Is the function f defined by

continuous at x = 0? At x = 1? At x = 2?

Find all points of discontinuity of f, where f is defined by

Find all points of discontinuity of f, where f is defined by

Find all points of discontinuity of f, where f is defined by

Find all points of discontinuity of f, where f is defined by

Find all points of discontinuity of f, where f is defined by

Find all points of discontinuity of f, where f is defined by

Is the function defined by

a continuous function?

Discuss the continuity of the function f, where f is defined by

Discuss the continuity of the function f, where f is defined by

Discuss the continuity of the function f, where f is defined by

For what value of is the function defined by

continuous at x = 0? What about continuity at x = 1?

Discuss the continuity of the cosine, cosecant, secant and cotangent functions,

Find the points of discontinuity of f, where

Determine if f defined by

is a continuous function?

Examine the continuity of f, where f is defined by

Find :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

Differentiate in three ways mentioned below

(i) By using product rule.

(ii) By expanding the product to obtain a single polynomial.

(iii By logarithmic differentiation.

Do they all give the same answer?

If

If, prove that

If, show that

Ifshow that

If, show that

If, show that

If, show that

Verify Rolle’s Theorem for the function

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

(i) 

(ii) 

(iii) 

, for some fixed and 

, for 

Find, if 

Find, if 

If, for, −1 < x <1, prove that

If  with  prove that

If and, find 

Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?

If, prove that 

If, show that