Lista de Exercícios - Cálculo I UFC PDF

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Lista de exercícios da disciplina de calculo I da UFC...

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Universidade Federal do Cear´ a - UFC Departamento de Matem´ atica C´ alculo Diferencial e Integral I 2014.1

Lista de Exerc´ıcios (1) Considere a fun¸c˜ao de dom´ınio R, definida por  2    x , se x < −1 f (x) =

(a) Esbo¸ce o gr´afico de f .

−x + 1,    log x,

se − 1 ≤ x < 1

se

x≥1

(b) Determine, se existirem: lim f (x), lim f (x), lim f (x), lim f (x) e lim f (x). x→+∞

x→−∞

x→−1

x→0

x→1

x2 + |x| , determine o dom´ınio de f (x), esbo¸ce o gr´afico no x seu dom´ınio de defini¸c˜ao, e determine, se existirem, os limites lim− f (x), lim f (x)

(2) Considerando f (x) =

x→0

e lim f (x). x→0

(3) Calcule, se existirem, os seguintes limites:  tan(x) 1 . (a) limπ x→ 2 2 x+3 (b) lim x→−3 |x − 3| − 6 x2 − 7x + 6 x→1 x3 − 3x + 2   1 1 − lim 1 − x3 x→1+ 1 − x √ x2 + x + 1 − 1 lim x→0 x √ 9+x−3 lim x→0 x √ √ x2 + 1 − 2 lim x→1 3x2 − 3x √ lim x − x2 + 1 x→0 h √ i lim ln( x2 + 2 + x)

(c) lim (d) (e) (f) (g) (h) (i)

x→−∞

1

x→0+

(j) (l) (m) (n) (o) (p)

√

lim √ 3

x2 − 3

8x3 + 1 2x − 1 lim 3 x→−1 x − x √ 3x2 + x − 2 lim x→−1 x3 + 1 1 − cos(x) lim x→0 x2 sin(x) lim x→π x − π   2−x2 lim π 1+x cos(ln(x))

x→+∞

x→+∞

(4) Verifique, justificando, se f ´e cont´ınua em x = xo .   1 − cos(x) , x 6= 0 ; xo = 0 (a) f (x) = x  1, x = 0.  sin(x)    , x −3  x2 − 9 ,   1 (c) f (x) = ; xo = −3 , ; x = −3  2     cos(πx) , x < −3 6 (5) Determine, se poss´ıvel, as constantes de modo que x2 + ax + 8 =8 x→6 x−6  2 x + ax + 8   , x < −1   x+1 (b) f (x) = c, x = −1     x2 + 5x, x > −1   x>1   3x − 2, (a) lim

(c) f (x) =

ax, x=1    5b − ax, x0

x − 1,

x...