integrate tan^-1t from o to x in fundamental theorem of Calculus find the increasing and concave up

Question

you have to integrate the problem using fundamental theorem of Calculus to find the intervals for the increasing and concave up 

Answer 1

The fundamental thereom of calculus is

 

df(x)/dx = limit (f(x+Δx) - f(x)) / Δx, as Δx goes to 0.

 

When we let apply this to tan^-1 [arctan], we get something like this:

 

d[arctan(x)]/dx = limit ( arctan(x+Δx) - arctan(x) ) / (Δx), as Δx goes to 0.

 

I would, starting with x=0 and Δx equals something small like 0.1, evaluate the derivative at various points (graph arctan(x) to see where it make sense to end ([0 to ?])).. then you can use your increasing/decreasing and concavity rules to finish the question.