Simple and short true-false tasks regarding Precalculus

0

$begingroup$

Here are few of the questions from the previous years' exams. I've chosen the ones I'm not sure about. It's a simple TRUE/FALSE task. Would anyone be able to verify my solution? Some of my answers are good, some are just random guess according to my intuition. I don't really need a detailed explanation... Thanks!

1. Domain of $f'$ is contained within domain of $f$. - TRUE


2. Boundary point of set A is also a cluster point of that set. - TRUE


3. Every increasing sequence and bounded above is convergent. - TRUE


4. Every increasing sequence and bounded below is convergent. - FALSE


5. Every increasing sequence is always bounded below. - TRUE


6. Every sequence is discontinuous function. - FALSE


7. Every sequence is continuous function. - TRUE


8. Every function integrable on $<a, b>$ is continuous on $<a, b>$. - FALSE


9. Function $f(x) = ln{|x|}$ is discontinuous at $0$. - TRUE


10. The continuity is necessary for differentiability. - TRUE


11. Function $f(x) = frac{x}{|x|}$ is monotonic. - FALSE

real-analysis calculus algebra-precalculus derivatives education

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asked Jan 29 at 19:11

wenoweno

37011

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I would say you missed 2, 9, 11
  $endgroup$
  – MPW
  Jan 29 at 19:15
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I do not know what is "cluster point" (language problem). For the rest: 1,3,4,5,6,7,8,10 are fine for me. 9 - for me is the function not defined at 0, so it cannot have a property there. 11 - f is monotonic!
  $endgroup$
  – user376343
  Jan 29 at 19:20
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  For 11: Constant functions are monotonic. Monotonicity is usually distinguished from strict monotonicity. For 9: it is neither continuous nor discontinuous at a point that is not in its domain; it is simply undefined there.
  $endgroup$
  – MPW
  Jan 29 at 19:21
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Monotonic signifies "doesn´t change from increasing to decreasing and vice versa".
  $endgroup$
  – user376343
  Jan 29 at 19:21
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @user376343 : Not quite, since constant functions can be regarded as increasing or decreasing anywhere you like (you would need to be more specific). A function $f:Xto Y$ is monotone if $f^{-1}(y)$ is connected (or empty) for each $yin Y$. In the case where $X$ and $Y$ are (subsets of) $mathbb R$, this means $f^{-1}(y)$ is a (possibly degenerate) interval for each $y$ in the image. We confirm this by noting that the image is ${-1,1}$ and the preimages are $f^{-1}(-1)=(-infty,0)$ and $f^{-1}(1)=(0,infty)$ which are both connected.
  $endgroup$
  – MPW
  Jan 29 at 19:26
  
  
  

  
  
  
  

 | 
show 4 more comments

0

$begingroup$

Here are few of the questions from the previous years' exams. I've chosen the ones I'm not sure about. It's a simple TRUE/FALSE task. Would anyone be able to verify my solution? Some of my answers are good, some are just random guess according to my intuition. I don't really need a detailed explanation... Thanks!

