Apply the eigenvalue method to find a general solution of thegiven system.

x'₁ = 5x₁ -9x₂, x'₂ =2x₁ - x₂

The initial mass of a certain species of fish is 7 million tons. The mass of fish, if left alone, would increase at a rate proportional to the mass, with a proportionality constant of 2/yr initially, but over time if the population gets too high it would reach a maximum sustainable population of 50 million tons.

1. Suppose, commercial fishing removes fish mass at a constant rate of 15 million tons per year. Set up the initial value problem which determines the fish mass, F(t), in millions of tons at time t in years.

2. For your harvesting model above, what will the long term behavior of the fish population be?

If you graph the function y = 3^x  and y = 3 ^–x  the graphs create a reflection about the y-axis.  Explain why this occurs.

Now, if you graph the function y = 3^x  and y = -(3^x)  the graphs create a reflection about the x-axis instead.  Explain why, in detail,  the reflection changes from being about the y-axis to being about the x-axis.

Now, know the answers to #11 and #12, if we take the base function y = 3^x what would cause a reflection about the origin?  Explain why this happens.

Prove that the dual space of l^1 is l^infinity

Queen Chloe is planning to build a castle inside of a rectangular moat. A river runs horizontally along the bottom of her land, and that river will form the bottom of the moat, but the other three sides must be dug by a ditch digging company. For the two vertical sections of the moat, the company will charge 1 gold piece per meter. But because the moat digging becomes easier further from the river, the company offers a discount to dig further from the river on the horizontal section. For the horizontal section, the price per meter is 1 gold piece divided by the vertical distance from the river.

If Queen Chloe has 12 gold pieces, what are the dimensions of the moat that will give her the maximum area for her castle? How do you know that these dimensions maximize the area?

Find the absolute maximum of g(x,y)=x^2+y^2-2y+1 on the disk x^2+y^2 less than or equal to 4. Solve 2 ways, parametrization and lagrange multipliers. Please solve using both methods!

What is impulse response, why it is important? How else is it called?

Set up an integral for the following scenarios:

Set up the integral in simplified form, do not integrate

a) Arc length of y = ln x , 2 ≤ x ≤ 4 .

b) The surface area generated by rotating y = sin x with respect to the x-axis, 0 ≤ x ≤ π .

c) The arc length of y = x 2 + 4 , 1≤ x ≤ 3 .

d) The surface area generated by revolving y = ln(cos x) about the xaxis, € 1≤ x ≤ π 4 .

Find (without using a calculator) the absolute extreme values of the function on the given interval.

f(x) = 3x² − x³ on [0, 3]

1A) Solve by substitution.

2x − y + 3 = 0 x2 +y2 −4x=0

1B)

Solving system by Gaussian Elimination and then by back substitution
3x − 3y + 6z = 6

x + 2y − z = 5 5x −8y +13z = 7

Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.

D is bounded by y = 1 − x² and y = 0; ρ(x, y) = 5ky

m=

(x bar ,y bar)=

13) For the given cost function C(x) = 48,400 + 600x + x²,

First, find the average cost function. Use it to find:

a) The production level that will minimize the average cost
x =     

b) The minimal average cost
$ =

14) Suppose a product's revenue function is given by R(q)=−4q²+400q, where R(q) is in dollars and q is units sold. Find a numeric value for the marginal revenue at 31 units.

MR(31) = ? $ per unit

Find the general solution:

dx/dt + x/(1+2t) = 5

Determine if exact equation, then solve:

x(y+2)y' = lnx+1 , y(1) = -1