Chapter-5. Continuity and Differentiability

If, show that 

If, prove that 

Using the fact that sin (A + B) = sin Acos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Using mathematical induction prove that for all positive integers n.

If, show that exists for all real x, and find it.

If and, find 

If  with prove that

If, for some  prove that

 is a constant independent of aand b.

If, for, - 1 < x<1, prove that

Find, if 

Find, if 

, for 

, for some fixed and 

, for some constant aand b.

Verify Mean Value Theorem, if in the interval [a, b], where a = 1 and b = 3. Find all for which 

Verify Mean Value Theorem, if  in the interval, where and.

If is a differentiable function and if does not vanish anywhere, then prove that.

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) 

Verify Rolle's Theorem for the function

If, show that

If, show that

If, show that

Ifshow that

If, show that

If findin terms of y alone.

If, prove that

Find the second order derivatives of the function.

Find the second order derivatives of the function.

Find the second order derivatives of the function.

Find the second order derivatives of the function.

Find the second order derivatives of the function.

Find the second order derivatives of the function.

Find the second order derivatives of the function.

Find the second order derivatives of the function.

Find the second order derivatives of the function.

Find the second order derivatives of the function.

If

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

x = sin t, y = cos 2t

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

x = a cos θ, y = b cos θ

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

If u, v and w are functions of x, then show that

in two ways-first by repeated application of product rule, second by logarithmic differentiation.

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?

Find the derivative of the function given by and hence find.

Find of function.

Find of function.

Find of function.

Find of function.

Differentiate the function with respect to x.

Differentiate the function with respect to x.

Differentiate the function with respect to x.

Differentiate the function with respect to x.

Differentiate the function with respect to x.

Differentiate the function with respect to x.

Differentiate the function with respect to x.

Differentiate the function with respect to x.

Differentiate the function with respect to x.

Differentiate the function with respect to 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 the following w.r.t. x:

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find  :

Find :

Prove that the greatest integer function defined byis not

differentiable at x = 1 and x = 2.

Prove that the function f given by

 is notdifferentiable at x = 1.

Differentiate the functions with respect to x.

Differentiate the functions with respect to x.

Differentiate the functions with respect to x.

Differentiate the functions with respect to x.

Differentiate the functions with respect to x.

Differentiate the functions with respect to x.

Differentiate the functions with respect to x.

Find all the points of discontinuity of f defined by.

Examine that  is a continuous function.

Show that the function defined by is a continuous function.

Show that the function defined by f (x) = cos (x²) is a continuous function.

Find the values of aand b such that the function defined by

Find the values of k so that the function fis continuous at the indicated point.

Find the values of k so that the function fis continuous at the indicated point.

Find the values of k so that the function fis continuous at the indicated point.

Find the values of k so that the function fis continuous at the indicated point.

Examine the continuity of f, where f is defined by

Determine if fdefined by

is a continuous function?

Find the points of discontinuity of f, where

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

Discuss the continuity of the following functions.

(a) f(x) = sin x + cos x

(b) f(x) = sin x - cos x

(c) f(x) = sin x x cos x

Is the function defined by continuous at x = π?

Show that the function defined by is discontinuous at all integral point. Here denotes the greatest integer less than or equal to x.

For what value of is the function defined by

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

Find the relationship between aand b so that the function fdefined by

is continuous at x = 3.

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

Is the function defined by

a continuous function?

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 fdefined by

continuous at x= 0? At x= 1? At x= 2?

Prove that the function is continuous at x= n, where n is a positive integer.

Examine the continuity of the function.

Prove that the functionis continuous at