021.10f.midterm.sol

021.10f.midterm.sol - HKUST MATHOZl Concise Calculus...

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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: (my-flu .._.-—-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 , SEW-NC .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:c-ln: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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