UNIVERSITY EXAMINATIONS 2018/2019 ACADEMIC YEAR
2nd YEAR EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE, BACHELOR OD EDUCATION AND BACHELOR OF ARTS
(INSTITUTIONAL BASED)
COURSE CODE/TITLE: SMA B202: CALCULUS III
END OF SEMESTER II DURATION: 2HOURS
DAY/TIME:THURSDAY:9.00-11.00 AM DATE:2/05/2019 (1st F/EW)
INSTRUCTIONS
Answer question ONE and any other TWO questions from section B
QUESTION ONE
3 x
y2
(a) Evaluate the iterated integral
1 1 x
dydx (5 marks)
x3
(b) Show that any differential function of the form w f (s) where s y bx is a solution of the
w w
partial differential equation b 0 b 0 is a constant ) (4 marks)
x y
(c) Find the Taylors series for f ( x) x up to polynomial of degree 3 for f ( x) about x 2
(8 marks)
1 1
(d) Evaluate lim (7 marks)
x 1
ln x x 1
(e) For the function f ( x) x 2 and g ( x) x 1 , test whether the Cauchy’s theorem holds on
2 3
the interval 1, 2 and find the approximate value of (5 marks)
(f) Let f ( x) x 1 . Show that f satisfies the hypothesis of the mean value theorem on the
3
interval 1, 2 and find all values of C in the interval whose existence is guaranteed by the
theorem. (5 marks)
(g) Verify Rolle’s Theorem for the polynomial f ( x) x x on the interval 1 x 0 and
3
0 x 1 , hence find the appropriate value of xi ( ) (6 marks)
Page 1 of 2
2nd YEAR EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE, BACHELOR OD EDUCATION AND BACHELOR OF ARTS
(INSTITUTIONAL BASED)
COURSE CODE/TITLE: SMA B202: CALCULUS III
END OF SEMESTER II DURATION: 2HOURS
DAY/TIME:THURSDAY:9.00-11.00 AM DATE:2/05/2019 (1st F/EW)
INSTRUCTIONS
Answer question ONE and any other TWO questions from section B
QUESTION ONE
3 x
y2
(a) Evaluate the iterated integral
1 1 x
dydx (5 marks)
x3
(b) Show that any differential function of the form w f (s) where s y bx is a solution of the
w w
partial differential equation b 0 b 0 is a constant ) (4 marks)
x y
(c) Find the Taylors series for f ( x) x up to polynomial of degree 3 for f ( x) about x 2
(8 marks)
1 1
(d) Evaluate lim (7 marks)
x 1
ln x x 1
(e) For the function f ( x) x 2 and g ( x) x 1 , test whether the Cauchy’s theorem holds on
2 3
the interval 1, 2 and find the approximate value of (5 marks)
(f) Let f ( x) x 1 . Show that f satisfies the hypothesis of the mean value theorem on the
3
interval 1, 2 and find all values of C in the interval whose existence is guaranteed by the
theorem. (5 marks)
(g) Verify Rolle’s Theorem for the polynomial f ( x) x x on the interval 1 x 0 and
3
0 x 1 , hence find the appropriate value of xi ( ) (6 marks)
Page 1 of 2
