Calculus.. what values of A will f(x) be continuous for all x?

Question

f(x)=
{a^2-x^2 x<0
{acosx 0<=x

I'm just lost, how should I start? thanks.

Answer 1

Function y = f(x) is continuous at point x=a if the following three conditions are satisfied :

i.) f(a) is defined ,
ii.)lim f(x) exists (i.e., is finite) ,
and
iii.) lim (x ->a) f(x) = f(a)

Applying them to your case:
(1) f(0) is defined, being a^2 and a in two cases.
(2)limit also exists here
(3) f(0)=f(0-)=f(0+)
therefore,
a^2=a
or a(a-1)=0
or a= 0 or 1.

Answer 2

Well, f is obviously cts when x≠0, so you just need to choose a so that f is cts at 0, at 0 you have f(0)=acos(0)=a and you set this equal to a^2-0^2=a^2. Thus it is cts when a^2=a or a= 0 or 1.


Vivek, I like how thorough you are, but as a fan of topology, I like to just pull out the pasting lemma on these ones . . . all you have to do is look at the intersection of the closures (or 0 in this case) :)

Answer 3

a^2 = a implies a=1

Answer 4

I suggest a calculus calculator.

Answer 5

http://www.compute.uwlax.edu/calc/

GO THERE, ITS A ONLINE CALCULUS CALCULATOR

Answer 6

I'm not much on math, but this site should help www.calc101.com

Answer 7

no value,