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2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04.2013 - A
NOME: No. USP: NOTA:
Questao Resposta Valor Questao Resposta Valor
01a. (a) (b) (c) (d) (e) 06a. (a) (b) (c) (d) (e)
02a. (a) (b) (c) (d) (e) 07a. (a) (b) (c) (d) (e)
03a. (a) (b) (c) (d) (e) 08a. (a) (b) (c) (d) (e)
04a. (a) (b) (c) (d) (e) 09a. (a) (b) (c) (d) (e)
05a. (a) (b) (c) (d) (e) 10a. (a) (b) (c) (d) (e)
Regras: 1 - UNICA SEM RASURA2 -3 -
01. Questao:x!0
x
4 + 5x- 1
2-px
2 + 4
x!1
x
3 - 1
(x- 1)2
+1-1
+1
02. Questao: f(x) = |x+ 1|-px
2 + 2
f [0, 1] y = 1 y = -1
f x = 1
2
y = 2
f [0, 1] x = 1 x = -1
f [0, 1]
f y = 1 y = -1
03. Questao:
f(x) =
�x+ 1, x ,
|x|+ 1, x .
g(x) =f(x)
|x|.
f Rf (-1, 0] g
f x = 0 g (0,1) g
f [0,1) g x = 0 g
f g R g
04. Questao: f : [a, b] ! R
f f
f M m (f) = [m,M]
c, d 2 [a, b] ⇠ 2 (min{f(c), f(d)},max{f(c), f(d)}) p 2 (c, d) f(p) = ⇠
f p 2 [a, b] f(p) = 0
(ii) (iii)
(i) (ii) (iii) (iv)
(i) (ii) (iv)
05. Questao: f : R ! R f(x) =
�a+ bx, x > 2
3, x = 2
b- ax
2
, x < 2
a, b 2 R
f a- b = 2
b- a = 2 f
a 2 (-1
2
, 0] b 2 [1, 2) f
f
a 2 (-1
2
, 0] b 2 [0, 1) f
06. Questao: f : R \ {0} ! R f(x) = x
1
|x|
x!-1f(x) =
u!0
-
u
u
f
f(0)
f
f
07. Questao:x!1
1
x
= 1
" > 0 � =
�1
2
,
"
2
�0 < |x- 1| < �
����1
x
- 1
���� < "
" > 0 � = {1, 2"} 0 < |x- 1| < �
����1
x
- 1
���� < "
" > 0 � =
�1
2
,
"
2
� ����1
x
- 1
���� < " 0 < |x- 1| < �
" > 0 � = {1, 2"}
����1
x
- 1
���� < " 0 < |x- 1| < �
x!1
1
x
= 1
x!1
+
1
x
= 1
x!1
-
1
x
= -1
08. Questao:x!0
⇣x
2 (3x)1- (x)
⌘
�x
2
�.
