2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04...
Transcript of 2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04...
![Page 1: 2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04 ...conteudo.icmc.usp.br/pessoas/andcarva/sma301/Prova02_A_B_C_D.pdf · 2. PROVA DE SMA301-CALCULO I - Professor Alexandre](https://reader033.fdocuments.co/reader033/viewer/2022041811/5e57c9283ca89c754272d4da/html5/thumbnails/1.jpg)
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)
![Page 2: 2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04 ...conteudo.icmc.usp.br/pessoas/andcarva/sma301/Prova02_A_B_C_D.pdf · 2. PROVA DE SMA301-CALCULO I - Professor Alexandre](https://reader033.fdocuments.co/reader033/viewer/2022041811/5e57c9283ca89c754272d4da/html5/thumbnails/2.jpg)
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)
![Page 3: 2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04 ...conteudo.icmc.usp.br/pessoas/andcarva/sma301/Prova02_A_B_C_D.pdf · 2. PROVA DE SMA301-CALCULO I - Professor Alexandre](https://reader033.fdocuments.co/reader033/viewer/2022041811/5e57c9283ca89c754272d4da/html5/thumbnails/3.jpg)
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)
![Page 4: 2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04 ...conteudo.icmc.usp.br/pessoas/andcarva/sma301/Prova02_A_B_C_D.pdf · 2. PROVA DE SMA301-CALCULO I - Professor Alexandre](https://reader033.fdocuments.co/reader033/viewer/2022041811/5e57c9283ca89c754272d4da/html5/thumbnails/4.jpg)
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)
![Page 5: 2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04 ...conteudo.icmc.usp.br/pessoas/andcarva/sma301/Prova02_A_B_C_D.pdf · 2. PROVA DE SMA301-CALCULO I - Professor Alexandre](https://reader033.fdocuments.co/reader033/viewer/2022041811/5e57c9283ca89c754272d4da/html5/thumbnails/5.jpg)
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
![Page 6: 2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04 ...conteudo.icmc.usp.br/pessoas/andcarva/sma301/Prova02_A_B_C_D.pdf · 2. PROVA DE SMA301-CALCULO I - Professor Alexandre](https://reader033.fdocuments.co/reader033/viewer/2022041811/5e57c9283ca89c754272d4da/html5/thumbnails/6.jpg)
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
![Page 7: 2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04 ...conteudo.icmc.usp.br/pessoas/andcarva/sma301/Prova02_A_B_C_D.pdf · 2. PROVA DE SMA301-CALCULO I - Professor Alexandre](https://reader033.fdocuments.co/reader033/viewer/2022041811/5e57c9283ca89c754272d4da/html5/thumbnails/7.jpg)
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
![Page 8: 2. PROVA DE SMA301-CALCULO I - Professor Alexandre 29.04 ...conteudo.icmc.usp.br/pessoas/andcarva/sma301/Prova02_A_B_C_D.pdf · 2. PROVA DE SMA301-CALCULO I - Professor Alexandre](https://reader033.fdocuments.co/reader033/viewer/2022041811/5e57c9283ca89c754272d4da/html5/thumbnails/8.jpg)
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