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

Use stars and bars to solve each counting problem. You may leave your answers as binomial coefficients.

(a) How many collections of 6 (not necessarily distinct) coins can be made from an infinite supply of pennies, nickels, dimes, and quarters?

(b) A social security number is a sequence of 9 digits. How many social security numbers are there

n1n2n3 . . . n9

such that ni ≤ ni+1 for i = 1 to 8?

For example, 024455888 would count but 254180419 would not count

Given the matrix with rows

[1,1,k 1]

[1,k,1 1]

[k,1,1 -2]

Find the reduced row echelon form of M, and explain how it depends on k. (b) Consider the linear system Ax = b for which the augmented matrix is A b = M. i. For what values of k is the system inconsistent? ii. For what values of k does the system have a unique solution?

Homework 1.1. (a) Find the solution of the initial value problem x' = x^(3/8) , x(0)=1 , for all t, where x = x(t). (b) Find the numerical solution on the interval 0 ≤ t ≤ 1 in steps of h = 0.05 and compare its graph with that of the exact solution. You can do this in Excel and turn in a printout of the spreadsheet and graphs.

Write each of the following permutations as a product of disjoint cycles:

(a) (1235)(413)

(b) (13256)(23)(46512)

(c) (12)(13)(23)(142)

1. In this problem, you can use the Matlab program posted on course website and Canvas (also given in the lecture) that computes the interpolation polynomial. We want to see how well a given function can be approximated by the interpolation polynomials. Let f be a function. We divide the the interval [−0.6,0.6] into subintervals of the same length h = 0.02. The gridpoints are −0.6 = x1 < x2 < ... < x61 = 0.6. Take N = 61 points (x1,y1),...(xN,yN) on the graph of f.

   1. (a) For f(x) = sinx, plot the graph of the interpolation P on the interval [−0.6,0.6]. Plot f and all of P on the same graph (for example, by using the command hold on). Does the interpolation polynomial approximate well the function f on the interval [−0.6, 0.6]?

   2. (b) The same questions as in Part (a) but for f (x) = 1+x .

   3. (c) We know that the error between f and P is estimated by

| f (x) − P (x)| ≤ max |f (n) | (∗)

n n−1 [a,b]
Let f (x) = 1 and [a, b] = [−0.6, 0.6]. Use Stirling approximation m√m! ≈ 1 (for large

1+x me m) to show that the right hand side of (∗) goes to infinity as n → ∞.

roblem 3 Find the solutions to the general cubic a x^3 +b x^2+c x +d=0 and the solutions to the general quartic a x^4+b x^3+c x^2+d x+e=0. Remember to put a space between your letters. The solutions to the general quartic goes on for two pages it is a good idea to maximize your page to see it. It is a theorem in modern abstract algebra that there is no solution to the general quintic in terms of radicals.

Please write it clearly! thank you!

A manufacturer is marketing a product made of an alloy material requiring a certain specified composition. The three critical ingredients of the alloy are manganese, silicon and copper. The specifications require 15 pounds of manganese, 22 pounds of silica and 39 pounds of copper for each ton of alloy to be produced. This mix ingredients require the manufacturer to obtain inputs from three different mining suppliers. Ore from the different suppliers has different concentrations of alloy ingredients, as detailed on the Table below:

Given this information, the supplier must determine how much ore to purchase from each supplier so that there is no waste of the alloy ingredients. A solution to the problem can be found by defining the following variables:

X_j = amount of ore purchased from supplier j

C_i = amount of ingredient i required per ton of alloy

A_ij = amount of ingredient i contained in each ton of ore shipped from supplier j

What amount of ore should be purchased from each supplier?

P4.4.2 Give a rigorous proof, using strong induction, that every positive natural has at least one factorization into prime numbers.

1. Are the following statements true or false? For each, explain why.

(a) 15 divides 5

(b) gcd(4, 16) = 2

(c) 46 ≡ 2 (mod 4)

(d) For any positive even integer n, gcd(12, n) = 2.

(e) For all integers x, if 10 does not divide x, then 10 does not divide 22x.

(f) For all integers y, if gcd(9, y) = 3, then gcd(9, y2 ) = 9.