(abstract algebra)

(a) Find d = (26460, 12600) and find integers m and n so that d is expressed in the form m26460 + n12600.

(b) Find d = (12091, 8439) and find integers m and n so that d is expressed in the form m12091 + n8439.

1. Explain the outcome of 3^4^5. In particular, what is the order of execution of the two exponentiation operations?

2. Write (5^4^3)−1 as a product of prime numbers.

3. The greatest common divisor of two integers a and b can be written as a linear combination (with integer coefficients k and ℓ) of a and b: gcd(a,b)=ka+ℓb.

   In Sage this is achieved with the command xgcd. Look in the help page of this command to write the greatest common divisor of 12214 and 2012 as an integer linear combination of these two numbers.

   Use Sage to verify your result.

4. What is the difference in Sage between 1/3+1/3+1/31/3+1/3+1/3 and 1.0/3+1.0/3+1.0/31.0/3+1.0/3+1.0/3? Explain.

1. Give your own example of a plane figure and its quadrature.
2. How does the quadrature of the triangle depend on the quadrature of the rectangle?
3. Draw a lune like Hippocrates.’ What is the relationship between the large semicircle and the semicircle that contains the lune?

"We want to verify that IP(·) and IP^-1(·) are truely inverse operations. We consider a vector x = (x1, x2, . . . ,x64) of 64 bit. Show that IPfive bits of x, i.e. for xi, i = 1,2,3,4,5.

Sketch the region of continuity for f (x; y) on a set of axes and sketch the region of

continuity for df/dy (x. y) on a separate set of axes. Apply Picard’s Theorem to determine whether the

solution exists and whether it is unique.

a) y'  = 2x²y + 3xy² ; y(1) = 2

b) y' = sqrt(2x - 3y) ; y(3) = 2

Let S be a subset of a vector space V . Show that span(S) = span(span(S)). Show that span(S) is the unique smallest linear subspace of V containing S as a subset, and that it is the intersection of all subspaces of V that contain S as a subset.