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(MATH014)[2010](s)final~4274^_19008.pdf
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HKUST
MATH 014 Calculus II
Final Examination Name:
24th May 2010 Student I.D.:
12:30-15:30 Tutorial Section:

1.
This is a closed book and notes examination.

2.
Do not open the exam until instructed to do so.


3.
When instructed to open the exam, please check that you have 11 pages of questions in addition to the cover page.

4.
Write your name, and other information in the space provided above.

5.
Show an appropriate amount of work for each problem. If you do not show enough work, you will get only partial credit.

6.
You may write on the backside of the pages, but if you use the backside, clearly indicate that you have done so.

7.
Please turn o. all phones and pagers and remove headphones.

8.
Cheating is a serious o.ense. Students caught cheating are subject to a zero score and other


penalties.
Question No.
Q. 1
Q. 2
Q. 3
Q. 4
Q. 5
Q. 6
Q. 7
Q. 8
Q. 9
Q. 10 Total Points
Points Out of
9
9
9
10
9
9
9
10
12
14
100

1. ([9 pts]) A function f : R R has continuous derivatives up to second order, and some of its

function values are given in the following table.

x
f(x)
f(x)
f(x)
Compute the following integrals:
(a) . 2 0 3f(x) 4+ [f(x)]2 dx

0 -2 -1 -3

3
6 4 3

2
2 0 2
[4 pts]
.
(b) 3 [f (x)cosx + f(x)cosx]dx. [5 pts]
0
1 1
2. ([9 pts]) Consider the integral dx.
0 1+2x
(a) Use the trapezoidal rule with n =4 subintervals to estimate the given integral. [3 pts]

b
(b) Recall that an error bound for the estimating of f(x)dx by trapezoidal rule has the form
a
K(b.a)3 (x)|
|ET| , if |f K. Determine how large should the number of intervals n
12n2
1 1 be taken in order to guarantee that the approximation of dx by trapezoidal rule is
0 1+2x accurate to within 0.00001. [6 pts]
3. ([9pts]) Recallthattheprobability of a random variable(random number) X hitting a number between a and b is given by
b
P{a X b}= f(x)dx
a
if f(x)is the probability density function of X. Suppose that
.
. 0 if x < 0,
.
.
f(x)= kx2(1.x) if 0 x 1,
.
.
. 0 if x > 1,

where k is a constant.
(a)
Find the constant k . [2 pts]

(b)
If F is the function de.ned by F(x)= P{X x}, .nd F(x)and sketch its graph. [5 pts]



11
(c) Find the probability that X , i.e, P{X }. [2 pts]
33
4. ([10 pts]) Consider the region enclosed bythe x-axis, the vertical line x = .2, and the parametric curve de.ned by
7
y 6x =2(cost + tsint), 5 4
y =2(sint .tcost),
3
where 0 t . 2
1x
0
(a)
Find the arc length of the parametric curve. [3 pts]

(b)
Find the area of the region. [7 pts]



5. ([9pts]) Twoclosedcurvesarede.nedby the polar equations r =4+2cos2 and r =4+2cos respectively. The graph of the second curve is shown in the .gure below.
(a) Sketch the curve r =4+2cos2. [3 pts]

(b) Express th