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Unformatted text preview: HKUST
MATHOZl Concise Calculus Midterm Examination Name:
12 Oct 2010 Student 1.1).: Lecture and Tutorial Section: Directions: 0 DO NOT open the exam until instructed to do so. 0 You may write on both sides of the examination papers.
0 You must Show the steps in order to receive full credits. o Electronic calculators are not allowed. _
—
_
_ Some formula:  sin($ + y) = 311123 cosy + cosrrsiny
o sin(:c H y) = sinmcos y — cosxsiny
0 (305(1: + y) = cosmcosy # sinwsiny 0 cost: — y) = cosmcosy ~I— sinmsiny __ tan m+tan y . taut]: + y) '— l—tanxtany
__ i tan :E—tan y
. 138'an y) _ 1+tanmtany o sin2$ = QSiIliL'COSﬂ: « (:0st :cos2x—sin2sc=260523:» 1 = 1—23'11122: 1. Let
f(;c) m 6””1. (a) What is domain of f?
(b) It is known that f is one—to—one on its domain. Compute. the inverse of f. \h‘jz Afgéq—ﬁ~ >0: (myflu .._.—7 :3 9%): Clanai’K/ . Let —\/1—3:2 if ~1<$SO? 2'
f(m)= ncosx if 0<$SGT
tank—37‘”) if 7r<$<27r. Complete the following table. Write down does not exist if the limit is 00, moo
or the limit does not exist. . Using the deﬁnition, calculate the derivative of the function W?) = 
m __l_ ﬂ, __.
f \w QCXhCW) " SPOL) \l m X+Li
j: {m “M 1::
(7K) Lg C) H H50 [5
€— l'm ”L X‘ CHM “:3 H“ i —~
I»? O h x wan) we 0 k >< been) 4. Evaluate each of the following limits, if it exists. (a) tan ms lim
m—rﬂ :13 (b) (a) , ﬁT‘MFX , SEWNC .L
\f w r hm , 7Q
(7030 7< 7‘90 CcmTVIC :3 km SYNNX L QTXKX L,
m g: 7’, g 741
X 7*
\(m .L; :0
X4? 60 7‘3” W )m
' . “— Mm
By The gmwmh m / Vs; 0Q 5. Compute the derivatives of the following functions. (a) f(x):cos:cln:1:, 0)) gm = (1+1)m, (C) Mac) = tan—1(ex) . (Ow) 5X?) = (109%ch 6. Consider the following implicit curve 334—x2y+y4=1. (3) Verify that the point (~1, 1) is on the curve.
(b) Compute the derivative dy/ (13:.
(c) Find the slope of the tangent line at (—1,1). 1 4 4t .
C09 NS: Vim” 1“ (er—(wtiahuf: I Cb) ' WNW: ...
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 Fall '10
 LUXNU
 Math

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