compare function tanx and x^2 when x belong to (0,pi/2)

Question

it is related to functions of calculus,monotonic functions

Answer 1

The derivatives of the two functions, tan(x) and x², are sec²(x) and 2x respectively.

The derivatives are the gradients of the functions. At x=0, the derivatives are 1 and 0. At x=π/2, they are infinite (vertical) and π. The functions both evaluate to zero at x=0. tan(x) has a steep rise to vertical as x→π/2, while x² becomes π²/4 (about 2.47), rising at the rate of π (about 3.14). By way of comparison, x² has a gradient=1 (when x=0.5), which is the gradient of tan(x) when x=0. At x=1 tan(x) has a gradient of about 1.3.