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UNIVERSIDADE FEDERAL DE SANTA CATARINA
CAMPUS DE JOINVILLE
EMB 5001 – CÁLCULO DIFERENCIAL E INTEGRAL I
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Unidade 1
Funções reais de variável real e funções elementares do cálculo
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1 – Seja f ( x) =

x2 − 4
, mostrar que: x −1

(a) f (0) = 4
(b) f (−2) = 0

1 − 4t 2
(c) f (1 / t ) = t −t2

x 2 − 4x x −3
15
(e) f (1 / 2) =
2

2
(f) f (t ) =

(d) f ( x − 2) =

t4 − 4 t 2 −1

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3x − 1
, mostrar que: x−7 5 f ( −1) − 2 f (0) + 3 f (5)
263
=−
7
98
[ f ( −1 / 2)]2 = 1
9
9x − 7 f (3 x − 2 ) =
3x − 9
− 22t 2 + 38t − 88 f (t ) + f ( 4 / t ) =
− 7t 2 + 53t − 28

2 – Se f ( x) =
(a)
(b)
(c)
(d)

f ( h ) − f ( 0)
20
= h 7(h − 7 )
11
(f) f [ f (5)] =
7
(e)

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3 – Se f ( x) =

ax + b e d = − a , mostre que f ( f ( x)) = x . cx + d

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4 – Dada Φ( x) =

1 x −1
, forme as expressões Φ (1 / x) e
.
Φ( x )
2x + 7

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2
5 – Dada a função f ( x) = x + 1 , mostrar que, para a ≠ 0 , f (1 / a ) =

f (a)
.
a2

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6 – Dada a função f ( x) = 1 / x , mostrar que f (1 + h) − f (1) =

−h
.
1+ h

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2
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7 – Seja f ( x) = ( x − 2)(8 − x ) para 2 ≤ x ≤ 8 .
(a) Determinar f (5) , f (−1 / 2) e f (1 / 2) .
(b)