1
12
7
6
09. Questao: p(x) = 32x
5 - 80x
4 - 80x
3 + 200x
2 + 18x- 45
p(x)
p(x)
p(x)
p(x)
p(x)
10. Questao: f(x) =
✓2+ (
x
3 - 2
x
2 + 1
)
◆1+ (x2)
f
0(x) =�1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) =⇣2+
⇣x
3-2
x
2+1
⌘⌘1+ (x2)
1+ (x2)
2+⇣
x
3-2
x
2+1
⌘⇣
x
3-2
x
2+1
⌘x
4+3x
2+4x
(x2+1)2 - 2x (x2)⇣2+
⇣x
3-2
x
2+1
⌘⌘�
f
0(x) =h ⇣
x
2-2
x
2+1
⌘ �1+ (x2)
�+ [1+ (x2)]
⇣2+
⇣x
3-2
x
2+1
⌘⌘i⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) = 6x
(x2+1)2
⇣x
2-2
x
2+1
⌘ �1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) = -2x (x2)�1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04.2013 - B
NOME: No. USP: NOTA:
Questao Resposta Valor Questao Resposta Valor
01a. (a) (b) (c) (d) (e) 06a. (a) (b) (c) (d) (e)
02a. (a) (b) (c) (d) (e) 07a. (a) (b) (c) (d) (e)
03a. (a) (b) (c) (d) (e) 08a. (a) (b) (c) (d) (e)
04a. (a) (b) (c) (d) (e) 09a. (a) (b) (c) (d) (e)
05a. (a) (b) (c) (d) (e) 10a. (a) (b) (c) (d) (e)
Regras: 1 - UNICA SEM RASURA2 -3 -
01. Questao:x!0
x
4 + 5x- 1
2-px
2 + 4
x!1
x
3 - 1
(x- 1)2
+1+1
-1
02. Questao: f(x) = |x+ 1|-px
2 + 2
f x = 1
2
y = 2
f [0, 1] y = 1 y = -1
f [0, 1] x = 1 x = -1
f y = 1 y = -1
f [0, 1]
03. Questao:
f(x) =
�x+ 1, x ,
|x|+ 1, x .
g(x) =f(x)
|x|.
f g R g
f [0,1) g x = 0 g
f x = 0 g (0,1) g
f (-1, 0] g
f R
04. Questao: f : [a, b] ! R
f f
f M m (f) = [m,M]
c, d 2 [a, b] ⇠ 2 (min{f(c), f(d)},max{f(c), f(d)}) p 2 (c, d) f(p) = ⇠
f p 2 [a, b] f(p) = 0
(i) (ii) (iii) (iv)
(ii) (iii)
(i) (ii) (iv)
05. Questao: f : R ! R f(x) =
�a+ bx, x > 2
3, x = 2
b- ax
2
, x < 2
a, b 2 R
f a- b = 2
b- a = 2 f
a 2 (-1
2
, 0] b 2 [0, 1) f
f
a 2 (-1
2
, 0] b 2 [1, 2) f
06. Questao: f : R \ {0} ! R f(x) = x
1
|x|f
f
f(0)
f
x!-1f(x) =
u!0
-
u
u
07. Questao:x!1
1
x
= 1
" > 0 � =
�1
2
,
"
2
� ����1
x
- 1
���� < " 0 < |x- 1| < �
" > 0 � = {1, 2"} 0 < |x- 1| < �
����1
x
- 1
���� < "
" > 0 � =
�1
2
,
"
2
�0 < |x- 1| < �
����1
x
- 1
���� < "
" > 0 � = {1, 2"}
����1
x
- 1
���� < " 0 < |x- 1| < �
x!1
1
x
= 1
x!1
+
1
x
= 1
x!1
-
1
x
= -1
08. Questao:x!0
⇣x
2 (3x)1- (x)
⌘
�x
2
�.
1
7
6
12
09. Questao: p(x) = 32x
5 - 80x
4 - 80x
3 + 200x
2 + 18x- 45
p(x)
p(x)
p(x)
p(x)
p(x)
10. Questao: f(x) =
✓2+ (
x
3 - 2
x
2 + 1
)
◆1+ (x2)
f
0(x) =�1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) =h ⇣
x
2-2
x
2+1
⌘ �1+ (x2)
�+ [1+ (x2)]
⇣2+
⇣x
3-2
x
2+1
⌘⌘i⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) =⇣2+
⇣x
3-2
x
2+1
⌘⌘1+ (x2)
1+ (x2)
2+⇣
x
3-2
x
2+1
⌘⇣
x
3-2
x
2+1
⌘x
4+3x
2+4x
(x2+1)2 - 2x (x2)⇣2+
⇣x
3-2
x
2+1
⌘⌘�
f
0(x) = 6x
(x2+1)2
⇣x
2-2
x
2+1
⌘ �1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) = -2x (x2)�1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04.2013 - C
NOME: No. USP: NOTA:
Questao Resposta Valor Questao Resposta Valor
01a. (a) (b) (c) (d) (e) 06a. (a) (b) (c) (d) (e)
02a. (a) (b) (c) (d) (e) 07a. (a) (b) (c) (d) (e)
03a. (a) (b) (c) (d) (e) 08a. (a) (b) (c) (d) (e)
04a. (a) (b) (c) (d) (e) 09a. (a) (b) (c) (d) (e)
05a. (a) (b) (c) (d) (e) 10a. (a) (b) (c) (d) (e)
Regras: 1 - UNICA SEM RASURA2 -3 -
01. Questao:
f(x) =
�x+ 1, x ,
|x|+ 1, x .
