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(MATH014)[2009](s)midterm~2617^_63438.pdf
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HKUST
MATH 014 Calculus II
Mid-Term Examination Name:
28th Mar 2009 Student I.D.:
10:30C12:30 Tutorial Section:

1.
Do not open the exam until instructed to do so.

2.
When instructed to open the exam, please check that you have 8 pages of questions in addition to the cover page.

3.
Write your name, and other information in the space provided above.

4.
Show an appropriate amount of work for each problem. If you do not show enough work, you will get only partial credit.

5.
This is a closed book and notes examination.

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. Points Out of
Q. 1 8
Q. 2 8
Q. 3 9
Q. 4 16
Q. 5 12
Q. 6 17
Q. 7 18
Q. 8 12





Total Points
1. ([8 pts]) The functionsf, g : R R have continuous derivatives up to second order, and some of their function values are given in the following table.
x
f (x)
f (x)
f (x)
g(x)
g (x)

(x)
g
0

-2
1
2
1
2
0 1 3 -2 1 -4 3 -1 2 5 0 -3 2 1 2 3 1 3 -2 5 4 4
Compute the following:
3
(a)
f (x)f (x)dx = [2 pts]

(b)
(x .2)g (x)dx = [3 pts]


1 2
0 3
(x).g(x)f (x)
(c) f (x)g dx = [3 pts]
0
2. ([8 pts])Rewritten version: Consider the function f (x)= kg(x) where k is some constant and the graph of g is given below.

(a)
For which constant k is the area bounded between the graph of f and the x-axis equal to 1? [2 pts]

(b)
If I is the interval consisting of all x values such that 2 f (x) 4, .nd the area under the graph of f over the interval I.


[2 pts]
x
(c) Consider the function de.ned by F (x)= f (x)dx. Sketch the graph of F (x). [4 pts]
.1
x 3 +x+1
3. ([9 pts]) Consider the graph of y = , where 0 x 6. Rotate the area between the

graph and the x-axis about the x-axis to form a solid region.
(a) Use Simpsons Rule with 6 subintervals to approximate the volume of the solid region.
[6 pts]
9
y
6
3
0
0 1 2 3 4 5 6
(b) How accruate is Simpsons Rule in this example? Give brief reason for your answer. [3 pts]

4. ([16 pts]) Evaluate the following integrals.

5
x
(a) dx. (Hint: let tan2 = u.) [8 pts]
3sinx .4cosx
4x .3
(b) dx [8 pts]
(x2 +6x +13)2
0
.x
5. ([12 pt]) Consider the graphof z = e2 , where x 0. Rotate the area below the graph over the interval[0, )about the z-axis togenerate a solid of revolution. (Note that the surface ofrevolution
.(x 2 +y 2 ) .x .y
thus obtained is given by the equation z = e= e2 e2 .)
(a) Find the volume of the solid of revolution. [5 pts]
z
z
y
x
(b) Consider the cross sectional area of the solid cut out by a plane perpendicular t