(1 point) A brick of mass 8 kg hangs from the end of a spring. When the brick is at rest, the spring is stretched by 3920 cm. The spring is then stretched an additional 2 cm and released with a downward force of F(t)=143cos(6t) NF(t)=143cos⁡(6t) N acts on it. Assume there is no air resistance. Note that the acceleration due to gravity, gg, is g=980g=980 cm/s22.

1. Find the spring constant  N/cm
   
2. Set up a differential equation that describes this system. Let y(t)y(t) to denote the displacement, in centimeters, of the brick from its equilibrium position, and give your answer in terms of y,y′,y′′y,y′,y″. Assume that positive displacement means the mass below the equilibrium position (when the spring stretched 3920 cm).
3. Solve the differential equation with initial conditions describing the motion/the displacement y(t)y(t) of the mass from its equilibrium position.

Solve the following problems using the two phase method:

max 3x1 + x2

S.t. x1 − x2 ≤ −1

−x1 − x2 ≤ −3

2x1 + x2 ≤ 4

x1, x2 ≥ 0

Please show all steps

8. Let G = GL2(R). (Tables are actually matrices)

(d) Determine whether A = {

} | a, b in R ) is a subgroup of G.

(e) Determine whether B = {

} | b, c in R ) is a subgroup of G.

(f) Determine whether C = {

} | d in R, d not equal 0) is a subgroup of G.

1)Let S be the set of all students at a college. Define a relation on the set S by the rule that two people are related if they live less than 2 miles apart. Is this relation an equivalence relation on S? Justify your answer.

2) Define another relation on the set S from problem 5 by defining two people as related if they have the same classification (freshman, sophomore, junior, senior or graduate student). Is this an equivalence relation on S? Justify.

1)Let S be the set of all students at a college. Define a relation on the set S by the rule that two people are related if they live less than 2 miles apart. Is this relation an equivalence relation on S? Justify your answer.

2) Define another relation on the set S from problem 5 by defining two people as related if they have the same classification (freshman, sophomore, junior, senior or graduate student). Is this an equivalence relation on S? Justify.

what do you understand by parttion values? write a short note on the role of parttion values in real life situations where these can be applied.

Find the zeroes and their order for the following. (Use Taylor series or derivatives)

sin(z)

A 2 liter tank of water contains 3 grams of salt at time t = 0 (in minutes). Brine with concentration 3t grams of salt per liter at time t is added at a rate of one liter per minute. The tank is mixed well and is drained at 1 liter per minute. At what positive time is there a minimum amount of salt and what is that amount?

Problem 11:

A summer resort took a poll of its 350 visitors to see which summer activities people preferred. The results are as follows:

112 of the resorts visitors liked to surf

121 of the resorts visitors liked to Jet Ski

145 of the resorts visitors liked to swim

18 of the resorts visitors liked to surf and Jet Ski

30 of the resorts visitors liked to surf and swim

25 of the resorts visitors liked to Jet Ski and swim

13 liked to participate in all three activities.

1. Draw a Venn Diagram:

2. How many people liked to do none of these activities?

3. How many people liked to surf and swim, but not Jet Ski?

4. How many people like to Jet Ski only?

5. How many people liked to surf or Jet Ski?

6. How many people did not like to swim?

f(x)= 9x^4-2x^3-36x^2+8x/3x^3+x^2-14

-Factor the numerator and denominator of f(x) completely. -Write the domain of f(x) in interval notation. -Locate all hole(s), if any, and write them in the form of coordinate pairs. -Locate all vertical asymptote(s), if any, and give their equations in the form x = c. For each one, describe what happens to f(x) as x approaches c from the left(-), and as x approaches c from the right (+). -Locate the horizontal/slant asymptote, if any, and give its equation in the form y = b (or y = mx+b). -Locate all x- and y-intercepts of f(x), if any, and give as coordinate pairs. -Construct a sign diagram and use test points to determine on which intervals f(x) is positive and negative. -Use all of this information to draw a sketch of the graph. Label all asymptotes, holes, and intercepts, as well as axes and tick marks.

Prove whether or not the set S is countable

a. S= {irrationals}

b. S= {terminating decimals}

c. S= [0, .001)

d. S= Q(rationals) x Q(rationals)

e. S= R(real numbers) x Z(integers)

Let f : N → N and g : N → N be the functions defined as ∀k ∈ N f(k) = 2k and g(k) = (k/2 if k is even, (k + 1) /2 if k is odd).

(1) Are the functions f and g injective? surjective? bijective? Justify your answers.

(2) Give the expressions of the functions g ◦ f and f ◦ g?

(3) Are the functions g ◦ f and f ◦ g injective? surjective? bijective? Justify your answers.

(Discrete Math) Prove that the equation 2x² + y² = 14 has no positive integer solutions.

REAL ANALYSIS I

Prove the following exercises (please show all your work)-

Exercise 1.1.2: Let S be an ordered set. Let A ⊂ S be a nonempty finite subset. Then A is bounded. Furthermore, inf A exists and is in A and sup A exists and is in A. Hint: Use induction.

Exercise 1.1.9: Let S be an ordered set and A is a nonempty subset such that sup A exists. Suppose there is a B ⊂ A such that whenever x ∈ A there is a y ∈ B such that x ≤ y. Show that sup B exists and sup B = sup A.