1. Domain of $f'$ is contained within domain of $f$. - TRUE


2. Boundary point of set A is also a cluster point of that set. - TRUE


3. Every increasing sequence and bounded above is convergent. - TRUE


4. Every increasing sequence and bounded below is convergent. - FALSE


5. Every increasing sequence is always bounded below. - TRUE


6. Every sequence is discontinuous function. - FALSE


7. Every sequence is continuous function. - TRUE


8. Every function integrable on $<a, b>$ is continuous on $<a, b>$. - FALSE


9. Function $f(x) = ln{|x|}$ is discontinuous at $0$. - TRUE


10. The continuity is necessary for differentiability. - TRUE


11. Function $f(x) = frac{x}{|x|}$ is monotonic. - FALSE

real-analysis calculus algebra-precalculus derivatives education

share|cite|improve this question

asked Jan 29 at 19:11

wenoweno

37011

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I would say you missed 2, 9, 11
  $endgroup$
  – MPW
  Jan 29 at 19:15
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I do not know what is "cluster point" (language problem). For the rest: 1,3,4,5,6,7,8,10 are fine for me. 9 - for me is the function not defined at 0, so it cannot have a property there. 11 - f is monotonic!
  $endgroup$
  – user376343
  Jan 29 at 19:20
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  For 11: Constant functions are monotonic. Monotonicity is usually distinguished from strict monotonicity. For 9: it is neither continuous nor discontinuous at a point that is not in its domain; it is simply undefined there.
  $endgroup$
  – MPW
  Jan 29 at 19:21
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Monotonic signifies "doesn´t change from increasing to decreasing and vice versa".
  $endgroup$
  – user376343
  Jan 29 at 19:21
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @user376343 : Not quite, since constant functions can be regarded as increasing or decreasing anywhere you like (you would need to be more specific). A function $f:Xto Y$ is monotone if $f^{-1}(y)$ is connected (or empty) for each $yin Y$. In the case where $X$ and $Y$ are (subsets of) $mathbb R$, this means $f^{-1}(y)$ is a (possibly degenerate) interval for each $y$ in the image. We confirm this by noting that the image is ${-1,1}$ and the preimages are $f^{-1}(-1)=(-infty,0)$ and $f^{-1}(1)=(0,infty)$ which are both connected.
  $endgroup$
  – MPW
  Jan 29 at 19:26
  
  
  

  
  
  
  

 | 
show 4 more comments

0

0

0

$begingroup$

Here are few of the questions from the previous years' exams. I've chosen the ones I'm not sure about. It's a simple TRUE/FALSE task. Would anyone be able to verify my solution? Some of my answers are good, some are just random guess according to my intuition. I don't really need a detailed explanation... Thanks!

1. Domain of $f'$ is contained within domain of $f$. - TRUE


2. Boundary point of set A is also a cluster point of that set. - TRUE


3. Every increasing sequence and bounded above is convergent. - TRUE


4. Every increasing sequence and bounded below is convergent. - FALSE


5. Every increasing sequence is always bounded below. - TRUE


6. Every sequence is discontinuous function. - FALSE


7. Every sequence is continuous function. - TRUE


8. Every function integrable on $<a, b>$ is continuous on $<a, b>$. - FALSE


9. Function $f(x) = ln{|x|}$ is discontinuous at $0$. - TRUE


10. The continuity is necessary for differentiability. - TRUE


11. Function $f(x) = frac{x}{|x|}$ is monotonic. - FALSE

real-analysis calculus algebra-precalculus derivatives education

share|cite|improve this question

asked Jan 29 at 19:11

wenoweno

37011

$endgroup$

Here are few of the questions from the previous years' exams. I've chosen the ones I'm not sure about. It's a simple TRUE/FALSE task. Would anyone be able to verify my solution? Some of my answers are good, some are just random guess according to my intuition. I don't really need a detailed explanation... Thanks!

1. Domain of $f'$ is contained within domain of $f$. - TRUE


2. Boundary point of set A is also a cluster point of that set. - TRUE


3. Every increasing sequence and bounded above is convergent. - TRUE


4. Every increasing sequence and bounded below is convergent. - FALSE


5. Every increasing sequence is always bounded below. - TRUE


6. Every sequence is discontinuous function. - FALSE


7. Every sequence is continuous function. - TRUE


8. Every function integrable on $<a, b>$ is continuous on $<a, b>$. - FALSE


9. Function $f(x) = ln{|x|}$ is discontinuous at $0$. - TRUE


10. The continuity is necessary for differentiability. - TRUE


11. Function $f(x) = frac{x}{|x|}$ is monotonic. - FALSE

real-analysis calculus algebra-precalculus derivatives education

real-analysis calculus algebra-precalculus derivatives education

share|cite|improve this question

asked Jan 29 at 19:11

wenoweno

37011

share|cite|improve this question

asked Jan 29 at 19:11

wenoweno

37011

share|cite|improve this question

share|cite|improve this question

asked Jan 29 at 19:11

wenoweno

37011

asked Jan 29 at 19:11

wenoweno

37011

asked Jan 29 at 19:11

wenoweno

37011

37011

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I would say you missed 2, 9, 11
  $endgroup$
  – MPW
  Jan 29 at 19:15
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I do not know what is "cluster point" (language problem). For the rest: 1,3,4,5,6,7,8,10 are fine for me. 9 - for me is the function not defined at 0, so it cannot have a property there. 11 - f is monotonic!
  $endgroup$
  – user376343
  Jan 29 at 19:20
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  For 11: Constant functions are monotonic. Monotonicity is usually distinguished from strict monotonicity. For 9: it is neither continuous nor discontinuous at a point that is not in its domain; it is simply undefined there.
  $endgroup$
  – MPW
  Jan 29 at 19:21
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Monotonic signifies "doesn´t change from increasing to decreasing and vice versa".
  $endgroup$
  – user376343
  Jan 29 at 19:21
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @user376343 : Not quite, since constant functions can be regarded as increasing or decreasing anywhere you like (you would need to be more specific). A function $f:Xto Y$ is monotone if $f^{-1}(y)$ is connected (or empty) for each $yin Y$. In the case where $X$ and $Y$ are (subsets of) $mathbb R$, this means $f^{-1}(y)$ is a (possibly degenerate) interval for each $y$ in the image. We confirm this by noting that the image is ${-1,1}$ and the preimages are $f^{-1}(-1)=(-infty,0)$ and $f^{-1}(1)=(0,infty)$ which are both connected.
  $endgroup$
  – MPW
  Jan 29 at 19:26
  
