using / for integral

Evaluate the double integral //R cos( (y-x)/(y+x) )dA where R is the trapezoidal region with vertices (1,0), (2,0), (0,2), and (0,1)

A manufacturer of animal feed makes two grades of food. Each bag of high grade feed contains 10kg of wheat brand and 5kg of maize, while each bag of low-grade feed contains 12kg of wheat brand and 3kg of maize. There are 1920kg of wheat brand and 780kg of maize currently available. The manufacture can make a profit of ¢12,000 on each bag of high grade and ¢10000 on each bag of the low-grade feed. Determine the number of bags of each grade to produce to maximize profit.

For a binary FSK system, s1(t)=A cos (ω0+Ω)t and s2(t)=A cos (ω0-Ω)t are defined.

a) Prove that the above signals are orthogonal if ΩT=nπ, where T is the bit interval and n is a positive integer.

b) Determine probability of error for the system

*(1)(a) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): z=c}.

(b) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): x=a}.

(c) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): y=b}.

*(2) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): z=kx+b} assuming both b and k are positive.

(a) For what value of k is this an hyperbola and for what value of k is this an ellipse?

(b) Plot one of each.

Justify your answer.

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.