Skip to main content

[Community Question] Calculus: Simple and short true-false tasks regarding Precalculus

One of our user asked:

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

Labels

Labels:

Comments

Post a Comment

Popular posts from this blog

[Community Question] Calculus: Manifold with boundary - finding the boundary

One of our user asked: I have the manifold with boundary $M:= \lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1\geq 0, x_1^2+x_2^2+x_3^2=1\rbrace \cup\lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1= 0, x_1^2+x_2^2+x_3^2\leq1\rbrace$ and I need to find the boundary of this manifold. I think it is $\lbrace (x_1,x_2,x_3) \in \mathbb R^n : x_1= 0, x_2^2+x_3^2=1\rbrace$ , the other option is that the boundary is the empty set? I think the first is right? Am I wrong?
Post a Comment
Read more

[Community Question] Linear-algebra: Are linear transformations between infinite dimensional vector spaces always differentiable?

One of our user asked: In class we saw that every linear transformation is differentiable (since there's always a linear approximation for them) and we also saw that a differentiable function must be continuous, so it must be true that all linear operators are continuous, however, I just read that between infinite dimensional vector spaces this is not necessarily true. I would like to know where's the flaw in my reasoning (I suspect that linear transformations between infinite dimensional vector spaces are not always differentiable).
Post a Comment
Read more