Questions regarding to calculus (please help)?

Question

Hey guys, I was just wondering if y = |x| can be considered as a bounded function? In addition to that, could you also tell me the definition of bounded function with an example.
My another question is if f(x) = 1/n, 1/n =x , n = 1, 2, 3, 4, and so on...is the function continuous, bounded, differentiable and integrable?
Thanks in advance.

Answer 1

You can look up the definition of a bounded function in any analysis book. y = |x| can't be bounded because as x-> infinity, so does y.

For your other question, since the function is defined only for the integers, it can't be be continuous which means it can't be differentiable. it is bounded because lim as x->inf, f(x) ->0. Since it is discrete, the Riemann Integral does not exist.

Answer 2

A bounded function is a function whose range is bounded That is,it has a maximum value and a minimum value for all x in the domain. An example is y=sin x which has an upper bound of +1 and a lower bound of -1 for all x in the designated domain.

y=|x| has a lower bound of 0 but it has no upper bound. Hence it is unbounded.

Your last question is unclear. I interpret it to be saying f(x) = x.
If this is correct, then it is continuous, unbounded, differentiable, and integrable.

Answer 3

A=?r^two When A=400?, ?r^two=400? r^two=four hundred r=20 Differentiating implicitly with admire to time t: dA/dt=2?r(dr/dt) As dA/dt=five, 2?r(dr/dt)=five When r=20, 2?(20)(dr/dt)=five dr/dt=five/40?=one million/(8?) cm/s