#### Chapters

Chapter 2 - Inverse Trigonometric Functions

Chapter 3 - Matrices

Chapter 4 - Determinants

Chapter 5 - Continuity and Differentiability

Chapter 6 - Application of Derivatives

## Chapter 5 - Continuity and Differentiability

#### Pages 159 - 161

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

Examine the following functions for continuity.

`f(x) = (x^2 - 25)/(x + 5), x != -5`

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

Examine the following functions for continuity.

f (x) = x – 5

Examine the following functions for continuity

`1/(x - 5), x != 5`

Examine the following functions for continuity

f(x) = | x – 5|

Prove that the function `f(x) = x^n` is continuous at *x* = *n*, where *n* is a positive integer

Is the function *f* defined by f(x)= `{(x, if x<=1),(5, if x > 1):}`

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

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

`f(x) = {(2x +3, if zx <=2),(2x - 3, if x > 2):}`

Find all points of discontinuity of *f*, where *f* is defined by `f(x) = {(|x|+3, if x<= -3),(-2x, if -3 < x < 3),(6x + 2, if x >= 3):}`

Find all points of discontinuity of *f*, where *f* is defined by `f(x) = {(|x|/x , if x != 0),(0, if x = 0):}`

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

`f(x) = {(x/|x|, ","if x < 0),(-1, ","if x >= 0):}`

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

`f(x) = {(x+1, "," if x >= 1),(x^2 + 1, ","if x < 1):}`

Find all points of discontinuity of f, where f is defined by `f(x) = {(x^3 - 3, if x <= 2),(x^2 + 1, if x > 2):}`

Find all points of discontinuity of *f*, where *f* is defined by `f(x) = {(x^10 - 1, ","if x <= 1),(x^2, ","if x > 1):}`

Is the function defined by

`f(x) = {(x+5, if x <= 1),(x -5, if x > 1):}` a continuous function?

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

`f(x) = {(3, ","if 0 <= x <= 1),(4, ","if 1 < x < 3),(5, ","if 3 <= x <= 10):}`

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

`f(x) = {(2x , ","if x < 0),(0, "," if 0 <= x <= 1),(4x, "," if x > 1):}`

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

`f(x) = {(-2,"," if x <= -1),(2x, "," if -1 < x <= 1),(2, "," if x > 1):}`

Find the relationship between *a* and *b* so that the function *f* defined by `f(x)= {(az + 1, if x<= 3),(bx + 3, if x > 3):}` is continuous at *x *= 3.

For what value of `lambda` is the function defined by

`f(x) = {(lambda(x^2 - 2x), "," if x <= 0),(4x+ 1, "," if x > 0):}` 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 point. Here [x] denotes the greatest integer less than or equal to *x*.

Is the function defined by `f(x) = x^2 - 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 the points of discontinuity of *f*, where

`f(x) = {((sinx)/x, "," if x < 0),(x + 1, "," if x >= 0):}`

Determine if *f* defined by

`f(x) = {(x^2 sin 1/x, "," if x != 0),(0, "," if x = 0):}` is a continuous function?

Examine the continuity of *f*, where *f* is defined by

`f(x) = {(sin x - cos x, if x != 0),(-1, "," if x = 0):}`

Find the values of *k *so that the function *f* is continuous at the indicated point.

`f(x) = {((kcosx)/(pi-2x), "," if x != pi/2),(3, "," if x = pi/2):} " at x =" pi/2`

Find the values of *k *so that the function *f* is continuous at the indicated point.

`f(x) = {(kx^2, "," if x<= 2),(3, "," if x > 2):} " at x" = 2`

Find the values of *k *so that the function *f* is continuous at the indicated point.

`f(x) = {(kx +1, if x<= pi),(cos x, if x > pi):} " at x " = pi`

Find the values of *k *so that the function *f* is continuous at the indicated point.

`f(x) = {(kx + 1, "," if x <= 5),(3x - 5, "," if x > 5):} " at x " = 5`

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

`f(x) = {(5, "," if x <= 2),(ax +b, "," if 2 < x < 10),(21, "," if x >= 10):}`

is a continuous function.

Show that the function defined by* f *(*x*) = cos (*x*^{2}) is a continuous function.

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

Examine sin |x| is a continuous function.

Find all the points of discontinuity of *f *defined by `f(x) = |x| - |x + 1|`.

