1. Sketch a connected 2-dimensional complex with three 2-cells, 8 1- cells, and 5 0-cells.

2. Draw three examples of a connected 1-dimensional complex with 7 0-cells and 6 1-cells.

3. Sketch a connected 1-dimensional complex with 7 0-cells and 7 1-cells.

4. Sketch a connected 1-dimensional complex with 7 0-cells and 8 1-cells.

5. Is it possible to construct a connected 1-dimensional complex with 8 0-cells and 6 1-cells? Explain.

T:R ->R3 T(x, y, z) = (2x + 5y − 3z, 4x + y − 5z, x − 2y − z) (a) Find the matrix representing this transformation with respect to the standard basis. (b) Find the kernel of T, and a basis for it. (c) Find the range of T, and a basis for it.

Let X be the set of all subsets of R whose complement is a finite set in R:

X = {O ⊂ R | R − O is finite} ∪ {∅}

a) Show that T is a topological structure no R.

b) Prove that (R, X) is connected.

c) Prove that (R, X) is compact.

1. The equation of the line with an x-intercept of 33 and a y-intercept of 44 can be written in the form y=mx+b where
the number m is:    
the number b is:    

Enter each answer as a reduced fraction (like 5/3, not 10/6) or as an integer (like 4 or -2).

2. You have filled your car with a full tank of gas, and could travel 485 miles. For every 17 miles you drive, you use 1 gallon of gass (17 miles per gallon).

Please write an equation in slope-intercept form to model this situation. Please use m for the the miles you've traveled, and g for the gallons of gas you use.    

The displacement of an object in a spring-mass system in free damped oscillation is:

2y''+12y'+26y=0 and has solution: y1 =−5e^(−3t)*cos(2t−0.5π)

if the motion is under-damped.

If we apply an impulse of the form f(t) = αδ(t−τ) then the differential equation becomes :

2y''+12y'+26y=αδ(t−τ) and has solution y =−5e^(−3t)cos(2t−0.5π)+αu(t−τ)w(t−τ) where w(t) = L(^−1)*(1/(2s^2+12s+26))

a. When should the impulse be applied? In other words what is the value of τ so that y1(τ) =0 . Pick the positive time closest to t = 0

b. What intensity should the impulse be so that the object is in equilibrium for t > τ (i.e. what should the value of α be so that y(t)=0 for t>τ ).

c. Write up your work for this problem including the inverse transform needed to write out w(t) and the completed form of the impulse function f(t)   

5. Find a matrix A of rank 2 whose nullspace N(A) has dimension 3 and whose transposed nullspace N(AT) has dimension 2.

In how many ways can you distribute 10 different balls into 4 different boxes, so there's no box with exactly 3 balls? Use inclusion-exclusion

Monica bought a $2000 4K Ultra HD TV for 10% down, with the balance to be paid with interest at 13% compounded monthly in six equal monthly payments. Construct the full amortization schedule for the debt. Calculate the total interest paid.

Show that the transformation w = z ^1/ 2 maps usually maps vertical and horizontal lines onto portions of hyperbolas. Investigate the lines x = a, y = b for the cases

1) a > 0, b > 0,

2) a = b =0

NATURAL DEDUCTION

1. Construct a proof that reveals the validity of the following inference. Make sure that you symbolize it first.

Tom was involved in the fight at the party Friday and stole money from Monica’s purse later that same night. The parole officer had told me that if he was involved in the fight then he’d have to go back to the rehab clinic and the social worker told me that if he stole the money then control over his savings account is going to be turned over to a representative from the office of juvenile corrections. Well, it follows that Tom is going back into rehab and he’s going to lose control over his savings.

.

Problem 4 [20 pts]: Playing Cards Let C represent a set of 52 playing cards with four suits (♥, ♦, ♣, ♠) each having 13 ranks (Ace,2,3,4,5,6,7,8,9,10,Jack,Queen,King). We define the following additional sets. F =Face cards (Jack, Queen, and King). R =Red cards. P =Ranks that are prime (2,3,5,7). J =One-eyed Jacks (Jack of Hearts, Jack of Spades).

1. Depict these sets as a Venn Diagram and show the cardinality of each distinct region. (The regions don’t have to be perfectly to scale - this can be hand-drawn.)

2. Using set notation, give an expression for the set of cards that are red or face cards or prime numbered or one-eyed Jacks and give its cardinality.

3. Give a set expression and cardinality for the set of cards that are not face cards or not prime-numbered cards.

4. Give a set expression and cardinality for the complement of the set of cards that are either red or prime-numbered but not one-eyed Jacks.

5. Give a set expression and cardinality for the set of cards that are either red non-prime cards or one-eyed Jacks, but not both.

Honeywell Bakery has been offering a new vegan cupcake and wants to know the demand in the future. For the past six weeks, the demand has been 60 cupcakes, 55 cupcakes, 75 cupcakes, 60 cupcakes, 80 cupcakes, and 75 cupcakes respectively. 1. a) Using a simple four-week moving average, calculate a forecast for next week. (4 points) b) If the forecast for week 6 was 85 cupcakes, then using single exponential smoothing with α=0.2, calculate a forecast for next week. (4 points) 2. a) Using simple linear regression, calculate the trend line for the historical data. X- Axis should represent the period (e.g. week 1 = 1, week 2 = 2, and so on), while the Υ axis is demand. (12 points) b) Calculate a forecast for next week. Is this forecast different from the ones calculated in 1a and 1b? If so, please explain the difference. (10 points)

Consider the following regression results of B for time series data. The dependent variable is the log of real consumption and the regressors are all lagged by one period. The coefficients are estimated by OLS method. *, **, and *** indicate that the coefficients are significant at the 10%, 5%, and 1% level, respectively. The F- test in the last row tests the joint hypothesis that all the coefficients except the constant are zero and their p-values are provided in the table. The empty space means that the variables are not included in the regression. (4) 3.1213*** 3.8876*** (1) Constant log(Real Income) 0.8683* Interest Rate Inflation Rate log(Stock Price) F-test 0.082 (2) (3) 3.3962*** 3.8115*** 0.6289*** 0.6341*** -0.2680** -0.2402** 0.5233 0.5230 -0.2504*** 0.6237 0.4775 0.064 0.095 0.044 (a) Why do we have to include lagg values of regressors in the equation? (b) Compare the results of equations (1) and (2) and make any comments. (c) Compare the results of equations (2) and (3) and make any comments. (d) Compare the results of equations (3) and (4) and make any comments.