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
You can get the exam summary answers for this course from 08039407882
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