COLLEGE OF EDUCATION
SCHOOL OF EDUCATION
CENTRE OF OPEN AND DISTANCE LEARNING
CONTINUOUS SPECIAL ASSESSMENT TEST
DEPARTMENT: Mathematics, Science and Physical Education
PROGRAMME: Diploma in education
SUBJECT: Mathematics
ACADEMIC YEAR: Intake 2021
MODULE CODE: MAT1242
MODULE TITLE: One Variable Real Analysis
NUMBER OF CREDITS: 10
TIME: 11:00-13:00
DURATION: 2hours
DATE: 16/09/2021
Instructions: Answer any 5 questions and total score is 50 marks
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Question one/10 marks
For each of the following statements or expressions answer by TRUE or FALSE as appropriate.
If the answer is False make it true
a) An anti derivative of
is
.
b) If a particle has a constant acceleration, then its position function is a cubic
polynomial.
c) If
then
is an inflection point of
.
d) The function
is neither an odd function nor an even function.
e) The value of
.
f) If the line
is a vertical asymptote of
Question two/10 marks
a) At which interval the following function is continuous
, then f is not defined at 1.
2 x  1 if x  1

f  x    x 2  1 if  1  x  1
 x  1 if x  1

b) Evaluate the following limit
c)
Evaluate the limits
and
and state with raison if the limit of
at
x  0 exist or not
Question three/10 marks
Given the following function
a) Find the domain and range of the function
Determine the largest intervals on which
b)
c)
d) Make a graph of the function
Question four/ 10 marks
a) Calculate the area of surface S ranging between the parabola
and the line
/5 marks
b) Evaluate
/5 marks
Question five/ 10 marks
a) Find the interval of convergence of the power series
b) Suppose that we want to construct a rectangle that has perimeter of 100m and whose area
is as large as possible. Find the length and the width of that rectangle.
Question six/ 10 marks
Consider the function
i) Find the range and domain of definition of
ii) Is
discontinuous? If yes, then determine discontinuity points.
iii) Determine coordinates of intersections with the -axis and with the -axis.
iv) Determine horizontal/vertical/slant asymptote of a function.
v) Find local extrema of a function - local minimum and local maximum.
vi) Find intervals of monotonicity - increasing/decreasing function, stationary points.
vii) Determine intervals of convexity and concavity of f(x), inflection points.
viii) Draw the graph of a function
Examiner: Immaculee UWINGABIRE
Moderator: Dr UMUGIRANEZA Odette & Jean Claude DUSHIMIMANA