Hello, ubhutto!
Square both sides: .
Let
Then we have: .
Hence: .
Back-substitute: .
hey everyone i had 2 questions
firstly
If y = sqrt(x^2 + 1) , then the derivative of y^2 with respect to x^2 is
i'm not sure where to start and its a mc question so i tried to first take the derivative of x^2 and with respect to dy/dx but i don't think that is right
secondly
the mean value theorem states that f(x) has to be continuous at [a,b], a and b being the end points, and f(x) has to be differentiable at (a,b), again a and b being the end points. if this is true, are the end points included in the mean value theorem
thanks alot
With regards to your second question: yes they are included. The conditions ask for to be continuous on . This is does not mean is continuous at and (it does mean should be defined at those points) so we must allow to be non-differentiable at and hence the open interval. For instance, the function:
is continuous on but discontinuous ( not differentiable) at -1 and 1. The conditions mean that we may apply the theorem on or any closed interval contained in .