  
  

  
  
  
  

 | 
show 4 more comments

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I would say you missed 2, 9, 11
  $endgroup$
  – MPW
  Jan 29 at 19:15
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I do not know what is "cluster point" (language problem). For the rest: 1,3,4,5,6,7,8,10 are fine for me. 9 - for me is the function not defined at 0, so it cannot have a property there. 11 - f is monotonic!
  $endgroup$
  – user376343
  Jan 29 at 19:20
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  For 11: Constant functions are monotonic. Monotonicity is usually distinguished from strict monotonicity. For 9: it is neither continuous nor discontinuous at a point that is not in its domain; it is simply undefined there.
  $endgroup$
  – MPW
  Jan 29 at 19:21
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Monotonic signifies "doesn´t change from increasing to decreasing and vice versa".
  $endgroup$
  – user376343
  Jan 29 at 19:21
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @user376343 : Not quite, since constant functions can be regarded as increasing or decreasing anywhere you like (you would need to be more specific). A function $f:Xto Y$ is monotone if $f^{-1}(y)$ is connected (or empty) for each $yin Y$. In the case where $X$ and $Y$ are (subsets of) $mathbb R$, this means $f^{-1}(y)$ is a (possibly degenerate) interval for each $y$ in the image. We confirm this by noting that the image is ${-1,1}$ and the preimages are $f^{-1}(-1)=(-infty,0)$ and $f^{-1}(1)=(0,infty)$ which are both connected.
  $endgroup$
  – MPW
  Jan 29 at 19:26
  
  
  

  
  
  
  

1

1

$begingroup$
I would say you missed 2, 9, 11
$endgroup$
– MPW
Jan 29 at 19:15

$begingroup$
I would say you missed 2, 9, 11
$endgroup$
– MPW
Jan 29 at 19:15

1

1

$begingroup$
I do not know what is "cluster point" (language problem). For the rest: 1,3,4,5,6,7,8,10 are fine for me. 9 - for me is the function not defined at 0, so it cannot have a property there. 11 - f is monotonic!
$endgroup$
– user376343
Jan 29 at 19:20

$begingroup$
I do not know what is "cluster point" (language problem). For the rest: 1,3,4,5,6,7,8,10 are fine for me. 9 - for me is the function not defined at 0, so it cannot have a property there. 11 - f is monotonic!
$endgroup$
– user376343
Jan 29 at 19:20

1

1

$begingroup$
For 11: Constant functions are monotonic. Monotonicity is usually distinguished from strict monotonicity. For 9: it is neither continuous nor discontinuous at a point that is not in its domain; it is simply undefined there.
$endgroup$
– MPW
Jan 29 at 19:21

$begingroup$
For 11: Constant functions are monotonic. Monotonicity is usually distinguished from strict monotonicity. For 9: it is neither continuous nor discontinuous at a point that is not in its domain; it is simply undefined there.
$endgroup$
– MPW
Jan 29 at 19:21

1

1

$begingroup$
Monotonic signifies "doesn´t change from increasing to decreasing and vice versa".
$endgroup$
– user376343
Jan 29 at 19:21

$begingroup$
Monotonic signifies "doesn´t change from increasing to decreasing and vice versa".
$endgroup$
– user376343
Jan 29 at 19:21

1

1

$begingroup$
@user376343 : Not quite, since constant functions can be regarded as increasing or decreasing anywhere you like (you would need to be more specific). A function $f:Xto Y$ is monotone if $f^{-1}(y)$ is connected (or empty) for each $yin Y$. In the case where $X$ and $Y$ are (subsets of) $mathbb R$, this means $f^{-1}(y)$ is a (possibly degenerate) interval for each $y$ in the image. We confirm this by noting that the image is ${-1,1}$ and the preimages are $f^{-1}(-1)=(-infty,0)$ and $f^{-1}(1)=(0,infty)$ which are both connected.
$endgroup$
– MPW
Jan 29 at 19:26

