= . , if cos , kx ≤π = > π at x = π . , if , if kx = > at x = . Find the values of a and b such that the function defined by , , if , ax b ≥ is a continuous function.
. Show that the function defined by f ( x ) = cos ( x ) is a continuous function. . Show that the function defined by f ( x ) = |cos x | is a continuous function.
. Examine that sin | x | is a continuous function. . Find all the points of discontinuity of f defined by f ( x ) = | x | – | x + |.
. . Differentiability Recall the following facts from previous class. We had defined the derivative of a real function as follows: Suppose f is a real function and c is a point in its domain.
The derivative of f at c is defined by f ( x ) x n sin x cos x tan x f ′ ( x ) nx n – cos x – sin x sec x provided this limit exists. Derivative of f at c is denoted by f ′ ( c ) or ( ( )) | c . The function defined by f ′ wherever the limit exists is defined to be the derivative of f . The derivative of f is denoted by f ′ ( x ) or ( ( )) or if y = f ( x ) by dy dx or y ′ .
The process of finding derivative of a function is called differentiation. We also use the phrase differentiate f ( x ) with respect to x to mean find f ′ ( x ). The following rules were established as a part of algebra of derivatives: ( ) ( u ± v ) ′ = u ′ ± v ′ ( ) ( uv ) ′ = u ′