Find both the maximum and minimum of the objective function y − 8x given these constraints. (If an answer does not exist, enter DNE.)

5x-2y≤13

y≥-4

y-7x≤31

2x+7y≤13

5. Consider the differential equation

xy^5/2 +1+x^2y^3/2dy/dx =0

1. (a) Show that this differential equation is not exact.

2. (b) Find a value for the constant a such that, when you multiply the d.e. through by xa, it becomes exact. Show your working. Do NOT solve the resulting differential equation.

6. Consider the differential equation

(D − 3)(D − 4)y = 0.

1. (a) Solve this d.e., showing brief working.

2. (b) How many solutions does this d.e. have? Justify your answer.

3. (c) How many independent solutions does this d.e. have? Provide a calculation to justify your answer.

Suppose the population of fish in a lake in month t is given by the equation F(t) = 800 − 500e^-0.2t

(a) Plot the equation for t in the interval [0,24].

(b) Is the function increasing, decreasing or neither? Is it concave up, down, or neither?

(c) As t → ∞, F (t) →( ). Fill in the blank and explain what this means in words.

Find the work done by the vector vield

F(x, y) =

3x+3x²y, 3y²x+2x³

on a particle moving first from

(−3, 0),

along the x-axis to (3, 0), and then returning along

y =

back to the starting point.

A graphing calculator is recommended.

For the limit

lim x → 2 (x³ − 3x + 8) = 10

illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1. (Round your answers to four decimal places.)

1. For the following function ?(?) = (?^2−8?+16) / (?^2−4)

a. Find the critical values

b. Use the FIRST DERIVATIVE TEST to determine the intervals where the function is INCREASING and DECREASING.

c. Find the RELATIVE EXTREMA of the function and state where they occur.

d. Find the ABSOLUTE EXTREMA of the function on the interval [−1, 1.75]

Show that a dilation σ by a factor of r scales all distances by a factor of r. That is, for all points A and B, we have:

|σ(A)σ(B)| = r|AB|

Assume that at a price of $2.00 per pound, the annual supply of coffee beans in Country A is 8 million pounds, while the demand is 10 million pounds. At a price of $3.00 per pound, the supply is 10.2 million pounds, and the demand is 8.6 million pounds. Assume that the price-supply and price-demand equations are linear.

1. Write an equation for each (price on the y-axis)

2. Find the equilibrium point (point of interception of the two linear equations)

3. Discuss the significance of the equilibrium point in this case

4. Graph the two equations in the same Cartesian system (upload)

first and second derivative of f(x)= (1-x)arctan(2x)

also find newtons method formula

Josh believes the Spanish club students at his school have an unfair advantage in being assigned to the Spanish class they request. He asked 500 students at his school the following questions: "Are you in the Spanish club?" and "Did you get the Spanish class you requested?" The results are shown in the table below:

Help Josh determine if all students at his school have an equal opportunity to get the Spanish class they requested. Show your work and explain your process for determining the fairness of the class assignment process.

Obtain the general solution to the following equation.

(x²+25)dy/dx + xy = 2x y(0)=4

The side of a rhombus forms two angles with its diagonals such that their difference is equal to 30 degrees. What are the angles of the rhombus?

Prove that the quadrilateral enclosed by the perpendicular bisectors of the sides of a rhombus is a rhombus.