Let⇀F(x,y) =xi+yj/(e^(x^2+y^2))−1. Let C be a positively oriented simple closed path that encloses the origin.

(a) Show that∫F·Tds= 0.

(b) Is it true that ∫F·Tds= 0 for any positively oriented simple closed path that does not pass through or enclose the origin? Justify your response completely.

Problem 5. During droughts, water for irrigation is pumped from the ground. When ground water is pumped excessively, the water table lowers. In California, lowering water tables have been linked to reduced water quality and sinkholes. A particular well in California’s Inland Empire has been monitored over many years. In 1994, the water level was 250 feet below the land surface. In 2000, the water level was 261 feet below the surface. In 2006, the water level was 268 feet below the surface. In 2009, the water level was 271 feet below the surface. In 2012, it was 274 feet below the surface. And in 2015, it was 276 feet below the surface. (a) Find a cubic (degree 3) polynomial model for this data on water level. First, define what x and y mean here, and write the data points you use. Then, find a cubic which is a best fit for this data, in the least squares sense. (b) Use your model to predict the water level of the well, in feet below the surface, in 2020. You may assume that the trends of 1994 to 2015 continue to 2020.

Disprove by counter-example that (? ∩ ?) ∪ (? ∩ ?) ⊆ ((? ∩ ?) ∪ ?) ∩ ?

Show by any valid method except Venn diagram that ((? ∪ ?) ∩ ?) ∩ ((? ∪ ?) ∩ ?) = ? ∩ ?

Show by universal generalization that ((? ∩ ?) ∪ ?) ∩ ? ⊆ (? ∩ ?) ∪ (? ∩ ?)

Use discrete math please and show all the work!

Thanks!

Solve x′=x−8y, y′=x−3y, x(0)=2, y(0)=1

U_n = {x ∈ Z_n^* | x & n are relatively prime}; w/ operator multiplication modulo(n)

show: U_n is a commutative group.

Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that

B = P⁻¹AP

is the diagonal form of A. Prove that

A^k = PB^kP⁻¹,

where k is a positive integer.

Use the result above to find A⁵

A =

A mass 7kg of stretches a spring 18cm. The mass is acted on by an external force of 5sin(t/2) N and moves in a medium that imparts a viscous force of 4N when the speed of the mass is 8cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 4cm/s, determine the position of the mass at any time t. Use 9.8m/s2 as the acceleration due to gravity. Pay close attention to the units.

u(t) = ? m

Question

Suppose you are given a 7 digit number which is a UPC. Prove that if a mistake is made when scanning the number, causing one digit to be read incorrectly, then you will be able to tell that an error has been made.

Extra information.

A number is a Universal Product Code (UPC) if its last digit agrees with the following computations:

• The sum of the odd position digits (not including the last) is M. That is we add the first digit to the third digit to the fifth digit etc.

• The sum of the even position digits (not including the last) is N. •

c = (3M + N)%10.

• If c = 0 then the check digit is 0.

• If c 6= 0, then the check digit is 10 − c.

An example with number 1231242.

1) you add all odd-positioned digits except the last one:

M=1+3+2=6

2) add all even positioned digits, not including the last one:

N=2+1+4=7

3) c=(3M+N)%10=(6*3+7)%10=5

4) the check digit is 10-5=5

So 1231242 is not a UPC.

However, if we change the last digit to be 5, then it will be UPC. That is 1231245 is a UPC

Answer each of the following questions in your submission for this part of the project:

1. List your 4 variables from the Consensus at School (opens in a new window) and state whether they are qualitative or quantitative.
2. Using your four variables, write your two questions you will analyze based on data from Census at School.
3. What is your population?
4. What type of sampling will you use to gather your data (stratified or simple random)?
5. What is cluster sampling and why would this type of sampling not work for this type of data gathering? What would it look like to use systematic sampling for this data?

Answer the following questions by entering your response in the space provided below.

1. What are your two questions you would like to analyze based on data from Census at School?
2. List your 4 variables and whether they are qualitative or quantitative.
3. What is your population?
4. What type of sampling will you use to gather your data (stratified or simple random)?
5. What is cluster sampling and why would this type of sampling not work for this type of data gathering? What would it look like to use systematic sampling for this data?

2. Given A = | 2 1 0 1 2 0 1 1 1 |.

(a) Compute eigenvalues of A.

(b) Find a basis for the eigenspace of A corresponding to each of the eigenvalues found in part (a).

(c) Compute algebraic multiplicity and geometric multiplicity of each eigenvalue found in part (a).

(d) Is the matrix A diagonalizable? Justify your answer

5. Compute a basis for each eigenspace of matrix A = | 6 3 −4 −1 | (a) Diagonalize A. (b) Compute A4 and A−3 . (c) Find a matrix B such that B2 = A. (or compute √ A.)

differential equation

12. Newton's law of cooling states that the temperature of an object changes at a rate pro-portional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 200F when freshly poured, and 1 min later has cooled to 190F in a room at 70F, determine when the coffee reaches a temperature of 150F.

A tank contains 70 kg of salt and 2000 L of water. Water containing 0.4kg/L of salt enters the tank at the rate 16L/min. The solution is mixed and drains from the tank at the rate 4L/min. A(t) is the amount of salt in the tank at time t measured in kilograms.
(a) A(0) =  (kg)
(b) A differential equation for the amount of salt in the tank is  =0=0. (Use t,A, A', A'', for your variables, not A(t), and move everything to the left hand side.)
(c) The integrating factor is
(d) A(t) =  (kg)
(e) Find the concentration of salt in the solution in the tank as time approaches infinity. (Assume your tank is large enough to hold all the solution.)
concentration =  kgL

1. Write a statement that satisfies the following:

• universal statement (for all, for every, etc.)
• uses variables and a set (integers, rational numbers, or real numbers)
• the predicate is a conditional (if...then...) statement
• the predicate contains at least one of the following terms: even, odd, prime, composite, rational, irrational
• the statement is true
• the statement is not taken from the textbook

2. Give the statement you would use to give a proof by contraposition.

3. Give the statement you would assume in a proof by contradiction.

4. Decide which proof method you think would be easiest to use to prove your original statement: direct proof, proof by contraposition, or proof by contradiction.

Evaluate the line integral, where C is the given curve, where C consists of line segments from (1, 2, 0) to (-3, 10, 2) and from (-3, 10, 2) to (1, 0, 1).

C zx dx + x(y − 2) dy