University of Ghana: Second Semester Examinations, 2013/2014 Bsc/Ba
University of Ghana: Second Semester Examinations, 2013/2014 Bsc/Ba
University of Ghana: Second Semester Examinations, 2013/2014 Bsc/Ba
Find
lim r(S)
s→0+
2. (a) Determine whether the following functions are continuous at a. Justify your
answer.
√
i. f (x) = x − 2 ; a = 1
( 2
x −1
if x 6= 1
ii. f (x) = x−1 ; a=1
3 if x = 1
( 2
x −x
if x 6= −1
iii. f (x) = x+1 ; a = −1
0 if x = −1
(b) Determine the intervals of continuity of the following functions.
i. p(x) = 4x4 − 3x2 + 1
x5 +6x+17
ii. f (x) = (x2 −9)
3x2 −6x+7
iii. g(x) = x2 +x+1
.
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3. Find the family of antiderivatives for the function
sin 2x
f (x) = .
2 cos2 x
x
4. Let f (x) = . Determine whether f is an even or odd function and hence or
1 + x4
otherwise evaluate Z π
2
f (x)dx.
−π
2
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SECTION B: Answer Any Three Questions
1. Sketch the curve
3x2
f (x) =
x2 − 1
indicating the following:
(a) intercepts
(b) Domain
(c) Asymptotes
(d) Behaviour of curve
(e) Intervals of increase and decrease
(f) Local maximum and minimum values
(g) Concavity
(h) Points of inflexion
x = sect
t
y = ln tan
2
dy
i. Find dx
.
d2 y
ii. Show that dx2
= −2 cot3 t csc t.
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3. (a) By using the substitution x = 3 + sin θ, show that,
π
Z 4 p Z
2
(−x2 + 6x − 8)dx = cos2 θdθ.
−π
2 2
(b) Evaluate Z
sin2 2x cos2 3x dx.
5. (a) Sand is dumped off a conveyor belt into a pile at the rate of 2 cubic feet per
minute. The sand pile is shaped like a cone whose height and base diameter
are always equal. At what rate is the height of the pile growing when the pile
is 5 feet high? (The volume of a cone is π3 r2 h where r is the radius of the base
and h is the height).
(b) Given y = e2x sin x, prove that
d2 y dy
2
− 4 + 5y = 0.
dx dx
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