1) . Solve the IVP:

y^''+6y^'+5y=0, y(0)=1, y^' (0)=3

2. Find the general solution to each of the following:

a) y^''+2y^'+5y=e^2x

b) y^''+2x/(x^2+1) y'=x

c) y^''+4y=1/(sin⁡(2x)) (use variation of parameters)

3) Cherillee Toys makes a fiberglass tricycle for kids that is composed of three major components: a handlebar-front wheel-pedal assembly, a seat, and frame unit, and rear wheels. The company has orders for 12,000 of these tricycles. The company does not have the resources available to manufacture everything needed for the completion of 12,000 tricycles so it has gathered purchase information for each component.

The following table shows relevant information:

Develop a linear programming model to advise the operations manager of Cherillee Toys how many of each component should be manufactured and how many should be purchased in order to provide 12,000 fully completed tricycles at the minimum cost.

PLEASE FINISH THIS PROBLEM USING EXCEL

Consider the linear transformation T : P1 → R^3 given by T(ax + b) = [a+b a−b 2a]

a) find the null space of T and a basis for it

(b) Is T one-to-one? Explain

(c) Determine if w = [−1 4 −6] is in the range of T

(d) Find a basis for the range of T and its dimension

(e) Is T onto? Explain

What is the ^VIX index and why is it so important? How do you explain the solutions and difference of Logarithmic and Polynomial solutions?

Let G = R\{0} and N = (0,∞). Show that G/N is isomorphic to the multiplicative group {1, −1}.

Let f : R → R be a function.

(a) Prove that f is continuous on R if and only if, for every open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is open.

(b) Use part (a) to prove that if f is continuous on R, its zero set Z(f) = {x ∈ R : f(x) = 0} is closed.

find the general solution

y'''-y''-4y'+4y=5-e^x-4e^2x

(UNDETERMINED COEFFICIENTS—SUPERPOSITION APPROACH)

Let A = Z and let a, b ∈ A. Prove if the following binary operations are (i) commutative, (2) if they are associative and (3) if they have an identity (if the operations has an identity, give the identity or show that the operation has no identity).

(a) (3 points) f(a, b) = a + b + 1

(b) (3 points) f(a, b) = a(b + 1)

(c) (3 points) f(a, b) = x² + xy + y²

A pharmacist has fluocinolone ointment with50% fluocinolone , and the other contains
15% fluocinolone. How many grams of each may be used to prepare 1950 g of a 20%
fluocinolone ointment?

Compute the nominal annual rate of interest (compounded monthly) at which $250.00 deposited at the end of each month for ten years will amount to $30 000.00.

How many sets of five numbers from 1 to 15 can you make in which exactly two of the numbers are divisible by 3?

Using Theorem 1, discuss lim n→∞ fn on B and C (as in Example (a)) for each of the following. (i) fn(x) = x n ; B = E1; C = [a, b] ⊂ E1. (ii) fn(x) = cos x + nx n ; B = E1. (iii) fn(x) = Xn k=1 xk; B = (−1, 1); C = [−a, a], |a| < 1. (iv) fn(x) = x 1 + nx ; C = [0, +∞). [Hint: Prove that Qn = sup 1 n 1 − 1 nx + 1 = 1 n .] (v) fn(x) = cosn x; B = 0, π 2 , C = h1 4 , π 2 ; (vi) fn(x) = sin2 nx 1 + nx ; B = E1. (vii) fn(x) = 1 1 + xn ; B = [0, 1); C = [0, a], 0 < a < 1.

THeorem 1

Theorem 1. Given a sequence of functions fm: A → (T, ρ′), let B ⊆ A and
Qm = sup
x∈B
ρ′(fm(x), f(x)).
Then fm → f (uniformly on B) iff Qm → 0.
Proof. If Qm → 0, then by definition
(∀ ε > 0) (∃ k) (∀m > k) Qm < ε.
However, Qm is an upper bound of all distances ρ′(fm(x), f(x)), x ∈ B. Hence
(2) follows.
Conversely, if
(∀ x ∈ B) ρ′(fm(x), f(x)) < ε,
then
ε ≥ sup
x∈B
ρ′(fm(x), f(x)),
i.e., Qm ≤ ε. Thus (2) implies
(∀ ε > 0) (∃ k) (∀m > k) Qm ≤ ε
and Qm → 0.
Examples.
(a) We have
lim
n→∞
xn = 0 if |x| < 1 and lim
n→∞
xn = 1 if x = 1.
Thus, setting fn(x) = xn, consider B = [0, 1] and C = [0, 1).
We have fn → 0 (pointwise) on C and fn → f (pointwise) on B, with
f(x) = 0 for x ∈ C and f(1) = 1. However, the limit is not uniform on
Saylor URL: http://www.saylor.org/courses/ma241/ The Saylor Foundation
230 Chapter 4. Function Limits and Continuity
C, let alone on B. Indeed,
Qn = sup
x∈C |fn(x) − f(x)| = 1 for each n.2
Thus Qn does not tend to 0, and uniform convergence fails by Theorem 1.
(b) In Example (a), let D = [0, a], 0 < a < 1. Then fn → f (uniformly) on
D because, in this case,
Qn = sup
x∈D |fn(x) − f(x)| = sup
x∈D |xn − 0| = an → 0.
(c) Let
fn(x) = x2 +
sin nx
n
, x ∈ E1.
For a fixed x,
lim
n→∞
fn(x) = x2 since

sin nx
n

≤
1
n → 0.
Thus, setting f(x) = x2, we have fn → f (pointwise) on E1. Also,
|fn(x) − f(x)| =

sin nx
n

≤
1
n
.
Thus (∀ n) Qn ≤ 1
n → 0. By Theorem 1, the limit is uniform on all of
E1.
Note 1. Example (a) shows that the pointwise limit of a sequence of con-
tinuous functions need not be continuous. Not so for uniform limits, as the
following theorem shows.

Show that any conditionally convergent series has a rearrangement that diverges.

Implementation in CLIPS programming language for the following problem

Acme Electronics makes a device called the Thing 2000. This device is available in five different models distinguished by the chassis. Each chassis provides a number of bays for optional gizmos and is capable of generating a certain amount of power. The following table sumarizes the chassis attributes:

Chassis --------- Gizmo Bays provided --- Power Provided----- Price($)
C100--------------------------- 1--------------------------- 4------------------ 2000
C200------------------------- 2 -------------------------- 5------------------ 2500
C300--------------------------- 3---------------------------7 ------------------3000
C400---------------------------2----------------------------8------------------ 3000
C500-------------------------- 4 ---------------------------9 ------------------3500

Each gizmo that can be installed in the chassis requires a certain amount of power to operate. The following table summarizes the gizmo attributes

Gizmo-------------------- Power Used --------------------Price($)

Zaptron ------------------------ 2 ------------------------------100
Yatmizer ----------------------- 6------------------------------ 800
Phenerator--------------------- 1 ------------------------------300
Malcifier----------------------- 3 ------------------------------200
Zeta-shield------------------- 4 ------------------------------150
Warnosynchronizer---------- 2 -------------------------------50
Dynoseparator---------------- 3 ------------------------------400

Given as input facts representing the chassis and any gizmos that have been selected, write a program that generates facts representing the number of gizmos used, the total amount of power required for the gizmo, and the total price of the chassis and all gizmos selected