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(math013)[2009](f)final~yhong^_10011.pdf
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HKUST
MATH 013 Calculus I
Final Examination Name:
18th Dec 2009 Student I.D.:
12:30C15:30 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 10 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.
dxp d sinx
p.1
= px= cosx
dx dx
dex d cosx

x
= e= .sinx
dx dx
d lnx 1 d tanx

2
= = secx
dx x dx d sin.1 x 1 d tan.1 x 1
= =
2
dx 1.xdx 1+x2
sin(A B)= sinA cos B sinB cosA cos(A B)= cosA cosB .sinA sinB.
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
10
9
12
9
9
8
10
8
16
100

1. (9=3+3+3pts) Considerafunctionde.nedby

c .x if x 1
f (x)= 2
x+ cx .2 if x> 1
where c is a constant.
(i) (3 points) For what constant c can f be afunction continuous onthe whole realline(., )? Give brief reason.
(ii)(3points) Usingthede.nitionofderivative,explainwhatit meansby sayingthat f isdi.erentiable at x =1.
(iii) (3 points) What is the constantc if the given function is di.erentiable at x =1?
2. (10 =6+2 +2points) Incertainchemicalreaction,theconcentration C of the product as a function of time t (in minutes)is given by
12t
C = C(t)= (mole/L).
6t +1
(i) (6 points) Show thatC = C(t)satis.es an equation of the form
dC

= k(2.C)a
dt
for some suitable k and a. Find also these constants, i.e., k and a.

(ii) (2 points) What happens to the concentration ast +?
dC
(iii) (2 points) What happens to the rate ofchange of the reaction, i.e., , as t +?
dt
3. (9=3+6points) Supposethatthe derivative of a di.erentiable function f (x)is

f (x)=(x .2)(x +1)2 (x .4) , where . <x< .
(a)
(3 points) At which value of x will the function f achieve a local maximum? Justify your answer for full credit.

(b)
(6 points) How manyin.ection points does the function f have? Justify your answer for full credit.


4. (12=3+3+3+3points) Hereisatableof somevaluesof threedi.erentiablefunctions f , g, and

h, where the derivatives f , g and h are assumed to be continuous.
xf (x) f (x) g(x) g (x) h(x) h (x)
0 .2

32
21 .11

102620 .1

5 .1

.


12
2323


Answer the following true/false, or computational questions: (An answer of true means true in all situations. An answer of false means there is a case where the assertion