g(x) =f(x)
|x|.
f g R g
f [0,1) g x = 0 g
f x = 0 g (0,1) g
f (-1, 0] g
f R
02. Questao: f : [a, b] ! R
f f
f M m (f) = [m,M]
c, d 2 [a, b] ⇠ 2 (min{f(c), f(d)},max{f(c), f(d)}) p 2 (c, d) f(p) = ⇠
f p 2 [a, b] f(p) = 0
(i) (ii) (iii) (iv)
(ii) (iii)
(i) (ii) (iv)
03. Questao: f : R ! R f(x) =
�a+ bx, x > 2
3, x = 2
b- ax
2
, x < 2
a, b 2 R
f a- b = 2
b- a = 2 f
a 2 (-1
2
, 0] b 2 [0, 1) f
f
a 2 (-1
2
, 0] b 2 [1, 2) f
04. Questao:x!0
x
4 + 5x- 1
2-px
2 + 4
x!1
x
3 - 1
(x- 1)2
+1+1
-1
05. Questao: f(x) = |x+ 1|-px
2 + 2
f x = 1
2
y = 2
f [0, 1] y = 1 y = -1
f [0, 1] x = 1 x = -1
f y = 1 y = -1
f [0, 1]
06. Questao:x!0
⇣x
2 (3x)1- (x)
⌘
�x
2
�.
1
7
6
12
07. Questao: p(x) = 32x
5 - 80x
4 - 80x
3 + 200x
2 + 18x- 45
p(x)
p(x)
p(x)
p(x)
p(x)
08. Questao: f(x) =
✓2+ (
x
3 - 2
x
2 + 1
)
◆1+ (x2)
f
0(x) =�1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) =h ⇣
x
2-2
x
2+1
⌘ �1+ (x2)
�+ [1+ (x2)]
⇣2+
⇣x
3-2
x
2+1
⌘⌘i⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) =⇣2+
⇣x
3-2
x
2+1
⌘⌘1+ (x2)
1+ (x2)
2+⇣
x
3-2
x
2+1
⌘⇣
x
3-2
x
2+1
⌘x
4+3x
2+4x
(x2+1)2 - 2x (x2)⇣2+
⇣x
3-2
x
2+1
⌘⌘�
f
0(x) = 6x
(x2+1)2
⇣x
2-2
x
2+1
⌘ �1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) = -2x (x2)�1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
09. Questao: f : R \ {0} ! R f(x) = x
1
|x|f
f
f(0)
f
x!-1f(x) =
u!0
-
u
u
10. Questao:x!1
1
x
= 1
" > 0 � =
�1
2
,
"
2
� ����1
x
- 1
���� < " 0 < |x- 1| < �
" > 0 � = {1, 2"} 0 < |x- 1| < �
����1
x
- 1
���� < "
" > 0 � =
�1
2
,
"
2
�0 < |x- 1| < �
����1
x
- 1
���� < "
" > 0 � = {1, 2"}
����1
x
- 1
���� < " 0 < |x- 1| < �
x!1
1
x
= 1
x!1
+
1
x
= 1
x!1
-
1
x
= -1
2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04.2013 - D
NOME: No. USP: NOTA:
Questao Resposta Valor Questao Resposta Valor
01a. (a) (b) (c) (d) (e) 06a. (a) (b) (c) (d) (e)
02a. (a) (b) (c) (d) (e) 07a. (a) (b) (c) (d) (e)
03a. (a) (b) (c) (d) (e) 08a. (a) (b) (c) (d) (e)
04a. (a) (b) (c) (d) (e) 09a. (a) (b) (c) (d) (e)
05a. (a) (b) (c) (d) (e) 10a. (a) (b) (c) (d) (e)
Regras: 1 - UNICA SEM RASURA2 -3 -
01. Questao:
f(x) =
�x+ 1, x ,
|x|+ 1, x .
