area under the curve and derivatives

Hello I need some help with my calculus revision. I have troubles with some of the exercises the teacher gave us. Could you please help me resolve this. My exam is coming up and I have a hard time with derivatives.

I have to find the derivative of this function: f (x) = log2(cos(x)) + 2 × arctan( x). EDIT: Ok I found for this one but I have a problem with another number :/

This one:

Area under the curve:

a) Area under the curve of y= x^3 between x= 2 and x= 4

b) Area under the curve of y= sin(x) between x= π / 3 and x= 2π / 3 (that's pi)

And I have this one too:

F(x) is defined by |x| ≤ 2 with:

f (x) =

−1/ x if − 2 ≤ x < −1

x2 if −1 ≤ x ≤1

2x −1 if 1 < x ≤ 2

a. show continuity of f(x) on interval [-2, 2]

b. Values of x in ]-2, 2[ if f'(x) does not exist? Clue: use f'(x) to decide if x exists at x = -1 et x = 1.

c. Find all x values as f'(x) = 0. For all these values find f"(x) if it exists.

d. Find all local maximums and minimums of f(x) in ]-2, 2[ and find the global max and min of f(x) in ]-2, 2[.

Thank you!

I may have some other questions but for now these are mine.

Re: Derivatives exercises

Hey Emy28.

Hint: for the first one, try using the fact that all continuous functions in an interval satisfy lim x->a f(x) = f(a) and show that all the piecewise limits are the same from each side.

This is establishing continuity for a piecewise function.