FUNDAMENTAL MATHEMATICS II
Question 1. Say that a sequence (an) is a Cauchy sequence (named after
the French mathematician Cauchy) if it has the following property:
For every > 0 there is a number M (depending on ) such that
|an - |am < for all n, m >= M.
(1) * Show that the sequences ( 1/n) and (n + 1/2n) and Cauchy sequences, but the sequence (n) is not.
(2) * Write down the definition of what it means for a sequence (an) to be convergent. Show that any convergent sequence is a Cauchy sequence.
(3) Show that every Cauchy sequence in R converges to a limit in R.
(4) * Give an example to show that there can be a Cauchy sequence
(an) of rational numbers that does not converge in Q.
Question 2. In this module we shall build upon the second definition of a
function being continuous that we had in Chapter 4, Definition
1.6. We recall that Definition here.
Definition. Say that the function f : R --> R is continuous at the point
a 2 R if for all > 0 there is a number > 0 with the property
|f(x) - f(a)| < e for all x with |x- a| < s
(1) * Use this Definition to show that the function
(a) f : R --> R given by f(x) = 2x is continuous at the point a 2 R
(b) g : R --> R given by g(x) = x2 is continuous at the point a 2 R
(c) : R --> R given by
s(x) =( 1 if x = 0
( 0 if x 6= 0
is not continuous at the point 0 e R.
(2) Prove that a function f : R ! R is continuous at a point a 2 R in
the sense of the definition above, if and only if it is continuous at a
in the sense that limx-->a f(x) = f(a).
Fundamental mathematics sequences are examined in the solution. It shows that every Cauchy sequence in R converges to a limit in R.
Probability, Combinations and Fundamental Principle of Counting: 7 Word Problems -
(Please see attached file for complete description)
Finite Math 6.3 - Please see attached file for complete description)
Multiplication Principle Problems
2. Commuter Passes. Five different types of monthly commuter passes are offered by a city's local transit authority for each of three different groups of passengers: youths, adults, and senior citizens. How many different kinds of passes must be printed each month?
4. Coin Tosses. A coin is tossed 4 times and the sequence of heads and tails is recorded.
a. Use the generalized multiplication principle to determine the number of outcomes of this activity.
b. Exhibit all the sequences by means of a tree diagram.
6. Commuter Options. Four commuter trains and three express buses depart from city A to city B in the morning, and three commuter trains and three express buses operate on the return trip in the evening. In how many ways can a commuter from city A to city B complete a daily round trip via bus and/or train?
9. Health-Care Plan Options. A new state employee is offered a choice of ten basic health plans, three dental plans, and two vision care plans. How many different health care plans are there to choose from if one plan is selected from each category?
10. Code Words. How many three-letter code words can be constructed from the first ten letters of the Greek alphabet if no repetitions are allowed?
14. Automobile Colors. The 2007 BMW is offered with a choice of 14 exterior colors (11 metallic and 3 standard), 5 interior colors, and 4 trims. How many combinations involving color and trim are available for the model?
19. License Plate Numbers. Over the years, the state of California has used different combinations of letters of the alphabet and digits on its automobile license plates.
a. At one time, license plates were issued that consisted of three letters followed by three digits. How many different license plates can be issued under this arrangement?
b. Later on, license plates were issued that consisted of three digits followed by three letters. How many different license plates can be issued under this arrangement?
22. Warranty Numbers. A warranty identification number for a certain product consists of a letter of the alphabet followed by a five-digit number. How many possible identifications numbers are there if the first digit of the five-digit number must be nonzero?
23. Lotteries. In a state lottery, there are 15 finalists eligible for the Big Money Draw. In how many ways can the first, second, and third prizes be awarded if no ticket holder can win more than one prize?View Full Posting Details