g(x) =f(x)
|x|.
f Rf (-1, 0] g
f x = 0 g (0,1) g
f [0,1) g x = 0 g
f g R g
02. Questao: f : [a, b] ! R
f f
f M m (f) = [m,M]
c, d 2 [a, b] ⇠ 2 (min{f(c), f(d)},max{f(c), f(d)}) p 2 (c, d) f(p) = ⇠
f p 2 [a, b] f(p) = 0
(ii) (iii)
(i) (ii) (iii) (iv)
(i) (ii) (iv)
03. Questao:x!0
x
4 + 5x- 1
2-px
2 + 4
x!1
x
3 - 1
(x- 1)2
+1-1
+1
04. Questao: f(x) = |x+ 1|-px
2 + 2
f [0, 1] y = 1 y = -1
f x = 1
2
y = 2
f [0, 1] x = 1 x = -1
f [0, 1]
f y = 1 y = -1
05. Questao:x!0
⇣x
2 (3x)1- (x)
⌘
�x
2
�.
1
12
7
6
06. Questao: p(x) = 32x
5 - 80x
4 - 80x
3 + 200x
2 + 18x- 45
p(x)
p(x)
p(x)
p(x)
p(x)
07. Questao: f(x) =
✓2+ (
x
3 - 2
x
2 + 1
)
◆1+ (x2)
f
0(x) =�1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) =⇣2+
⇣x
3-2
x
2+1
⌘⌘1+ (x2)
1+ (x2)
2+⇣
x
3-2
x
2+1
⌘⇣
x
3-2
x
2+1
⌘x
4+3x
2+4x
(x2+1)2 - 2x (x2)⇣2+
⇣x
3-2
x
2+1
⌘⌘�
f
0(x) =h ⇣
x
2-2
x
2+1
⌘ �1+ (x2)
�+ [1+ (x2)]
⇣2+
⇣x
3-2
x
2+1
⌘⌘i⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) = 6x
(x2+1)2
⇣x
2-2
x
2+1
⌘ �1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
f
0(x) = -2x (x2)�1+ (x2)
� ⇣2+
⇣x
3-2
x
2+1
⌘⌘ (x2)
08. Questao: f : R ! R f(x) =
�a+ bx, x > 2
3, x = 2
b- ax
2
, x < 2
a, b 2 R
f a- b = 2
b- a = 2 f
a 2 (-1
2
, 0] b 2 [1, 2) f
f
a 2 (-1
2
, 0] b 2 [0, 1) f
09. Questao: f : R \ {0} ! R f(x) = x
1
|x|
x!-1f(x) =
u!0
-
u
u
f
f(0)
f
f
10. Questao:x!1
1
x
= 1
" > 0 � =
�1
2
,
"
2
�0 < |x- 1| < �
����1
x
- 1
���� < "
" > 0 � = {1, 2"} 0 < |x- 1| < �
����1
x
- 1
���� < "
" > 0 � =
�1
2
,
"
2
� ����1
x
- 1
���� < " 0 < |x- 1| < �
" > 0 � = {1, 2"}
����1
x
- 1
���� < " 0 < |x- 1| < �
x!1
1
x
= 1
x!1
+
1
x
= 1
x!1
-
1
x
= -1
