V63_0121_0034_2008F_Midterm_Solutions

# V63_0121_0034_2008F_Midterm_Solutions - V63_O 0034 Calculus...

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Unformatted text preview: V63 _O\?.\_. 0034 Calculus I Midterm Exam Exercisel Compute the following limits; they might be ﬁnite, inﬁnite, or not exist. 2—4 Limit" 'I‘MK 0‘0 1) limmﬁo_2_m_x_ - W - 0-0-” = O a: ~3m—4 2. £332. x-” 4V l \$2_1 on x-‘o 2 ‘m _, -——— )11' +oox3+3\$+4 . x/a:+ —2 2) 11mm_.3—————— 33—3 1 1 4 1' _. —— ) 1mm 0:1: m2+m L —. 'L . J... , L 0.. X‘L_\ ‘- 44 a ~4ﬁﬂx xx = J 2 ' ~ 1 3- ' +041) 1 W m 1+ XL ~ ,o 1+ in i, ' via-hag ’< xaiw \IxH-L M¢~AH '1 x“ «L x+\-u 3 '2'. = \ HT : § ‘ Keg, X‘3 x43 x 3 w” *2' x43 (-63) (qt-H “A :M\( + - L‘m‘lwl* 2+1 - yq ‘5 x+l 2. xa) 2. 2% LI} | ‘ X.” _ l M \ M\ : —-‘—- ﬂA/‘N x X1+x X‘lxq: x 0”,“ 'xqo 7v” £&;\X+( OH ~ Exercise 2 What are the (horizontal or vertical) asymptotes of the graph of the function a: 1 10(93):??? ? £ not ée{{“€=l M x=\ MA ><"\ x-H x-H (X149. ‘ +50 . because, (:v. 70 near \I x-‘v ( M (“U =7. omA L‘m (Kt-“1 x-H x¢\ X'H x+l ﬂow 171 = 2% z ~—‘_._._— - *q-L‘ (x \\ XI; _‘* (x+‘\ (30“ *a-‘ (“44‘ (*4‘7. +00 + ao‘ ‘3‘; x-u _‘ . ‘ km (KI-n1 ’3‘“ (an mx‘ = -°° *a-\_ Xq-L W W a0. a4 \ .L KH '3 + H ____, _, K x 0 1w 1. L - ‘ 1. : " = O >< -“ — H- -\ ‘ 1‘... 4w xﬁ‘v‘! x‘ .L x’cl ‘ #3 + "H - 2’ = O M ,g- I (l - H1 - l ( -I\ KQ‘“ x‘ Exercise 3 Compute the derivatives of 1) (m + 1)6(:I: + 2)3 cos(a:) 2) ﬂ 3) (\$4 +1)1000 4) tan(cos(m) + 1) \ ‘3 ((x+|\c’(x*7—\3\ = Q)(><+|\S(x*2\3 + (x+{\6-3(x+z\z = = 3 (x+t\\$(x+2\1 [ mm + (x+|\\ - 3 (x+l\\$(x*2\z (n+5) 7.) MK ‘ ‘Mxvﬁ -WX'3:"L—'-x —gmx-x -wsx < :' = 3' \ X70 ()7 X 1 x 2‘ \ooo ‘Wn qqq M (xx-W = \000 (xhﬁ ‘ Li X3 = 4000 x3(x"+l\ ‘ - ‘ x A) 4a (mx +\\ = 7- (‘M 2 WM (/06 +‘\ our} (MK 1"} - I! A WW)" 1': 2~ , ~ 1/ x + t(mm(§-\) +21%» N“ k e Z Exercise 4 Find the tangent to the graph given by 2552 + my + y2 = 4 at the point (1,1). 2xL+x\1+\1?'=H‘ /%< Lt><+‘1+ X‘1'+Q~1\1'=O x='n ‘1‘“ 4+: + w‘m + 24mg ‘1‘03: —§ TanckemT \{ = -§x + 1, x=\,‘1=“ \:—%~+Jb 1w E =8 8 Exercise 5 1) Use transformations to sketch the graph of the function 1 HmSO me) = { W x2+% ifx>0 2) Where is this function continuous? 3) Where is it differentiable? m «awn-ma) ; , x L F E 2):”. E3 E W i s 3 i g i ‘””‘“““§K““ “Twmmmmmmwu E i i 2.3 ‘9 COMTihMoKS 0h (~Wﬁn U (41003 (“01’ Jeiined M '4, ‘ x:l: :‘ - (m z to @M i -1 I ~2. 2" U“: (MM = ‘5- W’ (:12?- W NW" x: \ ‘ HA: 2 t“ K>0 Max: new =o d: 4,103 4 x Exercise 6 A triangle ABC is deforming as time goes by in the following way: 0 the angles in A and B are equal 0 the distance AB is ﬁxed to 1 o the distance AC is equal to the distance BC is equal to t for any time t 2 O. 1) Make a picture. 2) Using the intemediate values theorem, show that there is a time t at which the distance BC is half the distance AB. A:o ‘55 A. For A, é-‘i There {s “o gkoln Trimale ‘ . A \$ ( lTiCXW%\e {heqwmlilﬁ\ , L 2.3 We «Jami Adscl: MAM: g-\ g a) L A O 7 A : : L which is V‘ol 99°“;le ' ' A l ‘5 c It: \ A \s C. =7. i i A is 6 M3 ,g s ...
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## This note was uploaded on 10/25/2010 for the course MATH V63.-0121- taught by Professor Staff during the Fall '08 term at NYU.

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V63_0121_0034_2008F_Midterm_Solutions - V63_O 0034 Calculus...

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