1) Suppose that a function f(x) is defined for all real values of x, except x = x_o. Can anything be said about lim x → x 0 f ( x ) ?

Give reasons for your answer.
2) If x⁴ ≤ f(x) ≤ x² for x in [-1,1] and x² ≤ f(x) ≤ x⁴ for x < -1 and x > 1, at what points c do you automatically know lim x → c f ( x ) ? What can you say about the value of the limits at these points (x = +/-1) and at x = 0?

3) Explain why the following statement is true or false:
If g is continuous and increasing on its entire domain, then g(x₂) > g(x₁) when x₁ < x₂

Identify the letter name, the quality, and quantity of each of the following propositions. Also, state whether their subject and predicate terms are distributed or undistributed.

Some residents of Manhattan are not people who can afford to live there.

Produce an equivalent proposition without implications (">") and without not's ("-") using DeMorgan's Laws and the implication rule.

-Ǝ x Ǝy ∀z [ [ ( -X>Z ) > (Z > -Y)] + -(Y + Z )]

1. Use a Laplace transform to solve the initial value problem: 9y" + y = f(t), y(0) = 1, y'(0) = 2

2. Use a Laplace transform to solve the initial value problem: y" + 4y = sin 4t, y(0) = 1, y'(0) = 2

Let x,y ∈ R3 such that x = (x₁,x₂,x₃) and y = (y₁,y_2,y₃) determine if <x,y>= x1y1+2x₂y₂+3x₃y₃   

0. is an inner product

a loan of 17000 is to be repaid by 4 equal payments due in 2,5,6and 9 years respectively. determine the size of the equal payments if the loan was granted at 12% p.a. compounded semi- annually.

Use the elimination method to find a general solution for the given linear​ system, where differentiation is with respect to t.

x'=9x-2y+sin(t)

y'=25x-y-cos(t)

solve the IVP

11/32y''+2/10y'+132/5y=6cos(3t), y(0)=0, y'(0)=0

At one of its factories, a manufacturer of college logo sweatshirts makes two styles: crewneck pullover and hooded. Each pullover sweatshirt takes 5 minutes to cut out and 20 minutes to assemble and finish. Each hooded sweatshirt takes 5 minutes to cut out and 30 minutes to assemble and finish. The plant has enough workers to provide at most 3,750 minutes per day for cutting and at most 19,500 minutes per day for assembly and finishing. The profit on each pullover sweatshirt is $6 and the profit on each hooded sweatshirt is $8. How many of each sweatshirt should be produced each day to obtain maximum profit? Find the maximum daily profit.

#1 Part 1: Find the payback in years (to the nearest hundredths place) for the following cash flow with a WACC of 4%:

Time Period                 Cash Flow                   Cumulative Out of Pocket

       0                                -100                                         -100

       1                                   40                                           -60

       2                                   50                                           -10

       3                                   20                                          +10

       4                                   70                                          +80

#1 Part 2: Find the discounted payback in years (to the nearest hundredths place) for the following cash flow with a WACC of 12%. Hint: interpolation must be used and I have started the table for you.

Time Period                 Cash Flow                   PV of Cash Flow        Cumulative

         0                              -100                              -100                           -100

         1                                  40                             35.71                        -64.29

         2                                  50                                    ?                                 ?

         3                                  20                                    ?                                 ?

         4                                  70                                    ?                                 ?                       

Reminder, your payback numbers are in units of years.

#2 Calculate the MIRR of the cash flows of the project below. Assume both the finance rate and the reinvestment rate are 5%

Time Period                 Cash Flow

         0                             -100

         1                                20

         2                                80

         3                                90

The problem below involves three variables. Solve it with the simplex method, Excel, or some other technology.

Patio Iron makes wrought iron outdoor dining tables, chairs, and stools. Each table uses 8 feet of a standard width wrought iron, 2 hours of labor for cutting and assembly, and 2 hours of labor for detail and finishing work. Each chair uses 6 feet of the wrought iron, 2 hours of cutting and assembly labor, and 1.5 hours of detail and finishing labor. Each stool uses 1 footof the wrought iron, 1.5 hours for cutting and assembly, and 0.5 hour for detail and finishing work, and the daily demand for stools is at most 16. Each day Patio Iron has available at most 156 feet of wrought iron, 70 hours for cutting and assembly, and 50 hours for detail and finishing. The profits are $60 for each dining table, $48 for each chair, and $36 for each stool.

Suppose Patio Iron wants to maximize its profits each day by making dining tables, chairs, and stools.

Let x be the number of dining tables, y be the number of chairs, and z be the number of stools made each day.

Let f be the maximum profit (in dollars). Form the profit equation that needs to be maximized.

f = ____

Since Patio Iron has available at most 156 feet of wrought iron, form the constraint inequality for the total feet of wrought iron used for dining tables, chairs, and stools.

___ ≤ 156

Since Patio Iron has available at most 70 hours for cutting and assembly, form the constraint inequality for the total hours spent on cutting and assembling dining tables, chairs, and stools.

___ ≤ 70

Since Patio Iron has available at most 50 hours for detail and finishing work, form the constraint inequality for the total hours spent on detailing and finishing dining tables, chairs, and stools.

___ ≤ 50

Since the daily demand for stools is at most 16,  ---Pick one--- x ≤ 16 or y ≤ 16 or z ≤ 16 .

How many of each item should be made each day to maximize profit? Solve with the simplex method, Excel, or some other technology.

tables ___

chairs ___

stools ___

Find the maximum profit.

$ ___

Consider a transformation T : R2×2 → R2×2 such that T(M) = MT .
This is infact a linear transformation. Based on this, justify if the following
statements are true or not. (2)
a) T ◦ T is the identity transformation.
b) The kernel of T is the zero matrix.
c) Range T = R2×2
d) T(M) =-M is impossible.

Solve each of the following ODEs. If initial conditions are given, give the unique solution.

d) y''' − 5y'' + 2y' + 8y = 0, y(0) = 2, y'(0) = −1, y''(0) = −5

e) y''' + 9y'' + 27y' + 27y = 0, y(0) = 2, y'(0) = 0, y''(0) = 3

a) x^2y'' + xy' + 4y = 0, x > 0

b) x^2y'' − 6y = 0, x > 0, y(1) = y'(1) = 5

c) x^2y'' + 5xy' + 4y = 0, x > 0, y(1) = 5, y'(1) = −9

A page is to have a total area of 96 square inches. The top and bottom margins will be 1 inch each, and the left and right margins will be 1_^1⁄2 inches each as shown in the figure below.

Required:

1. a) What should be the overall dimensions “x” and “y” that will maximize the area of the space

   inside the margins (the printed area)?

2. b) Also give the proof that these values of “x” and “y” will maximize this area.