NCERT solutions for माठेमटिक्स पार्ट १ अँड २ [इंग्रजी] इयत्ता १२ chapter 5 - Continuity and Differentiability [Latest edition]

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

Examine the following function for continuity:

f(x) = `(x^2 - 25)/(x + 5)`, x ≠ −5

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

Examine the following function for continuity:

f(x) = x – 5

Examine the following function for continuity:

f(x) = `1/(x - 5)`, x ≠ 5

Examine the following function 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 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"  x<=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) = `{(ax + 1", if"  x<= 3),(bx + 3", if"  x > 3):}` is continuous at x = 3.

For what value of λ is the function defined by f(x) = `{(λ(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 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 function:

f(x) = sin x × cos x

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 value 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 value 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 value 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 = π

Find the value 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.

Differentiate the function with respect to x.

sin (x² + 5)

Differentiate the function with respect to x.

cos (sin x)

Differentiate the function with respect to x.

sin (ax + b)

Differentiate the function with respect to x.

`sec(tan (sqrtx))`

Differentiate the function with respect to x.

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

Differentiate the function with respect to x. 

cos x³ . sin² (x⁵)

Differentiate the function with respect to x. 

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

Differentiate the function with respect to x.

`cos (sqrtx)`

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

Find `bb(dy/dx)` in the following:

2x + 3y = sin x

Find `bb(dy/dx)` in the following:

2x + 3y = sin y

Find `bb(dy/dx)` in the following:

ax + by² = cos y

Find `bb(dy/dx)` in the following:

xy + y² = tan x + y

Find `bb(dy/dx)` in the following:

x² + xy + y² = 100

Find `bb(dy/dx)` in the following:

x³ + x²y + xy² + y³ = 81

Find `bb(dy/dx)` in the following:

sin² y + cos xy = k

Find `bb(dy/dx)` in the following:

sin² x + cos² y = 1

Find `bb(dy/dx)` in the following:

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

Find `bb(dy/dx)` in the following:

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

Find `bb(dy/dx)` in the following:

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

Find `bb(dy/dx)` in the following:

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

Find `bb(dy/dx)` in the following:

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

Find `bb(dy/dx)` in the following:

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

Find `bb(dy/dx)` in the following:

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

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^5)`

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

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 + x^{log 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 `bb(dy/dx)` for the given function:

x^y + y^x = 1

Find `bb(dy/dx)` for the given function:

y^x = x^y

Find `bb(dy/dx)` for the given function:

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

Find `bb(dy/dx)` for the given function:

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:

1. By using the product rule.
2. By expanding the product to obtain a single polynomial.
3. By logarithmic differentiation.

Do they all give the same answer?

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.

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

x = 2at², y = at⁴

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

x = sin t, y = cos 2t

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

x = 4t, y = `4/y`

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

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

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

x = a cos θ, y = b cos θ

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

x = a (θ – sin θ), y = a (1 + cos θ)

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

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

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

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

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

x = a sec θ, y = b tan θ

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

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

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

Find the second order derivative of the function.

x² + 3x + 2

Find the second order derivative of the function.

x . cos x

Find the second order derivative of the function.

log x

Find the second order derivative of the function.

x³ log x

Find the second order derivative of the function.

e^x sin 5x

Find the second order derivative of the function.

e^6x cos 3x

Find the second order derivative of the function.

tan^–1 x

Find the second order derivative of the function.

log (log x)

Find the second order derivative 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 x²y₂ + xy₁ + 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 (x² + 1)² y₂ + 2x (x² + 1) y₁ = 2

Find the second order derivative of the function.

x²⁰

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 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) = x² – 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 the function with respect to x:

(3x² – 9x + 5)⁹

Differentiate the function with respect to x:

sin³ x + cos⁶ x

Differentiate the function with respect to x:

`(5x)^(3cos 2x)`

Differentiate the function with respect to x:

`sin^(–1)(xsqrtx), 0 ≤ x ≤ 1`

Differentiate the function with respect to x:

`(cos^(-1)  x/2)/sqrt(2x+7)`, −2 < x < 2

Differentiate the function with respect to x:

`cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx) - sqrt(1-sinx))], 0 < x < pi/2`

Differentiate the function with respect to x:

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

Differentiate the function with respect to x:

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

Differentiate the function with respect to x:

`(sin x - cos x)^((sin x - cos x)), pi/4 < x < (3pi)/4`

Differentiate the function with respect to x:

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

Differentiate the function with respect to x:

`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) + y  sqrt(1+x) = 0`, for, −1 < x < 1, prove that `dy/dx = -1/(1+ x)^2`.

If (x – a)² + (y – b)² = c², 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`.