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 :/
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[.
I may have some other questions but for now these are mine.
Re: Derivatives exercises
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.