$begingroup$
@user376343 : Not quite, since constant functions can be regarded as increasing or decreasing anywhere you like (you would need to be more specific). A function $f:Xto Y$ is monotone if $f^{-1}(y)$ is connected (or empty) for each $yin Y$. In the case where $X$ and $Y$ are (subsets of) $mathbb R$, this means $f^{-1}(y)$ is a (possibly degenerate) interval for each $y$ in the image. We confirm this by noting that the image is ${-1,1}$ and the preimages are $f^{-1}(-1)=(-infty,0)$ and $f^{-1}(1)=(0,infty)$ which are both connected.
$endgroup$
– MPW
Jan 29 at 19:26

 | 
show 4 more comments

1 Answer
1

active

oldest

votes

1

$begingroup$

1. Right


2. Wrong (What is the boundary of ${0}$ in $mathbb R$? What is its set of cluster points?)


3. Right


4. Right


5. Right


6. Right


7. Right


8. Right


9. Wrong (it isn't defined there)


10. Right


11. Wrong (think about what this function is)

share|cite|improve this answer

answered Jan 29 at 19:18

José Carlos SantosJosé Carlos Santos

171k23132240

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  It's up to you.
  $endgroup$
  – José Carlos Santos
  Jan 29 at 19:23
  

  
  
  
  

add a comment | 

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1 Answer
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1 Answer
1

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oldest

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active

oldest

votes

1

$begingroup$

1. Right


2. Wrong (What is the boundary of ${0}$ in $mathbb R$? What is its set of cluster points?)


3. Right


4. Right


5. Right


6. Right


7. Right


8. Right


9. Wrong (it isn't defined there)


10. Right


11. Wrong (think about what this function is)

share|cite|improve this answer

answered Jan 29 at 19:18

José Carlos SantosJosé Carlos Santos

171k23132240

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  It's up to you.
  $endgroup$
  – José Carlos Santos
  Jan 29 at 19:23
  

  
  
  
  

add a comment | 

1

$begingroup$

1. Right


2. Wrong (What is the boundary of ${0}$ in $mathbb R$? What is its set of cluster points?)


3. Right


4. Right


5. Right


6. Right


7. Right


8. Right


9. Wrong (it isn't defined there)


10. Right


11. Wrong (think about what this function is)

share|cite|improve this answer

answered Jan 29 at 19:18

José Carlos SantosJosé Carlos Santos

171k23132240

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  It's up to you.
  $endgroup$
  – José Carlos Santos
  Jan 29 at 19:23
  

  
  
  
  

add a comment | 

1

1

1

$begingroup$

1. Right


2. Wrong (What is the boundary of ${0}$ in $mathbb R$? What is its set of cluster points?)


3. Right


4. Right


5. Right


6. Right


7. Right


8. Right


9. Wrong (it isn't defined there)


10. Right


11. Wrong (think about what this function is)

share|cite|improve this answer

answered Jan 29 at 19:18

José Carlos SantosJosé Carlos Santos

171k23132240

$endgroup$

1. Right


2. Wrong (What is the boundary of ${0}$ in $mathbb R$? What is its set of cluster points?)


3. Right


4. Right


5. Right


6. Right


7. Right


8. Right


9. Wrong (it isn't defined there)


10. Right


11. Wrong (think about what this function is)

share|cite|improve this answer

answered Jan 29 at 19:18

José Carlos SantosJosé Carlos Santos

171k23132240

share|cite|improve this answer

share|cite|improve this answer

answered Jan 29 at 19:18

José Carlos SantosJosé Carlos Santos

171k23132240

answered Jan 29 at 19:18

José Carlos SantosJosé Carlos Santos

171k23132240

answered Jan 29 at 19:18

José Carlos SantosJosé Carlos Santos

171k23132240

171k23132240

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  It's up to you.
  $endgroup$
  – José Carlos Santos
  Jan 29 at 19:23
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  It's up to you.
  $endgroup$
  – José Carlos Santos
  Jan 29 at 19:23
  

  
  
  
  

1

1

$begingroup$
It's up to you.
$endgroup$
– José Carlos Santos
Jan 29 at 19:23

$begingroup$
It's up to you.
$endgroup$
– José Carlos Santos
Jan 29 at 19:23

add a comment | 

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