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(MATH021)[2009](f)final~id-^_10399.pdf
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HKUST
MATH021 Concise Calculus
Final Examination Name:
17th Dec 2009 Student I.D.:
Tutorial Section:
Seat Number:

Directions:
.
DO NOT open the exam until instructed to do so.

.
You may write on both sides of the examination papers.

.
You must show the steps in order to receive full credits.

.
Electronic calculators are not allowed.


Question No. Points Out of
1 3
2 4
3 4
4 4
5 10
6 10
7 10
8 10
9 10
10 10

Some formula:
.
sin(.. + ..) = sin .. cos .. + cos .. sin ..

.
sin(.. . ..) = sin .. cos .. . cos .. sin ..

.
cos(.. + ..) = cos .. cos .. . sin .. sin ..

.
cos(.. . ..) = cos .. cos .. + sin .. sin ..


tan ..+tan ..
. tan(.. + ..)=
1.tan .. tan .. tan ...tan ..
. tan(.. . ..)=
1+tan .. tan ..
.
sin 2.. = 2 sin .. cos ..

.
cos 2.. = cos2 .. . sin2 .. = 2 cos2 .. . 1=1 . 2 sin2 ..


1. (a) What is an even function?
(b) If .. is an even di.erentiable function (so that .. does have a derivative), eval-uate .. (0).
2. (a) For each .. > 0, evalualte


.......
.
..=0
(b) Is your result in part (a) valid if .. 0?
3. Let

1
..(..)= . 1 for all .. =0M.
..
Evaluate the area of the region bounded by the graph of .., the ..-axis and the two vertical lines de.ned by .. =1/.. and .. = ...
4. For each of the following in.nite series and improper integral, determine if it con-verges or diverges

(a) ..=1 ......... ,
+
(b) 0 sin .. .... .
5. (a) Let .. be a di.erentiable function. .. is a number. Evaluate

..(.. + ..) . ..(.. . ..)
lim .
..0 ..
(b) Let .. be a function so that
....(..) =1 . ..(..) for all ... Evaluate .. (0).
(c) Evaluate

1 . cos ..
lim .
..0 .. sin ..
(d) Let
..(..) = cos2 .. sin2 ...
Evaluate .. (0).
(e) Let

..2 +2..
.(..)= .
(.. + 1)2
Evaluate . (0).
6. Let
.... ..(..) = for all ...
1 . ..2
Sketch the graph of ... Your sketch should reveal all the relative extremums and asymptotes of ...
7. Let
../2 ....,.. = cos .. .. sin.. .. .... .
0
(a) Use integration by parts to establish the following reduction formula for .. 2
and .. 2
.. . 1

....,.. = ....,...2 .
.. + ..
(b)
Use integration by parts to establish a reduction formula involving ....,0 and .....2,0 for .. 2.

(c)
Use (a) and (b), or otherwise, to determine ..6,4.


8. If a triangle has sides of length .., .. and .., one can determine its area by the so-called Herons formula

.. = ..(.. . ..)(.. . ..)(.. . ..) ,
where .. =(.. + .. + ..)/2.
Find the biggest possible area of an isosceles triangle (that is, two of its edges have the same length) whose perimeter is 1 meter.
9. Evaluate the following integrals
(a)
1
....
....
2 . ....
0
(b)
1 ..(..2 . 1)2/3 ....
0
(c)

0
.... .1 (.. . 1)(..2 + 1)
(d) The improper integral
1
.. ln .. ....