Name:
April 2008
Mathematics 101
Page 2 of 11 pages
Marks
[21]
1.
ShortAnswer Questions.
Put your answer in the box provided but show your work also.
Each question is worth 3 marks, but not all questions are of equal diFculty. ±ull marks will
be given for correct answers placed in the box, but at most 1 mark will be given for incorrect
answers. Unless otherwise stated, simplify your answer as much as possible.
(a)
Evaluate
x
3
−
2
x
√
x
dx
.
Answer
(b) Evaluate
π
0
(4 sin
θ
−
3 cos
θ
)
dθ
. You must simplify your answer
completely
.
Answer
(c)
Express lim
n
→∞
1
n
n
i
=1
1
1 + (
i/n
)
2
as a de²nite integral.
Do not
evaluate this integral.
Answer
Continued on page 3
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
April 2008
Mathematics 101
Page 3 of 11 pages
(d) Write down the Trapezoidal Rule approximation
T
3
for
4
1
x
cos(
π/x
)
dx
. Leave your
answer expressed as a sum involving cosines.
Answer
(e)
Let
f
(
x
) =
kx
2
(1
−
x
) if 0
≤
x
≤
1 and
f
(
x
) = 0 if
x <
0 or
x >
1. For what value of
the positive constant
k
is
f
(
x
) a probability density function?
Answer
(f)
Find the ±rst three nonzero terms in the powerseries representation in powers of
x
(i.e.
the Maclaurin series) for
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 AKOS
 Toricelli

Click to edit the document details