5. For each piecewise linear function, graph the function and find:      2 lim x f x      2 lim x f x (a) { ? + 3 ?? ? < 2 2? + 1 ?? ? ≥ 2 (b) ?(?) = { ? − 2 ?? ? < 2 ? + 1 ?? ? ≥ 2 6. For each piecewise linear function, graph the function and find:    x  0 lim f x    x  0 lim f x   x 0 lim f x   (a) ?(?) = −|?| + 4 (b) ?(?) = − 2|?| �

Problem 1 [20 pts]: An experiment consists of tossing a coin 6 times. Let X be the random variable that is the number of heads in the outcome. Find the mean and variance of X.

Thank You

Show that a set is convex if and only if its intersection with any line is convex. Show that a set is affine if and only if its intersection with any line is affine.

Factor the following polynomials as a product of irreducible in the given polynomial ring.

A. p(x)=2x^2+3x-2 in Q[x]

B. p(x)=x^4-9 in Q[x]

C. p(x)=x^4-9in R[x]

D. p(x)=x^4-9in C[x]

E. f(x)=x^2+x+1 in R[x]

F. f(x)=x^2+x+1in C[x]

Let A =

(a) Calculate the matrix exponential e^(At). (Hint: It might help to write down the power series expansions for the hyperbolic functions
cosh(t) =(e^t + e^(−t))/2
and sinh(t) =(e^t −e^(−t))/2
and then try to write eAt in terms of these two functions.)
(b) Use your matrix from part (a) to solve the nonhomogeneous initial value problem
x' =

x +

, x(0) =

. (Hint: You might need the identity cosh^2(t)−sinh^2(t) = 1.)

If a set K that is a subset of the real numbers is closed and bounded, then it is compact.

Show that (0, 1) and (−1, 1) have the same cardinality.

Let set E be defined as E={?∙?? +? | [?]∈R2}, where ?? is the natural exponential?

function, please show E is a vector space by checking all the 10 axioms. (Notice: you may use the properties of vector addition and scalar multiplication in R2)

If an undamped spring-mass system with a mass that weighs 6 lb and a spring constant 9 lb/in is suddenly set in motion at t=0 by an external force of 99cos(20t) lb, determine the position of the mass at any time. Assume that g=32 ft/s2. Solve for u in feet.

Suppose I is an interval in R. Prove from the definition of outer Lebesgue measure that m^∗ (I) = I.

Bonus Problem:

Find the number of spanning trees in K_m,n.

Note: I need how to get the right formula-not just oh this is the formula you need.

Cucina Corp. signed a new installment note on January 1, 2018, and deposited the proceeds of $51,000 in its bank account. The note has a 3-year term, compounds 6 percent interest annually, and requires an annual installment payment on December 31. Cucina Corp. has a December 31 year-end and adjusts its accounts only at year-end

1. Use an online application, such as the loan calculator with annual payments at mycalculators.com, to generate an amortization schedule. Enter that information into an amortization schedule with the following headings: Year, Beginning Notes Payable, Interest Expense, Repaid Principal on Notes Payable, and Ending Notes Payable.
2. Prepare the journal entries on (a) January 1, 2018, and December 31 of (b) 2018, (c) 2019, and (d) 2020.
3. If Cucina Corp.’s year-end were March 31, rather than December 31, prepare the adjusting journal entry would it make for this note on March 31, 2018?

(1) In each part, decide whether or not the described relation is an equivalence relation.

Explain your answers.

(a)

A

is the set of all residents of the United States. The relation

∼

on

A

given by:

x

∼

y

if

x, y

are in the same state at noon on November 3rd.

(b)

U

is the set of all undergraduates at the University of Oregon. The relation

∼

on

U

given by:

x

∼

y

if

x, y

are in a class together.

(c)

P

is the set of all people on Earth. The relation

∼

on

P

given by:

x

∼

y

if

x

and

y

have birthdays on the same day.

(d)

P

is the set of all people on Earth. The relation

∼

on

P

given by:

x

∼

y

if

x

and

y

have birthdays on the same day of June.

1) What permutation group is pentagon D5 a subgroup of ?why?

2) What is a generating set of pentagon D5? Why? Is it a minimal generating set?Why or why not?

3) What is not a generating set of pentagon D5? Why?