# Thread: Help with a few questions

1. ## Help with a few questions

Hi all,

Im kinda new here and have been set some work outside of class to try and do, I have read through all of my examples and notes that I have and can't seem to find anything usefull to help me with these...

I believe they all involve differentiation and integration, so I have put them together. I have managed to do 3b luckily.. but not 100% sure its right.

Anyway here it is:

Any help is appreciated!

Thanks, Ramdrop

2. What kind of help do you want? What definition of "continuous" are you using? Do you know that "f(x) is continuous at x= a if and only if, given a sequence $\displaystyle \{a_n\}$ that converges to a, $\displaystyle f(a_n)$ converges to f(a)"? If so then consider a sequence made entirely of rational numbers and a sequence made entirely of irrational numbers, both converging to a.

As far as "differentiable" is concerned, the derivative at x= 0 is defined as $\displaystyle \lim_{h\to 0} \frac{f(h)- f(0)}{h}= \lim_{h\to 0}\frac{f(h)}{h}$. Again, you can look at all sequences $\displaystyle \{a_n\}$ that converge to 0.

For (5) an obvious "appropriate function" is x- 1- ln x. What is the derivative of the that? What does that tell you?

Since (6) says "By considering primitives for $\displaystyle \frac{1}{1+ x^2}$", what is a "primitive" (integral) for that function. You might also want to note that $\displaystyle \frac{1}{1+ x^2}= \frac{1}{1- (-x^2}$ and think about the sum of a geometric series. That will allow you to write $\displaystyle \frac{1}{1+ x^2}$ as a power series. What is the term-by-term integral of that?

3. Originally Posted by HallsofIvy
For (5) an obvious "appropriate function" is x- 1- ln x. What is the derivative of the that? What does that tell you?
I assume you move it over the the other side, so you have
0 (lessthan equal to) x - 1 - ln(x): If i differentiate that, I get 1 - 1/x.

If i was to move that back say, 1/x (lessthan equal to) 1. Multiply through by x, to get 1 on the left, and x on the right.
So: 1 (lessthan equal to) x - for all x>0..

Or am i going completely wrong?
Since (6) says "By considering primitives for $\displaystyle \frac{1}{1+ x^2}$", what is a "primitive" (integral) for that function. You might also want to note that $\displaystyle \frac{1}{1+ x^2}= \frac{1}{1- (-x^2}$ and think about the sum of a geometric series. That will allow you to write $\displaystyle \frac{1}{1+ x^2}$ as a power series. What is the term-by-term integral of that?
Here I would just work out the integral of the function $\displaystyle \frac{1}{1+ x^2}$ . I think I could do thatm but again, unsure where to go after..

The first question (3a) makes absolute no sence whatsoever..btw, continuious in the form of, YOu can draw the function without removing your pen off of the paper

4. Originally Posted by ramdrop
I assume you move it over the the other side, so you have
0 (lessthan equal to) x - 1 - ln(x): If i differentiate that, I get 1 - 1/x.

If i was to move that back say, 1/x (lessthan equal to) 1. Multiply through by x, to get 1 on the left, and x on the right.
So: 1 (lessthan equal to) x - for all x>0..

Or am i going completely wrong?

Here I would just work out the integral of the function $\displaystyle \frac{1}{1+ x^2}$ . I think I could do thatm but again, unsure where to go after..
Yes, now, as I asked before, what is the integral?

The first question (3a) makes absolute no sence whatsoever..btw, continuious in the form of, YOu can draw the function without removing your pen off of the paper
I don't believe that "removing your pen off of the paper" is a mathematical definition! At best it is a rough analogy. Doesn't your text book define "continuous"?

5. The Integral i got by using substitution, using $\displaystyle x = tan(theta)$ and have:
$\displaystyle {integral}d(theta)$.... which = $\displaystyle arctan(theta) + C$

Which coinsidently is tan^-1(x).. which is shown below oO now i have to figure where to go after.. ;x

And the best definition that I can find, is that it continues forever, and when drawing the function, you should be able to draw the function without removing your pen off of the paper.

Hope thats close or. .?

6. Your questions look like they come from an assignment that counts towards your final grade. MHF policy is to not knowingly help with such questions. You can pm me and discuss this if you want.