For the differential equation dy/dx=sqrt(y^2−36) does the existence/uniqueness theorem guarantee that there is a solution to this equation through the point

1. (1,6)

2. (4,42)

3. (−2,38)

4. (7,−6)

Use euler method to approiximate y(1)

y' is given , solve for y. approiximate y(1)

Create a table to show the approximate value of y(1) for each choice of h.

y’ = -15y y(0) = 1

h = .25

h = .125

h = .0625

h = .03125

Prove using induction:

If F is any field and f(x)=p1(x)p2(x)...pn(x) is a nonconstant polynomaial of the field, f an element of F, and p1,...,pn are irreducible factors of the field.

Then, there exists a field L such that f factors into linear factors over L.

Hint: start with p1(x) to prove that F is a subset of some K1=F[x]/((p1)) , then induct.

0.3 The Fibonacci numbers Fn are defined by F1 = 1, F2 = 1 and for n >2, Fn = F sub (n-1) + F sub (n-2). Find a formula for Fn by solving the difference equation.

Find the Upper and Lower Darboux sums for the following functions.
(i) f(x) = −x − 1 on [0, 3], n = 3. [10]
(ii) f(x) = 1 + 2x 0n [0, 1] , n = 3

Find the Upper and Lower Darboux sums for the following functions.
(i) f(x) = −x − 1 on [0, 3], n = 3. [10]
(ii) f(x) = 1 + 2x 0n [0, 1] , n = 3

(e) Find the first 4 terms of the Taylor series for the following functions
(i) ln x centered at x0 = 1 . [8]
(ii) sin x centered at x0 =
π
4

Solve the following initial value problems:

a) ut+xux= -tu, x is in R, t>0; u(x,0) =f(x), x is in R.

c) ut+ux=-tu, x is in R, t>0; u(x,0)=f(x), x is in R

d)2ut+ux = -2u, x,t in R, t>0; u(x,t)=f(x,t) on the straight line x = t, where f is a given function.

Use Newton-Raphson to find the real root to five significant figures 64x^3+6x^2+12-1=0

Prove that x^8 - 14x^4 + 25 is irreducible

This problem is also a Monte Carlo simulation, but this time in the continuous domain: must use the following fact: a circle inscribed in a unit square

has as radius of 0.5 and an area of ?∗(0.52)=?4.π∗(0.52)=π4.

Therefore, if you generate num_trials random points in the unit square, and count how many land inside the circle, you can calculate an approximation of ?

For this problem, you must create code in python

(A) Draw the diagram of the unit square with inscribed circle and 500 random points, and calculate the value of ?

Prove the following statements!

1. There is a bijection from the positive odd numbers to the integers divisible by 3.

2. There is an injection f : Q→N.

3. If f : N→R is a function, then it is not surjective.

Prove the following statements!

1. If A and B are sets then

(a) |A ∪ B| = |A| + |B| − |A ∩ B| and

(b) |A × B| = |A||B|.

2. If the function f : A→B is

(a) injective then |A| ≤ |B|.

(b) surjective then |A| ≥ |B|.

3. For each part below, there is a function f : R→R that is

(a) injective and surjective.

(b) injective but not surjective.

(c) surjective but not injective.

(d) neither injective nor surjective

Prove the following statements!

1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r when 24|(k−r). If g : S→S is defined by

(a) g(m) = f(7m) then g is injective and

(b) g(m) = f(15m) then g is not injective.

2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is injective.

3. Let f : A→B and g : B→C be surjective. Then g ◦ f : A→C is surjective.

4. There is a surjection f : A→B such that f −1 : B→A is not a function.