#### Page 166

Differentiate the functions with respect to *x*.

sin (x^{2} + 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*.

`sec(tan (sqrtx))`

Differentiate the functions with respect to *x*.

`(sin (ax + b))/cos (cx + d)`

Differentiate the functions with respect to *x*.

`cos x^3. sin^2 (x^3)`

Differentiate the functions with respect to *x*.

`2sqrt(cot(x^2))`

Differentiate the functions with respect to *x*.

`cos (sqrtx)`

Prove that the function *f *given by `f(x) = |x - 1|, x in R` is notdifferentiable at *x* = 1.

#### Page 169

Find `dy/dx`

2x + 3y = sin x

Find `dy/dx`

2x + 3y = sin y

Find `dy/dx`

ax + by^{2} = cos y

Find `dy/dx`

xy + y^{2} = tan x + y

Find `dx/dy`

x^{2} + xy + y2 = 100

Find `dy/dx`

x^{3} + x2y + xy^{2} + y^{3} = 81

Find `dy/dx`

sin^{2} y + cos xy = Π

Find `dy/dx`

sin^{2} x + cos^{2} y = 1

Find `dy/dx`

`y = sin^(-1)((2x)/(1+x^2))`

Find `dy/dx`

`y = tan^(-1) ((3x -x^3)/(1 - 3x^2)), - 1/sqrt3 < x < 1/sqrt3`

Find `dy/dx`

`y = cos^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`

Find `dy/dx`

`y = sin^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`

Find `dx/dy`

`y = cos^(-1) ((2x)/(1+x^2)), -1 < x < 1`

Find `dy/dx`

`y = sin^(-1)(2xsqrt(1-x^2)), -1/sqrt2 < x < 1/sqrt2`

Find `dy/dx`

`y = sec^(-1) (1/(2x^2 - 1)), 0 < x < 1/sqrt2`

#### Pages 147 - 174

Differentiate the following w.r.t. *x*:

`e^x/sinx`

Differentiate the following w.r.t. x: `e^(sin^(-1) x)`

Differentiate the following w.r.t. *x*: `e^(x^3)`

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*:

`e^x + e^(x^2) + ....+ e^(x^3)`

Differentiate the following w.r.t. *x*:

`sqrt(e^(sqrtx)), x > 0`

Differentiate the following w.r.t. *x*: log (log x), x > 1

Differentiate the following w.r.t. *x*:

`cos x/log x, x >0`

Differentiate the following w.r.t. *x*:

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

#### Pages 178 - 179

Differentiate the function with respect to *x*.

cos x . cos 2x . cos 3x

Differentiate the function with respect to *x*.

`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`

Differentiate the function with respect to *x*.

`(log x)^(cos x)`

Differentiate the function with respect to *x*.

`x^x - 2^(sin x)`

Differentiate the function with respect to *x*.

(x + 3)^{2} . (x + 4)^{3} . (x + 5)^{4}

Differentiate the function with respect to *x*.

`(x + 1/x)^x + x^((1+1/x))`

Differentiate the function with respect to *x*.

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

Differentiate the function with respect to *x*.

`(sin x)^x + sin^(-1) sqrtx`

Differentiate the function with respect to *x*.

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

Differentiate the function with respect to *x*.

`x^(xcosx) + (x^2 + 1)/(x^2 -1)`

Differentiate the function with respect to *x*.

`(x cos x)^x + (x sin x)^(1/x)`

Find `dy/dx` of function

x^{y} + y^{x} = 1

Find `dy/dx` of Function y^{x} = x^{y}

Find `dy/dx` of Function

(cos x)^{y} = (cos y)^{x}

Find `dy/dx` of function

xy = e^{(x – y)}

Find the derivative of the function given by f (x) = (1 + x) (1 + x^{2}) (1 + x^{4}) (1 + x^{8}) and hence find f ′(1).

