=========================preview======================
(math013)[2009](s)midterm~2548^_10391.pdf
Back to MATH013 Login to download
======================================================
HKUST
MATH 013 Calculus I
Mid-Term Examination Name:
24th Oct, 2009 Student I.D.:
10:00-12:00 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.
You may use an ordinary scienti.c calculator, but calculators with graphical, or symbolic calculation functions are NOT allowed.

8.
Please turn o. all phones and pagers and remove headphones.

9.
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
8 10 10 10 12 10 8 10 10 12 100
1. ([7 pts]) Findall complex roots of the following equations.

6
(a)
Solve the equation u= .i. [4 pts]

(b)
Using the rootsin(a), solvethe equation(z .1)6 + i(z +1)6 =0. [4 pts]


2. ([7 pts]) The graphof a function f is shown in the following grids:
(a) Sketch the graph of y = .1 f (x)+1 [3pts]
2
y = f(x)
y
1 1 x
(b) Sketch the graph of y = f .1(x + 2), where y = f .1(x)is the inverse function of f . y y = f(x) [3 pts]
1 1 x
(c) Sketch roughly the graph of y = f (x) y y = f(x) [4 pts]
1 1 x

3. ([9 pts]) The functionsf and g aredi.erentialablefunction ontheinterval(., ). A table of some of their function values is given as:
x
f (x)
f (x)
g(x)
g (x)

-8 -1 -5 9 4 -6 -3 -3 5 3 -4 -1 -1 2 2
Let h be the function de.ned by h(x)= x.
.
g
(a) (.4)
h + f
-2 0 0 -1 -1 0 1 2 -1 -1
1 3 3 1 -3 2 5 5 5 -5 3 6 7 9 -7
4 8 8 12 -9
Compute the following derivatives:
[4 pts]
(b)
(f .g) (.4), where f .g is the composition of f and g. [3 pts]

(c)
(f .(g +2h)) (.4) [3 pts]


4. ([10 pts]) Solve the following problems.

(a) Suppose y = tekt, where k is a constant. Determine the constant k for which the function y = tekt satis.es the equation [4 pts]
d2y dy
.8+16y =0 .
dt2 dt
(b) Sandispouredinto a conicalpile with theheight of the coneequal totwicethebaseradius. If the sand is poured at a constant rate of 5 m3/s, .ndthe rate at which the radius is increasing when the height reaches 1 m. [6 pts] 5. ([12pts]) Supposethefunctionf is continuous anddi.erentiable on theinterval(0, ). Determine if the following statements are true or false.
(a) If f has a horizontal asymptote, then lim f (x)=0. [3pts]
x+
Circle your answer: True False
Brief reason:

(b) If lim f (x)=+, then x =0 is a vertica