Question 1 - Infinite Sequences.

(a). Determine an infinite sequence that satisfies the following . . .

(i) An infinite sequence that is bounded below, decreasing, and convergent

(ii) An infinite sequence that is bounded above and divergent

(iii) An infinite sequence that is monotonic and converges to 1 as n → ∞

(iv) An infinite sequence that is neither increasing nor decreasing and converges to 0 as n → ∞

(b). Given the recurrence relation an = an−1 + n for n ≥ 2 where a1 = 1, find a explicit formula for an and determine whether the sequence converges or diverges

(c). Find a explicit formula for an given that {an}∞ n=1 generates the infinite sequence 1, − 1 9 , 1 25 , − 1 49 , . . . Does the above infinite sequence converge or divergence?

6.

For this problem we will use the fact that:

?????? = ??????? − ????

? ? = ? ? − ?(?)

A company produces and sells copies of an accounting program for home

computers. The total weekly cost (in dollars) to produce x copies of the program

is ? ? = 8? + 500, and the weekly revenue for selling all x copies of the program

is ? ? = 35? − 0.1?).

a. Find a function, ?(?), for the profit of producing and selling x copies.

b. How many programs must be sold each week for the profit to be $1200?

c. How many programs do they need to sell to maximize their profit? What is

the maximum profit?

Question 2: AHP
A tourism company want to evaluate four hotels and select the best one using four criteria project's COST, CLEANNESS and DISTANCE and SIZE of the HOTEL. Assume that the company prefers; cleanness two times more than size, cost two times more than distance, and distance 1.5 times more than size.

1. Calculate the weights of each criteria
2. Generate the pair-wise comparison of hotels based on each criterion using the scale of Saaty (1-9)
3. Calculate the score of each hotel based on each criterion
4. Calculate the score of each hotel. Justify the best selection for the company.
5. Calculate CI, and CR. What does the value of CT means?

c) Let R be any ring and let ??(?) be the set of all n by n matrices. Show that ??(?) is a ring with identity under standard rules for adding and multiplying matrices. Under what conditions is ??(?) commutative?

There was a maths tutoring organisation called Pony Lovers’ Maths (PLM), which provides tuition for children in school-based mathematics by using ponies as the basis for most of the calculations. This business is a not-for-profit social enterprise. The classroom will house a maximum of 7 people at a time (6 children and 1 teacher) for 3 hours each work day (5 days per week) during school term and holiday periods, but NOT on Public Holidays (10 working days per year). That is, 50 weeks per year.
Part of the workplace health and safety (WHS) requirements to operate is to make available toilet facilities. Also, the Child Protection Act 1999 of Queensland1 requirement is to require for all children to be supervised while that present on the premises. Although there are toilet facilities in another building about 100 metres away in another building, the teacher needs to remain with the other children. Consequently, PLM needs to consider the cost of building a toilet within PLM complex. The classroom and 2 stables has a total roof area of 16 metres long by 6 metres wide. The full roof area is used for collecting rainwater that fills a 25,000 litre tank. There is no other source of water for the building. The Toowoomba Regional Council does not provide sewerage to the site due to its distance away from the city. The building has solar-generated electricity that is backed-up by the electricity grid when the solar panels are not in use at night.
OPTION 1: to build a water closet toilet (the type of toilet normally used in Australian buildings). Because there is no sewerage provided by the council, a septic system (provided by eco-safe waste water treatment) would need to be purchased and installed for $12,500, and the cost of running the electrical pump is around $30 per year (based on non-solar sources). The Eco-safe system is treated by J & K Wastewater services for $250 each year, plus an average of $60 each year for replacement parts and maintenance. The water from this wastewater tank is not suitable for drinking but can be used to irrigate the paddock and generate grass growth. This will reduce the amount of money needed to buy packaged horse-feed, to the value of $225 per year on average.
The link below gives further information.
https://www.ecosafe.com.au/
OPTION 2: to build an Earth closet toilet; a self-contained self-composting, 4 chamber toilet at a cost of $2,254. The system uses a 12 volt/5 Watt fan to keep odours away. This 5 Watt per hour fan runs at all times, and is charged at 30 cents per KWh during the 12 hours of night when solar panels are not operating. Installment also requires the installment of an odour vent pipe ($80 cost plus $100 to install) and an underground drain and gravel pit for draining the liquid waste ($600 total cost of digging, gravel, parts plus installation). The gravel pit must be maintained and not have any additional soil over the pit, which will reduce the usefulness of this area of land. Professional maintenance will cost $200 each year.
In addition, you will need to include the cost of one pack of “Nature Quick Microbes” at $18 for each chamber of the system. These Microbes will need to be added to each chamber twice each year. You will also need 2 bottles a year of “Nature Flush Enzymes concentrate” for $33 each. The link that describes more about this system is given below. (You should select the 4 chamber option to see the price of $2,254 for “Nature Loo Excelet with Chamber Screen” in the link in Footnote 2, below)2:

Statistic Element                                                     January   February   March   April   May   June   July   August   September   October   November   December
Mean rainfall (mm) for years 1869 to 2007             132.1        121.1        94.6     61.9   58.4   56.8     52      39.5        46.7             72.2         89.5             120
Other Information is Below:
Litres per sq. metre of roofing for 1mm of rain           1
Sq. metres area of roof                                              96
Monthly drinking water usage for horses (Litres (L)) 3200 L
Other water usage per month (Litres (L))                   1000 L
Water usage Per Year for flushing toilet in Waste Water System

                                                                                   35,000 L
Rainwater tank capacity (Litres (L))                           25,000 L
Beginning volume of rainwater as at 1 October (Litres (L)) 70%
                                                                                   17,500 L
Using the accepted rainwater collection ratio of 1 millimetre (mm) of rainwater creates 1 litre of rainwater collection per square metre of roofing and PLM’s buildings floor area is 96 square metres, calculate the potential average monthly rainwater collection assuming there is an average monthly rainfall of 79 mm.

2. Calculate the costs to set up each of the 2 toilet system options AND to run the system for the first 12 months after installation.
(5 marks for water closet system; 5 marks for Earth closet system = 10 marks)

Find the characteristics and characteristic coordinates, and reduce the following equation to canonical form:

for y>0 only

Uxx+yUyy=0

An operation called the logical implication, is written as x → y and corresponds to English statements “x implies y” or “if x then y”. The implication operator outputs 0 if the first operand is 1 and the second operand is 0, and outputs 1 otherwise.

1. Imagine a gate that implements the logical implication operator. Is this gate a Universal Gate (i.e. can any arbitrary combinational function be implemented if you had an infinite supply of gates that implement the logical implication operator)? If it is, write the expressions for NOT and OR, ∼ x and x|y in terms of this implication operator. In your expressions, use the character sequence -> to represent this operator, i.e. x → y is repesented in plain text by x -> y.

2. Simplify the following boolean logic expressions that make use of the logical implication operator → into simplest possible expressions that you can (in terms of the constants 0 and 1, and the variables present in the given expressions), while only making use of the three standard boolean logic operators ∼, & and |.

• (a → b)&(a →∼ b)

• (a → b)|(b → c)

• (a|b) → (a&b)

Consider Sturm-Liouville problem

−u`` + q(x)u = λu , x ∈ (a, b),

u(a) = 0, u(b) = 0.

please can you find the eigenvalues and eigenfunction?

1. Draw a 3-D sketch of the pallet unit load on a wood pallet. Include dimensions for a box and the pallet unit load.

2. Show your calculations for the quantity of boxes in a unit load.

3. Show your calculations for the number of pallets per trailer load.

4. Show your calculations for the quantity of boxes per trailer load.

5. Explain why these results are important to a producer who ships goods to customers by freightliner.

Box dimension: 12” x 10” x 6” O.D. Pallet dimensions: 48” x 42” x 6” Constraints: (1) No pallet overhang (2) Pallet unit load <= 48” high, including the pallet, for storage in a rack system Trailer dimensions: 53’ long x 8’ 6” wide (b/w the hinges) x 9’ high

Analyticity of trigonometric functions (a) Directly from the definition, construct the Taylor Series centered at x = 0 for the function f(x) = cos(x). (b) Show that this series converges for all x ∈ R. (c) Show that this series converges to cos(x) for all x ∈ R.  

DIRECTIONS: Show all the work in the space provided. Box the final answers, and follow the indicated directions.

1. Solve the DE:

y"-3y'+2y=e^xsinx

The graph of a function

y = g(x)

on the domain

−8 ≤ x ≤ 8

consists of line segments and semicircles of radius 2 connecting the points

(−8, 0), (−4, 4), (0, 4), (4, 4), (8, 0).

(a) What is the range of g?

0 < y < 60 ≤ y ≤ 4    0 ≤ y ≤ 20 < y < 40 ≤ y ≤ 6

(b) Where is the function increasing? (Select all that apply.)

−8 ≤ x ≤ −2−2 ≤ x ≤ 22 ≤ x ≤ 44 ≤ x ≤ 8−8 ≤ x ≤ 8

Where is the function decreasing? (Select all that apply.)

−8 ≤ x ≤ −2−2 ≤ x ≤ 22 ≤ x ≤ 44 ≤ x ≤ 8−8 ≤ x ≤ 8

(c) Find the multipart formula for y = g(x) if

(d) If we restrict the function to the smaller domain

−6 ≤ x ≤ 0,

what is the range?

0 ≤ y ≤ 62 ≤ y ≤ 6    0 ≤ y ≤ 42 ≤ y ≤ 40 ≤ y ≤ 2

(e) If we restrict the function to the smaller domain

0 ≤ x ≤ 4,

what is the range?

2 ≤ y ≤ 40 ≤ y ≤ 2    0 ≤ y ≤ 64 ≤ y ≤ 60 ≤ y ≤ 4

Find an optimal parenthesization of matrices whose sequence of dimensions is: <5, 10, 12, 5, 50>. Please write out both the m[·, ·] and s[·, ·] tables.