=========================preview======================
(MATH013)[2010](f)final~=yqp24^_19492.pdf
Back to MATH013 Login to download
======================================================
HKUST
MATH 013 Calculus I
Final Examination Name:
18th Dec 2010 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.
Question No.
Points Out of
Q. 1
10
Q. 2
6
Q. 3
10
Q. 4
9
Q. 5
12
Q. 6
9
Q. 7
10
Q. 8
10
Q. 9
6
Q. 10
Q. 11
Total Points
1. (10=3+3+4points) Afunction f is de.ned by
. . x2 .9
.
if x< .3
f(x)= x2 +2x .3
.
.
asin(x)+b if x .3
where a, b are some constants.
(a)
(3 points) Determine allvertical and horizontal asymptotes of f, whenever existing.
(b)
(3 points) What values of a and b can make the function f continuous everywhere? Give brief reason.
(c)
(4points)Arethere anyvalues of a and bwhich can make f afunctiondi.erentiable everywhere (i.e.,f (x)exists everywhere)? Show your reasoning for full credit.
22)2
2. (6points) Acurveisde.nedby theequation(x+y=16(x2 .y2). Find allpointsonthe curve at which the tangent line is horizontal.
3. (10 points) Two carts, Aand B, are connected by a rope 20 m long that passes through a pulley 8 m above a location Q on the .oor. If cart B is being pulled away from Q at a speed of 0.8 m/s, how fast is cart A moving toward Q at the instant when cart B is 6 m from Q?
4. (9=3+6points) Considertheequationlnx = sin x .
2
x
(a)(3points) Findtheequationof thetangentlinetothegraph of thefunction y =lnx.sin at x =1.
(b) (6 points) Apply Newtons method to .nd the solution of the equation lnx = sin x , which
2
is accurate to .ve decimal places. (Show some of your steps and keep 7 decimal digits in your calculation.)
5
5. (12=4+4+4points) Apolynomialfunction f(x)=5+160x3 .3xis given.
(a)(4points) Findalllocal maximumorlocal minimumof f.Justifyyour answerforfull credit.
(b)
(4 points) Findall in.ection points of f. Justify your answer for full credit.
(c)
(4 points) How manyreal roots does f have? Justify your answer for full credit.
6. (9=2+3+4points) Theareaunderthegraph of thefunction y = tanx over the interval[0,
4
]
is shown below.
(a)(2points) Usingthe midpoints of .ve subintervals of equal length, estimate thede.nite
.
4
integral I = tanxdx by a Riemann sum of rectangular areas.
0
y
y = tanx