MTH341 Tma Solutions

Admin_Louis — Fri, 18 Jun 2021 13:10:19 +0000

MTH341

Question: A function \\(f: E \\rightarrow R\\) defined on a set \\(E \\subset R\\) is said to be _________ on E if \\( \\forall x_1, x_2 \\in (x_1 < x_2 \\Rightarrow f(x_1) \\geq f(x_2)\\)).
Answer: increasing

Question: A function \\(f: E \\rightarrow R\\) defined on a set \\(E \\subset R\\) is said to be _________ on E if \\( \\forall x_1, x_2 \\in (x_1 < x_2 \\Rightarrow f(x_1) > f(x_2)\\)).
Answer: decreasing

Question: What is the intervals in which the function f defined on R by f(x) = \\(2x^3 – 30x^2 +144x + 7 \\forall x \\in R\\) is decreasing?
Answer: [4, 6]

Question: Let \\(f: R \\rightarrow R\\) be defined as f(x) = x for \\(0 leq x < 1\\) and f(x) = 1 for \\(x \\geq 1\\). When is f(x) continuous?
Answer: x = 1

Question: A function \\(f: E \\rightarrow R\\) defined on a set \\(E \\subset R\\) is said to be _________ on E if \\( \\forall x_1, x_2 \\in (x_1 < x_2 \\Rightarrow f(x_1) < f(x_2)\\)).
Answer: increasing

Question: Let a function f be defined on an interval I. If f is derivable at a point \\(c \\in I\\), then it is ________ at c.
Answer: continuous

Question: Let \\(f : R \\rightarrow R\\) be a function defined as f(x) = \\(x^n \\forall x \\in R\\) where n is a fixed positive integer. What is the differentiability of f at any point \\( x \\in R.\\)?
Answer: f\'(x) = \\(nx^{n-1}\\)

Question: What is the intervals in which the function f defined on R by f(x) = \\(2x^3 – 30x^2 +144x + 7 \\forall x \\in R\\) is increasing?
Answer: \\(]-\\infty, 4] and [6, \\infty[\\)

Question: Let \\(f: R \\rightarrow R\\) be defined as f(x) = x for \\(0 leq x < 1\\) and f(x) = 1 for \\(x \\geq 1\\). When is f(x) not derivable?

Answer: x = 1

Question: Let f be a real function defined on an open interval [a, b]. Let c be a point of this interval so that a < c < b. The function f is said to be differentiable at the point x = c if____ exists and is finite.
Answer: \\(limx\\rightarrow c \\frac{f(x) – f(c)}{x – c}\\)

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MTH341 : REAL ANALYSIS II (2014)

Admin_Louis — Mon, 29 Jul 2019 01:17:13 +0000

NATIONAL OPEN UNIVERSITY OF NIGERIA

14/16 AHMADU BELLO WAY, VICTORIA ISLAND, LAGOS

SCHOOL OF SCIENCE AND TECHNOLOGY

MARCH/APRIL 2014 EXAMINATION

COURSE CODE: MTH341
COURSE TITLE: REAL ANALYSIS II
TIME ALLOWED: 3HOURS
INSTRUCTION: ANSWER ANY FOUR QUESTIONS

1(a) State and prove Langrange’s mean value theorem.                           6marks

(b) If a and b (a < b) are real numbers, then show that there exists a real number c
between a and b such that                             4marks
(c) Verify Cauchy’s mean value theorem for the functions f and g defined as
,       .
4marks

2. (a)   Define and explain a Monotonic Functions.                                                               5marks

(b)   Separate the intervals in which the function, f, defined on R by , , is increasing or decreasing.                                                                                 5marks

(c)    Show that the function f, defined on R by ,
is increasing in every interval.                                                                             4marks

3(a)     Let be the function given by , . Show that   is continuous at but it is not derivable at the same point.                                                       4 marks
(b)      Prove that a function defined as , x ≠ 0; and f(0) = 0, is continuous but not derivable at the origin.                                                                       4 marks
(c)       Show that is continuous but not derivable at x = 0 and x = 1
6 marks

4(a)     Let a function f be defined on an interval I. Show that If f is derivable at a point , then it is continuous at c.                                                                                               5marks

(b)      Let f and g be two functions both defined on an interval I. If these are derivable at then show that f – g is also derivable at x = c and (f – g)’(c) = f’(c) – g’(c).
5marks

(c)        Find the derivative at a point y0 of the domain of the inverse function of the function f, where f(x) = sin x, .                                                                                    4marks

5 (a)       State the Rolle’s theorem and give its geometrical interpretation.           7marks

(b)        Verify Rolle’s theorem for the function f defined by

(i)                     , .                                                                       3marks
(ii)                   f(x) = sin x,                                                                                   4marks

6. (a) Show that a necessary condition for f(c) to be an extreme value of a function f is that f’(c) = 0, in case it exists.                                                                        4marks

(b) Examine the function f given by ;   for extreme values. 4marks

(c)   Examine the polynomial function given by for local maximum and minimum values.                       6marks

7 (a)   Using Maclaurin’s theorem, prove that
4marks
(b) Find the Maclaurin Series expansion of
(i)     (ii)     (iii)                                         10marks
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MTH341 Tma Solutions

MTH341 : REAL ANALYSIS II (2014)