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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  00” = 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, '
viahag ’<
xaiw
\IxHL M¢~AH '1 x“ «L x+\u
3 '2'. = \ HT : §
‘ Keg, X‘3 x43 x 3 w” *2' x43 (63) (qtH “A
:M\( +  L‘m‘lwl* 2+1 
yq ‘5 x+l 2. xa) 2. 2%
LI}  ‘ X.” _ l M \ M\ : —‘—
ﬂA/‘N x X1+x X‘lxq: x 0”,“ 'xqo 7v” £&;\X+( OH ~ Exercise 2
What are the (horizontal or vertical) asymptotes of the graph of the function a: 1
10(93):??? ? £ not ée{{“€=l M x=\ MA ><"\
xH xH
(X149. ‘ +50 . because, (:v. 70 near \I
x‘v (
M (“U =7. omA L‘m (Kt“1
xH x¢\
X'H x+l
ﬂow 171 = 2% z ~—‘_._._— 
*qL‘ (x \\ XI; _‘* (x+‘\ (30“ *a‘ (“44‘ (*4‘7. +00
+ ao‘ ‘3‘;
xu _‘ . ‘
km (KIn1 ’3‘“ (an mx‘ = °°
*a\_ XqL W W
a0. a4
\ .L
KH '3 + H
____, _, K x 0
1w 1. L  ‘ 1. : " = O
>< “ — H \ ‘
1‘... 4w xﬁ‘v‘! x‘
.L
x’cl ‘ #3 + "H  2’ = O
M ,g I (l  H1  l
( I\ KQ‘“ x‘ Exercise 3
Compute the derivatives of 1) (m + 1)6(:I: + 2)3
cos(a:) 2) ﬂ 3) ($4 +1)1000 4) tan(cos(m) + 1) \
‘3 ((x+\c’(x*7—\3\ = Q)(><+\S(x*2\3 + (x+{\63(x+z\z = = 3 (x+t\$(x+2\1 [ mm + (x+\\  3 (x+l\$(x*2\z (n+5) 7.) MK ‘ ‘Mxvﬁ WX'3:"L—'x —gmxx wsx < :' = 3' \ X70
()7 X 1 x 2‘
\ooo ‘Wn qqq
M (xxW = \000 (xhﬁ ‘ Li X3 = 4000 x3(x"+l\
‘  ‘ x
A) 4a (mx +\\ = 7 (‘M 2 WM
(/06 +‘\ our} (MK 1"}
 I!
A WW)" 1': 2~ , ~ 1/ x + t(mm(§\) +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 «awnma)
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‘ 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
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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.
 Fall '08
 staff
 Calculus

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