Differentiate (x^{2} – 5x + 8) (x^{3} + 7x + 9) in three ways mentioned

(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?d below:

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

`d/dx(u.v.w) = (du)/dx v.w+u. (dv)/dx.w + u.v. (dw)/dx`

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

#### Page 181

If *x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

`x = 2at^2, y = at^4`

If *x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

*x* = sin *t*, *y* = cos 2*t*

If *x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = 4t, y = 4/y

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

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

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

*x* = *a* cos *θ*, *y* = *b* cos *θ*

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = a (θ – sin θ), y = a (1 + cos θ)

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

`x = (sin^3t)/sqrt(cos 2t), y = (cos^3t)/sqrt(cos 2t)`

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

`x = a(cos t + log tan t/2), y = a sin t`

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = a sec θ, y = b tan θ

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

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

if `x = sqrt(a^(sin^(-1)))`, y = `sqrt(a^(cos^(-1)))` show that `dy/dx = - y/x`

#### Pages 183 - 184

Find the second order derivatives of the function.

x^{2} + 3x + 2

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.

x^{3} 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 = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`

If y = cos^{–1} x, Find `(d^2y)/dx^2` in terms of y alone.

If y = 3 cos (log x) + 4 sin (log x), show that x^{2} y_{2} + xy_{1} + y = 0

If y = Ae^{mx} + Be^{nx}, show that `(d^2y)/dx^2 - (m+ n) (dy)/dx + mny = 0`

If y = 500e^{7x} + 600e^{–7x}, show that `(d^2y)/(dx^2) = 49y`

If e^{y} (x + 1) = 1, show that `(d^2y)/(dx^2) =((dy)/(dx))^2`

If y = (tan^{–1} x)^{2}, show that (x^{2} + 1)^{2} y_{2} + 2x (x^{2} + 1) y_{1} = 2

Find the second order derivatives of the function. `x^20`

#### Page 186

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

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?

f (x) = [x] for x ∈ [5, 9]

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?

f (x) = [x] for x ∈ [– 2, 2]

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?

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) = x^{2} – 4x – 3 in the interval [a, b], where a = 1 and b = 4.

Verify Mean Value Theorem, if f (x) = x^{3} – 5x^{2} – 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.

#### Pages 191 - 192

Differentiate w.r.t. x the function (3x2 – 9x + 5)9

Differentiate w.r.t. x the function sin^{3} x + cos^{6} x

Differentiate w.r.t. x the function (5x)^{3cos 2x}

Differentiate w.r.t. x the function `sin^(–1)(xsqrtx ), 0 ≤ x ≤ 1`

Differentiate w.r.t. x the function `(cos^(-1) x/2)/sqrt(2x+7), -2 < x < 2`

Differentiate w.r.t. x the function `cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx) - sqrt(1-sinx))]`, ` 0 < x < pi/2`

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 (sin x – cos x) ^{(sin x – cos x)}, `pi/4 < x < (3pi)/4`

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 `x^(x^2 -3) + (x -3)^(x^2)`, for x > 3

Find `dy/dx` ,if y = 12 (1 – cos t), x = 10 (t – sin t), `-pi/2< t< pi/2`

Find `dy/dx` , if y = sin^{–1} x + sin^{–1} `sqrt(1-x^2)`, 0 < x < 1

if `xsqrt(1+y) + ysqrt(1+x) = 0`, for, −1 < *x* <1, prove that `dy/dx = 1/(1+ x)^2`

If (x – a)^{2} + (y – b)^{2} = c^{2}, for some c > 0, prove that

`[1+ (dy/dx)^2]^(3/2)/((d^2y)/dx^2)` is a constant independent of *a* and *b*.

If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`

If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`

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

Using mathematical induction prove that `d/(dx) (x^n) = nx^(n -1)` 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 continuos everywhere but not differentiable at exactly two points? Justify your answer ?

if y = `[(f(x), g(x), h(x)),(l, m,n),(a,b,c)]`, prove that `dy/dx` =`|(f'(x), g'(x), h'(x)),(l,m, n),(a,b,c)|`

if `y = e^(acos^(-1)x)`, -1 <= x <= 1 show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`

#### Textbook solutions for Class 12

## NCERT solutions for Class 12 Mathematics chapter 5 - Continuity and Differentiability

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Concepts covered in Class 12 Mathematics chapter 5 Continuity and Differentiability are Derivative - Exponential and Log, Concept of Differentiability, Proof Derivative X^n Sin Cos Tan, Infinite Series, Higher Order Derivative, Algebra of Continuous Functions, Continuous Function of Point, Mean Value Theorem, Second Order Derivative, Derivatives of Functions in Parametric Forms, Logarithmic Differentiation, Exponential and Logarithmic Functions, Derivatives of Implicit Functions, Derivatives of Inverse Trigonometric Functions, Derivatives of Composite Functions - Chain Rule, Concept of Continuity.

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