Cálculo Diferencial E integrais 1,2 e 3 (48)

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Questão 3) (1.5)(a)Seja~; R+_ R dadapor f(x) = -x -lnx. Seja9 a inversadefi admitindoque9
sejaderivávelatésegundaotdem,calculeg'(-l) eg"(-l).
x2
(b) (1.5)Mostreque cosx:t.1- 2" paratodox >O.
,
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~(~)=-7, - eJl =~ J ~ -~(':» -f.m ty'D»)
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Questão 3) (a) (1.5)SejaI : IR~-+IRdadapor I(x) =x + lnx. Sejag a inversade li admitindoqueg
sejaderivávelatésegundaordem,calculeg'(I) e gl/(I).
x2
(b)(1.5)Mostrequecosx>1- - paratodox >O.2
o.) l '""~C?L) <. j ("')) =- ;lI..
~(,,) =1tt .e.,~ ==) d=5(,,)-t~(:'i"'\)
j -::l =') .1 = y!)t e..(5(") =-~j( .).=~1- _ -
sf-l~)= cl bb)l"ct(~b'»)
"'J cl')
j =- ~\''1)-t ~ -) J' (~)=~ ~)
J(")) .1t-Y~)
---
.,
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VCUvn.h V\'W? l/lan r {td:>o , tlJL ) o .
\
~(,l) '" -~)l -t':>(. V7t>O
;) Y- ,,7,,y, , ".),,:, !'" ~ ~1 ~) -/""'" ~ -1 =-'~"'<+~~-i < 'YL>O
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C"V'a~l ~ [O,t-tP (
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-- -
Questão 4)(a) (1.0) SejamXo E]a,b[ e J :]a,b[--+]R. urnafunçãocontínuaem]a,b[e derivávelemtodo
x:f xo. Mostre que, se lim f'(x) = l E ]R, então f é derivável em Xo e P(xo) =l.x-+xo
(
2
)
tg",
(b) (1.0)Calcule lim - .",-0+ X
(c\ )
5e~o
Qç
>"0
1 (\1 ~[ \ ~~o) ) (~
)< t-"" f(~) - P(~J
'li: - ~
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=- O
G~Jl VIIAli dk
elN'\ )(0)
JcC~) \\~ FL~' = O
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k.
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3 \i f'(1 llM t/(1 { ( F°'-
. - -
)( -,'lu
G'('fl
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)<.-" y.u
- -
Questão4) (a) (1.0)SejamxQ E]a,b[ e f :]a,b[ IRumafunçãocontínuaemJa,b[e derivávelemtodo
x =I xo.Mostreque,se lim f'(x) = l E IR,entãoJ éderivávelemXoe J'(xo) =l.x-+xo
(
3
)
tgX
(b) (1.0)Calcule lim - .
x-o+ X
;-
\rvM e
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_ exf C tP . l.. (~11-
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lbl 1élM.-i~
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e
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y.. .-"/ OIr-
liV\-> d(,) = +v.J
~-'t0+
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C\ H ; V"I
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