Expert: Scotto - 12/13/2007

Question
1)How do I locate extrema on a closed interval?
2) What is differentiability?
3) What is Intermediate Value Theorem
4) What is implicit differentiation

5) and how do I solve these equations.. I don't need you to solve this! I would like to know which Trig. identities to use.
secXcscX = 2cscX   and   
cos2X + sinX =1 (the two  between cos and X is a square) In this case can I not move 1 to the other side? Is th is like a proof?

and the last question... (sorry for many many questions...)
6) How do I solve limits..??
  FOr an example...
lim      x2-4
x->-2    ----
        x3+8  This is as graph approaches -2, whatis its limit? (the equation is x square minus 4 all over x cube plus 8)

Please help...
I don't have a calculus book because I left it at my school. It is a boarding school...

Thank you..




I've been looking for the answers on the internet... but I couldn't understand... Can you explain these in simple words?

Answer
1)How do I locate extrema on a closed interval?
Solve the derivative for f'(x) = 0 and check the value of the function at endpoints of the interval.

2) What is differentiability?
If the function is continuous and smooth, it has a derivative.

3) What is Intermediate Value Theorem?
If a function is continuous on the interval (a,b), then if any value c is chosen between f(a) and f(b), then there exists some point x between a and b such that f(x)=c.

4) What is implicit differentiation?
If you have a function where y is part of the function, for example, 25 = x^2 + y^2, then differentiating we get
0 = 2x +2yy'.

5) It should be cos^2(x) + sin^2(x) = 1  (I think you just forgot to square sin in your question).  Also, sec(x) = 1/cos(x) and csc(x) = 1/sin(x) should come in handy.  I myself always find problems easier if I reduce everything to sin(x) and cos(x) and go from there.

6) Note that the top and bottom are both divisible by
(x-(-2)) = (x+2).

I hope that's good enough - good luck.