Analysis 1 (MATS2101)

Mathematics and Computer Science - MCS

Semester: First Semester

Level: 200

Year: 2017

Answers (200 )
University of Bamenda
Facility of Science
Department of Mathematics and Computer Science.
Continuous Assessment
Course Title: Analysts I .
Course Code : MAT2101.
Duration: 2 hrs.
Course Instructor: Bime Markdonal. G
Instructions: Answer all questions. You are reminded of the necessity for good English and orderly
presentation of your answers.
1. Let S ⊂ ℝ. Answer the following with respect to S.
(a) (5 Marks) Define the following:
(i) Limit point (ii) The closure of S (iii) An interior point
(iv) Least upper bound (v) Greatest lower bound
(b) (5 Marks) When is S said to be
(i) Bounded (ii) Open (iii) Closed (iv) An interval in E
(v) Dense in M
(c) (4 Marks) Define the following and give examples:
(i) A function (ii) An inductive set
(d) (3 Marks) Is it true, that if a is a limit, point of S then a ∈ S? Justify.
(e) i. (4 Marks) Is the set S = {x ∈ ℝ : x
2
< x} bounded above (below)? if so, find its supremum
(infimum).
ii. (2 Marks) Is S having a maximum element? Justify.
2. (4 Marks) Let a be rational and b irrational, show that if a ≠0, then ab is also irrational.
3. (a) (2 Marks) State the principal of Mathematical induction
(b) (4 Marks) Show that 2
3n + 1
+ 5 is always a multiple of 7, ∀n ∈ ℕ.
4. Find the domain and range, and plot the graph of the function determine by the given formula:
(a) (4 Marks) f(x) =
|
|
(b) (4 Marks) g(x) = √4 −
(c) (4 Marks) h(x) = √
− 4
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