Mathematics

Problem

Consider the concept of derivative in differential calculus. a. (i) State the definition of the derivative of a function

f(x)

at a point

x=a

. b. (ii) Explain, without calculation, how the derivative gives the slope of the tangent to the curve

y=f(x)

at the point

(a,f(a))

. c. (iii) For the function

y=\tan x

on the interval

0<x<\pi

, describe, with reasons based on the sign of

\dfrac{dy}{dx}

, the portions of the interval where the curve is increasing and where it is decreasing.