(a) Give an example of a collection of closed sets whose union is not closed.

(b) Give an example of a collection of open sets whose infinite intersection is not open.

Thank you!

Find all the subgroups of the group of symmetries of a cube. Show all steps.

Hint: Label the diagonals as 1, 2, 3, and 4 then consider the rotations to get the subgroups.

For each pair a, b with a ∈ R − {0} and b ∈ R, define a function fa,b : R → R by fa,b(x) = ax + b for each x ∈ R.

(a) Prove that for each a ∈ R − {0} and each b ∈ R, the function fa,b is a bijection.

(b) Let F = {fa,b | a ∈ R − {0}, b ∈ R}. Prove that the set F with the operation of composition of functions is a non-abelian group. You may assume that function composition is associative.

A soccer ball has 32 faces, each of which is a regular pentagon or hexagon. Because of the angles involved, exactly three faces meet at each corner. Without looking at the ball, determine how many of each type of face there are.

Find the solution of the following differential equations

(x(x-1))dy-(xy+2x³-x²-2y)dx=0

2.) Use the method of Lagrange multipliers to find the maximum and minimum values of the function ?(?, ?) = ??^2 − 2??^2 given the constraint ?^2 + ?^2 = 2 along with evaluating the critical points of the function, find the absolute extrema of the function ?(?, ?) = ??^2 − 2??^2 in the region ? = {(?, ?)|?^2 + ?^2 ≤ 2}.

1. Recall that the set {0,1}∗ is the set of all finite-length binary strings. Let f:{0,1}∗→{0,1}∗ to be f(x1x2…xk)=x2x3…xkx1. That is, f takes the first bit of a string x and moves it to the end of x, so for example a string 100becomes 001; if |x|≤1, then f(x)=x Also, suppose that g:{0,1}∗→{0,1}∗ is a function such that g(x1…xk)=0x1…xk (that is, gg puts an extra 0 in front of the given string, so for example g(100)=0100. Everywhere in this question we will refer to these f and g.
   1. Then  f(0011010)= and f(1)=
   2. Then g(000)=andg(1)=
   3. Then f−1(0011010)=
   4. Is f one-to-one? Is it onto? Is it a bijection?
   5. Is g one-to-one? Onto? Bijection?
   6. Calculate f(g(100101)) and g(f(100101)).
   7. Which of the following is true for these ff and gg? Justify your answer.
      1. ∀x∈{0,1}∗,f(g(x))=g(f(x))
      2. ∀x∈{0,1}∗,f(g(x))≠g(f(x))
      3. Neither

1.Prove the following statements:

.

(a) If bn is recursively defined by bn =bn−1+3 for all integers n≥1 and b0 =2,

then bn =3n+2 for all n≥0.

.(b) If cn is recursively defined by cn =3cn−1+1 for all integers n≥1 and c0 =0,

then cn =(3n −1)/2 for all n≥0.

.(c) If dn is recursively defined by d0 = 1, d1 = 4 and dn = 4dn−1 −4dn−2 for all integers n ≥ 2,

then dn =(n+1)2n for all n≥0.

Question 1. Go to random.org. This website is a random number generator. Use it to generate three numbers a, b, c between -10 and 10. Now let your a, b and c be the coefficients of the quadratic function

f(x)=ax2 +bx+c.

(For example, if the numbers you generated happened to be a = 2,b = 12, c = −1, your function for the rest of the question would bef(x) = 2x2 +12x−1.)

1. (a) Put f(x) into “standard” or “vertex” formf(x)=a(x−h)2 +k.

2. (b) Identify the location of the vertex and determine whether it is a local minimum or local maximum.

3. (c) From the “standard form” determine whether the equationf(x) = 0

   has any real solutions. If it has real solutions then find them.

   Otherwise, explain how you know that it has no real solutions.

4. (d) Use the quadratic formula to confirm what you found in (c). If the equation has no real solutions then use the quadratic formula to

   find the complex solutions.

Question 2. By changing one of your coefficients a, b, c in Question 1, create a quadratic that has a different number of real roots. That is, if your function had no real roots then your goal is to change it to one which does have real roots. If your function did have real roots then your goal is to build one which has no real roots. Once you have created a quadratic function which is what you are looking for (either

with or without real roots), repeat Question 1 using it.

Prove that all parallel lines have the same vanishing point.

if L1(x) = x is a solution to the ODE:

(1-x)L'' - 2xL' +2L = 0

a.) Show that L1(x) = x is a solution to the ODE

2.) Use reduction of order to find another solution L2(x)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = x³ + y³ − 3x² − 3y² − 9x

Solve y''-y=e^t(2cos(t)-sin(t)) with initial condition y(0)=y'(0)=0

Use induction to prove

Let f(x) be a polynomial of degree n in Pn(R). Prove that for any g(x)∈Pn(R) there exist scalars c0, c1, ...., cn such that g(x)=c0f(x)+c1f′(x)+c2f′′(x)+⋯+cnf(n)(x), where f(n)(x)denotes the nth derivative of f(x).