Solution of Chapter 5. Continuity and Differentiability (Mathemetics Part-I Book)

Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

Examine the continuity of the function f (x) = 2x² – 1 at x = 3.

Examine the following functions for continuity.

f (x) = x – 5

Examine the following functions for continuity.

Examine the following functions for continuity.

Examine the following functions for continuity.

f (x) = | x – 5|

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

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

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

Find the relationship between a and b so that the function f defined by

is continuous at x = 3.

For what value of λ is the function defined by

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

Show that the function defined by g(x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

Is the function defined by f (x) = x² – sin x + 5 continuous at x = π?

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 . cos x

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

Find all 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 the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29.

Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29.

Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29.

Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29.

Find the values of a and b such that the function defined by

is a continuous function.

Show that the function defined by f (x) = cos (x²) 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 + 1|.

Differentiate the functions with respect to x.

sin (x² + 5)

Differentiate the functions with respect to x.

cos (sin x)

Differentiate the functions with respect to x.

sin (ax + b)

Differentiate the functions with respect to x.

Differentiate the functions with respect to x.

Differentiate the functions with respect to x.

cos x³ . sin² (x⁵)

Differentiate the functions with respect to x.

Differentiate the functions with respect to x.

cos(√x)

Prove that the function f given by f (x) = | x – 1|, x ∈ R is not differentiable at x = 1.

Prove that the greatest integer function defined by f (x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.

Find dy/dx in the following:

2x + 3y = sin x

Find dy/dx in the following:

2x + 3y = sin y

Find dy/dx in the following:

ax + by² = cos y

Find dy/dx in the following:

xy + y² = tan x + y

Find dy/dx in the following:

x² + xy + y² = 100

Find dy/dx in the following:

x³ + x²y + xy² + y³ = 81

Find dy/dx in the following:

sin² y + cos xy = π

Find dy/dx in the following:

sin² x + cos² y = 1

Find dy/dx in the following:

Find dy/dx in the following:

Find dy/dx in the following:

Find dy/dx in the following:

Find dy/dx in the following:

Find dy/dx in the following:

Find dy/dx in the following:

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:

sin (tan^–1 e^–x)

Differentiate the following w.r.t. x:

log (cos e^x)

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

log (log x), x > 1

Differentiate the following w.r.t. x:

Differentiate the following w.r.t. x:

cos (log x + e^x), x > 0

Differentiate the functions given in w.r.t. x.

cos x . cos 2x . cos 3x

Differentiate the functions given in w.r.t. x.

Differentiate the functions given in w.r.t. x.

(log x)^{cos x}

Differentiate the functions given in w.r.t. x.

x^x – 2^{sin x}

Differentiate the functions given in w.r.t. x.

(x + 3)². (x + 4)³. (x + 5)⁴

Differentiate the functions given in w.r.t. x.

Differentiate the functions given in w.r.t. x.

(log x)^x + x^{log x}

Differentiate the functions given in w.r.t. x.

(sin x)^x + sin^–1 √x

Differentiate the functions given in w.r.t. x.

x^{sin x} + (sin x)^{cos x}

Differentiate the functions given in w.r.t. x.

Differentiate the functions given in w.r.t. x.

Find dy/dx of the functions.

x^y + y^x = 1

Find dy/dx of the functions.

y^x = x^y

Find dy/dx of the functions.

(cos x)^y = (cos y)^x

Find dy/dx of the functions.

xy = e^{(x – y)}

Find the derivative of the function given by f (x) = (1 + x) (1 + x²) (1 + x⁴) (1 + x⁸) and hence find f ′(1).

Differentiate (x² – 5x + 8) (x³ + 7x + 9) 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 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.

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

x = 2at², y = at⁴

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

x = a cos θ, y = b cos θ

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

x = sin t, y = cos 2t

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

x = 4t, y = 4/t

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

x = cos θ – cos 2θ, y = sin θ – sin 2θ

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

x = a (θ – sin θ), y = a (1 + cos θ)

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

x = a sec θ, y = b tan θ

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ)

If x and y are connected parametrically by the equations given in without eliminating the parameter, Find dy/dx.

If, show that

Find the second order derivatives of the function

x² + 3x + 2

Find the second order derivatives of the function

x²⁰

Find the second order derivatives of the function

x . cos x

Find the second order derivatives of the function

log x

Find the second order derivatives of the function

x³ log x

Find the second order derivatives of the function

e^x sin 5x

Find the second order derivatives of the function

e^6x cos 3x

Find the second order derivatives of the function

tan^–1 x

Find the second order derivatives of the function

log (log x)

Find the second order derivatives of the function

sin (log x)

If y = cos^–1 x, Find d²y/dx² in terms of y alone.

If y = 3 cos (log x) + 4 sin (log x), show that x² y₂ + xy₁ + y = 0

If y = Ae^mx + Be^nx, show that

If y = 500e^7x + 600e^–7x, show that .

If e^y (x + 1) = 1, show that=

If y = (tan^–1 x)², show that (x² + 1)² y₂ + 2x (x² + 1) y₁ = 2

Verify Rolle’s theorem for the function f (x) = x² + 2x – 8, x ∈ [– 4, 2].

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

(i) f (x) = [x] for x ∈ [5, 9]

(ii) f (x) = [x] for x ∈ [– 2, 2]

(iii) f (x) = x 2 – 1 for x ∈ [1, 2]

If f : [– 5, 5] → R is a differentiable function and if f′(x) does not vanish anywhere, then prove that f (– 5) ≠ f (5).

Verify Mean Value Theorem, if f (x) = x² – 4x – 3 in the interval [a, b], where a = 1 and b =4.

Verify Mean Value Theorem, if f (x) = x³ – 5x² – 3x in the interval [a, b], where a = 1 and b = 3. Find all c ∈ (1, 3) for which f′(c) = 0.

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.

Differentiate w.r.t. x the function

(3x² – 9x + 5)⁹

Differentiate w.r.t. x the function

sin³ x + cos⁶ x

Differentiate w.r.t. x the function

(5x)^{3cos 2x}

Differentiate w.r.t. x the function

Differentiate w.r.t. x the function

Differentiate w.r.t. x the function

Differentiate w.r.t. x the function

(log x)^{log x}, x > 1

Differentiate w.r.t. x the function

cos (a cos x + b sin x), for some constant a and b.

Differentiate w.r.t. x the function

Differentiate w.r.t. x the function

x^x + x^a + a^x + a^a, for some fixed a > 0 and x > 0

Differentiate w.r.t. x the function

, for x > 3

Find dy/dx, if y = 12 (1 – cos t), x = 10 (t – sin t),

Find dy/dx, if , 0 < x < 1

If for , – 1 < x < 1, prove that

If (x – a)² + (y – b)² = c², for some c > 0, prove that is a constant independent of a and b.

If cos y = x cos (a + y), with cos a ≠ ± 1, prove that

If x = a (cos t + t sin t) and y = a (sin t – t cos t), find d²y/dx².

If f (x) = |x|³, show that f″(x) exists for all real x and find it.

Using mathematical induction prove that for all positive integers n.

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

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

If , prove that