GREAT ZIMBABWE UNIVERSITY
.
HMAT 101
GARY MAGADZIRE SCHOOL OF AGRICULTURE AND NATURAL SCIENCES
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE ‘
RESEARCHBSc HONOURS IN MATHEMATICS EXAMINATIONS: PART 1 SEMESTER 1BSc HON
EXAMINATION: ASSIGNMENT 2
HMAT 101: CALCULUS I
DUE DATE: 30/09/19
Time : OF YOUR CHOICE hours
Candidates may attempt ALL questions
A1. Evaluate each of the following integrals:
Z
√
(a)
x2 2x3 + 4dx,
Z
1
x ln(x + 3)dx.
(b)
[5]
[5]
0
Z
(c)
3
x2 e2x dx.
[5]
Z
1
dx,
a2 + x 2
R
(e) x2 sin xdx,
Z
1 − x + 2x2 − x3
(f)
dx.
x(x2 + 1)2
Z 1
dx
p
(g)
,
(x + 2)(3 − x)
−1
Z
x2 + 2x − 1
(h)
dx.
2x3 + 3x2 − 2x
(d)
[4]
[5]
[6]
[6]
[8]
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HMAT 101
A2. Suppose that the amount of money in a bank account after t years is given by,
t2
A(t) = 2000 − 10te5− 8 .
Determine the minimum and maximum amount of money in the account during the
first 10 years that it is open.
[4]
A3.
1
is not uniformly continuous in 0 < x < 1.
x
(b) Show that if y = tan−1 x, then
(a) Prove that f (x) =
(1 + x2 )
[4]
d2 y
dy
= 0.
+ 2x
2
dx
dx
[4]
A4.
(a) Show that if f (x) and g(x) are continuous at x0 then f (x)g(x) is also continuous
at x0 .
[3]
x+2
(b) Use the definition of a derivative to find the derivative of f (x) =
.
[5]
x−2
(c) Let
f (x) = x3 + 2x2 − x − 1.
Find a number x0 ∈ (−2, 1) so that the tangent to the graph of y = f (x) is
horizontal at x = x0 .
[5]
A5.
(a) If x2 y + y 3 = 2, find
d2 y
,
dx2
at the point (1; 1).
(b) If exy − ln y = x2 + 1, find
[4]
d2 y
.
dx2
[5]
(c) Prove the formula
0
0
0
(f (x)g(x)) = f (x)g(x) + g (x)f (x),
assuming that g and f are differentiable.
A6.
[6]
(a) State and prove the Mean Value Theorem.
[5]
3
(b) Verify the Mean Value Theorem for f (x) = x − 12x on [−1, 3].
[4]
(c) Verify the Rolle’s theorem for f (x) = x2 (1 − x)2 , 0 ≤ x ≤ 1.
[4]
(d) Verify the Rolle’s Theorem given that,
f (x) = (x − a)m (x − b)n ,
where m and n are positive integers and where a < b.
[6]
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HMAT 101
(e) Use the Mean Value Theorem or otherwise to prove that if 0 < a < b, then
a
b
b
1−
< ln
<
−1 .
b
a
a
[5]
(f) Hence or otherwise show that
1
1
< ln(1.2) < .
6
5
[3]
END OF QUESTION PAPER
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