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(MATH013)[2010](f)midterm~=pcidkjd_^_45882.pdf
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HKUST
MATH 013 Calculus I
Mid-Term Examination Name:
27th Oct, 2010 Student I.D.:
19:00-21:00 Tutorial Section:
1.
Do not open the exam until instructed to do so.
2.
This is a closed book and notes examination.
3.
When instructed to open the exam, please check that you have 8 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.
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 Total Points
Points Out of
8 10 12 12 18 16 12 12 100
1.i
5
1. ([8 pts]) Findall complex roots of the equation z=.
1+i
5x2 +6
2. ([10 pts]) Given a functiony = f(x)= .
x2 +2x .8
(a)
Find the domain of f. [5 pts]
(b)
Find all horizontal and vertical asymptotes of f. [5 pts]
3. ([12 pts]) The number of rabbits in a 200 day period is modeled by the following graph of a di.erentiable function q(t), 0 t 200.
q (rabbits)
2000
1000
t 100 200 (days)
(a)
What is the value of the derivative function q (t)when the number of rabbits is largest? [1 pts]
(b)
What does q (t)represent, and what is its unit? [2 pts]
(c)
Using thegivengraph,estimatethesize of therabbitpopulation whenitsderivativeislargest, and also the size of the rabbit population when its derivative is smallest respectively. [4 pts]
(d)
Sketch roughly the graph of q (t). Show the scales in your axes. [5 pts]
4. Choose coordinate axes and scales and then draw the graph of a function f on the grid satisfying all of the following conditions on the interval .1 <x< 5: [12 pts]
(a) f(0)=1, f (x)< 0 for .1 <x< 0, and lim f(x) =3
x0+
(b) f is continuous for 0 <x 2, f (x)> 0 for 0 <x< 1, and f(2) =2
f(2+h).f(2) 1
(c) lim = .
h0+ h 2
(d) f is continuous for 2 x< 5 and f (4) =0
5. ([18 pts]) Two functions f and g di.erentialable function on the interval .4 <x< 4 are given with the following table of function values:
x f(x) f (x) g(x) g (x) -3 -1 1 1 1
Compute the following derivatives:
-2 2 3 -4 3
-1 3 -1 -2 2 0 0 0 -1 -1
(a) F (2), where the functionF is de.ned by F(x)=
1 -2 2 -2 -1 2 3 1 1 -3
f(x)
.
f(x).2g(x)
(b) G (1), where the functionG is de.ned by G(x)= g(x2 .1) .
3 1 -2 3 -2
[6 pts]
[6 pts]
(c) H (.1), where the function H is de.ned by H(x)= fg(x)+x2 f(x) . [6 pts]
6. ([16