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(MATH021)[2008](f)final~id-^_10397.pdf
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HKUST MATH021 Concise Calculus
Final Examination (Version 1) Name:
13th Dec 2008 Student I.D.:
12:30-15:30 Lecture Section:

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

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Please write your Name, ID number, and Lecture Section in the space provided above.

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You may use a HKEA approved calculator, but graphical calculators are NOT allowed.

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All mobile phones and electronic equipments other than calculators should be switched o. during the examination.

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This is a closed book examination.

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When instructed to open the exam, please check that you have 9 pages of questions in addition to the cover page.

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Answer all questions.

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Show the working steps of your answers for full credit.

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Cheating is a serious o.ense. Students caught cheating are subject to a zero score as well as additional penalties.


Trigonometric Identities
cos2 . + sin2 . =1
1 + tan2 . = sec2 .
1 + cot2 . = csc2 . sin 2. = 2 sin . cos . cos 2. = 2 cos2 . . 1=1 . 2 sin2 .
2 tan .
tan 2. =
1 . tan2 . sin(A + B) = sin A cos B + cos A sin B sin(A . B) = sin A cos B . cos A sin B cos(A + B) = cos A cos B . sin A sin B cos(A . B) = cos A cos B + sin A sin B
1
sin A cos B = sin(A + B) + sin(A . B)
2
1
cos A cos B = cos(A + B) + cos(A . B)
2
1
sin A sin B = cos(A . B) . cos(A + B)
2
tan A + tan B
tan(A + B)=
1 . tan A tan B tan A . tan B
tan(A . B)=
1 + tan A tan B
Question No. Points Out of
Q. 1-10 30
Q. 11 20
Q. 12 10
Q. 13 10
Q. 14 10
Q. 15 10
Q. 16 10
Total Points 100

Part I: Multiple choice questions.
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Only one answer in each question is correct.

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Each question is worth 3 point.

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Put your answers to the multiple choice questions in the boxes below. No credit will be given if you show your answers in other places.


Question 1 2 3 4 5 6 7 8 9 10 Total
Answer e a b b d a c a d c

1. Which of the following is the graph of a function y = f (x) in the xy plane?
(a)
The unit circle centered at the origin

(b)
The square with vertices (1, 1), (.1, 1), (.1, .1) and (1, .1)

(c)
The triangle with vertices (.1, 0), (1, 1), (0, .1)

(d)
All of the above


(e) None of the above
Solution None of the above.

2. Let a, b, c be positive numbers. Evaluate (1 . e.ax) cos2 bx
lim 2 .
x.0 xecx
(a) a (b) b (c) c (d)0 (e)1 Solution The limit is a, by LH.opitals rule:
.ax
(1 . e.ax) cos2 bx aecos2 bx . 2b(1 . e.ax ) cos bx sin bx
lim =lim = a
x.0 xecx2 x.0 ecx2 +2cx2ecx2
3. How many local maximum should f (x) have if its derivative is f (x)=(x + 1)(x . 2)3(x + 4)2(x . 5) ?
(a)0 (b)1 (c)2 (d)3 (e)4 Solution One local maximum f (2) at the critical point x = 2, by .rst derivative test.
4
4. Find .x 2 +2x . 3.dx.
0
95 8674 6458
(a) (b) (c) (d) (e)
3 33 33 Solution x2 +2x . 3=(x . 1)(x +3) = 0 x =1 or x = .3.
41 4
.x 2