Determine whether the sequence is inc, dec, etc:

[MarkSA]

Hello,

My questions say:
Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

What is the proper way to determine these qualities of the sequence? I'm not really sure what I should show for work on these. Some are very obviously increasing/decreasing or bounded.

One of the more difficult ones I think is: a(sub n) = n/[n^2 + 1]

What I was doing was taking the limit to find out if the sequences are bounded.. since if it's convergent it must be bounded by the number the limit evaluates to, correct? I'm not really sure how to check for increasing,decreasing,monotonic though on some of the more difficult ones.

 

[stapel]

You're in calculus; use those tools! :wink:

What do the first and second derivatives tell you about the function f(x) = x / (x[sup:25qeqyl2]2[/sup:25qeqyl2] + 1) for x > 0?

Eliz.

 

[MarkSA]

If the first derivative is always >0 from even exponents, then I suppose it would be increasing. I'm not sure how the 2nd derivative factors into it. All I remember about that is using it to determine concave up/down.

I suppose I could solve the limit and if it's convergent then it must be bounded. If it's divergent it is unbounded? Any other way to determine boundedness?

How would I determine if it's not monotonic? I think I would get some first derivatives that aren't very obvious as to always increasing/decreasing.

 

[stapel]

If the first derivative is always >0 from even exponents...

Sorry, but I don't know what this means...? You don't have an even polynomial function; you have a rational function. Apply the Quotient Rule, find the derivative, set this "equal to zero", and find the critical points.

I'm not sure how the 2nd derivative factors into it.

If the second derivative if positive on an interval, what does this say about the original function? If a critical point occurs in an interval on which the second derivative is negative, what sort of extrema occurs at that critical point? If a function is non-negative on an interval (say, for all x > 2), the first derivative is always negative, and the second derivative is always positive, what can you say about whether or not the function is increasing or decreasing, and whether or not there is some lower bound on the values?

Eliz.

 

[PAULK]

Hello,

My questions say:
Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

What is the proper way to determine these qualities of the sequence? I'm not really sure what I should show for work on these. Some are very obviously increasing/decreasing or bounded.

One of the more difficult ones I think is: a(sub n) = n/[n^2 + 1]

What I was doing was taking the limit to find out if the sequences are bounded.. since if it's convergent it must be bounded by the number the limit evaluates to, correct? I'm not really sure how to check for increasing,decreasing,monotonic though on some of the more difficult ones.

..........................................................
Look at the difference between consecutive terms:

a[n+1] - a[n].

If you can prove that is always positive, your sequence is monotone increasing.
If you can prove that is always negative, your sequence is monotone decreasing.

In this case,
n + 1 n
a[n+1] - a[n] = ----------- - ---------
(n+1)^2 + 1 n^2 + 1

After some routine hackwork, you find that to be:

- n^3 - n^2 - n + 1
-----------------------------
(n^2 + 2n + 2)(n^2 + 1)

Now the bottom is always +, for we always have n >= 1. All you have to do is show (in this case) that the top is negative. Not too hard for n >= 1, so your sequence is monotone decreasing.

 

[MarkSA]

Sorry, but I don't know what this means...? You don't have an even polynomial function; you have a rational function. Apply the Quotient Rule, find the derivative, set this "equal to zero", and find the critical points.

I was speaking in general since this was a trick i had to use in the past once or twice.. but after taking the derivative on this one I see that it has the -x^2 + 1 in the numerator. Normally I would have ignored this if we were still doing the section on properties of derivatives, but I see now that since n>=1 in the sequence, this means that the derivative will always be <= 0 (or decreasing..)

If second derivative is positive in the interval, it's concave up I believe, opposite for concave down. I seem to recall critical points being inflection points but there may be a caveat to that i'm forgetting. That's all I remember about the 2nd derivative properties...
"If a critical point occurs in an interval on which the second derivative is negative, what sort of extrema occurs at that critical point? " <-- i'm not sure about this one

But thanks I think I have the gist of this now. I was able to solve I believe all of the homework problems of this type by examining the derivative and solving the limits.

 

[stapel]

I think I have the gist of this now. I was able to solve I believe all of the homework problems of this type by examining the derivative and solving the limits.

Excellent!

Eliz.