NCERT Exemplar solutions for Mathematics Exemplar [English] Class 12 chapter 5 - Continuity And Differentiability [Latest edition]

Find the value of the constant k so that the function f defined below is continuous at x = 0, where f(x) = `{{:((1 - cos4x)/(8x^2)",", x ≠ 0),("k"",", x = 0):}`

Discuss the continuity of the function f(x) = sin x . cos x.

If f(x) = `{{:((x^3 + x^2 - 16x + 20)/(x - 2)^2",", x ≠ 2),("k"",", x = 2):}` is continuous at x = 2, find the value of k.

Show that the function f defined by f(x) = `{{:(x sin  1/x",", x ≠ 0),(0",", x = 0):}` is continuous at x = 0.

Given f(x) = `1/(x - 1)`. Find the points of discontinuity of the composite function y = f[f(x)]

Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0

Differentiate `sqrt(tansqrt(x))` w.r.t. x

If y = tan(x + y), find `("d"y)/("d"x)`

If e^x + e^y = e^x+y , prove that `("d"y)/("d"x) = -"e"^(y - x)`

Find `("d"y)/("d"x)`, if y = `tan^-1 ((3x - x^3)/(1 - 3x^2)), -1/sqrt(3) < x < 1/sqrt(3)`

If y = `sin^-1 {xsqrt(1 - x) - sqrt(x) sqrt(1 - x^2)}` and 0 < x < 1, then find `("d"y)/(dx)`

If x = a sec³θ and y = a tan³θ, find `("d"y)/("d"x)` at θ = `pi/3`

If x^y = e^x–y, prove that `("d"y)/("d"x) = logx/(1 + logx)^2`

If y = tanx + secx, prove that `("d"^2y)/("d"x^2) = cosx/(1 - sinx)^2`

If f(x) = |cos x|, find f'`((3pi)/4)`

If f(x) = |cos x – sinx|, find `"f'"(pi/6)`

Verify Rolle’s theorem for the function, f(x) = sin 2x in `[0, pi/2]`.

Verify mean value theorem for the function f(x) = (x – 3)(x – 6)(x – 9) in [3, 5].

If f(x) = `(sqrt(2) cos x - 1)/(cot x - 1), x ≠ pi/4` find the value of `"f"(pi/4)`  so that f (x) becomes continuous at x = `pi/4`

Show that the function f given by f(x) = `{{:(("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "if"  x ≠ 0),(0",",  "if"  x = 0):}` is discontinuous at x = 0.

Let f(x) = `{{:((1 - cos 4x)/x^2",",  "if"  x < 0),("a"",",  "if"  x = 0),(sqrt(x)/(sqrt(16) + sqrt(x) - 4)",", "if"  x > 0):}`. For what value of a, f is continuous at x = 0?

Examine the differentiability of the function f defined by
f(x) = `{{:(2x + 3",",  "if"  -3 ≤ x < - 2),(x + 1",",  "if"  -2 ≤ x < 0),(x + 2",",  "if"  0 ≤ x ≤ 1):}`

Differentiate `tan^-1 (sqrt(1 - x^2)/x)` with respect to`cos^-1(2xsqrt(1 - x^2))`, where `x ∈ (1/sqrt(2), 1)`

The function f(x) = `{{:(sinx/x + cosx",",  "if" x ≠ 0),("k"",",  "if" x = 0):}` is continuous at x = 0, then the value of k is ______.

The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at ______.

The number of points at which the function f(x) = `1/(x - [x])` is not continuous is ______.

The function given by f (x) = tanx is discontinuous on the set ______.

Let f(x)= |cosx|. Then, ______.

The function f(x) = |x| + |x – 1| is ______.

The value of k which makes the function defined by f(x) = `{{:(sin  1/x",",  "if"  x ≠ 0),("k"",",  "if"  x = 0):}`, continuous at x = 0 is ______.

The set of points where the functions f given by f(x) = |x – 3| cosx is differentiable is ______.

Differential coefficient of sec (tan^–1x) w.r.t. x is ______.

If u = `sin^-1 ((2x)/(1 + x^2))` and v = `tan^-1 ((2x)/(1 - x^2))`, then `"du"/"dv"` is ______.

The value of c in Rolle’s Theorem for the function f(x) = e x sinx, x ∈ π [0, π] is ______.

The value of c in Mean value theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is ______.

The number of points at which the function f(x) = `1/(log|x|)` is discontinuous is ______.

If f(x) = `{{:("a"x + 1,  "if"  x ≥ 1),(x + 2,  "if"  x < 1):}` is continuous, then a should be equal to ______.

The derivative of log₁₀x w.r.t. x is ______.

If y = `sec^-1 ((sqrt(x) + 1)/(sqrt(x + 1))) + sin^-1((sqrt(x) - 1)/(sqrt(x) + 1))`, then `"dy"/"dx"` is equal to ______.

The derivative of sin x w.r.t. cos x is ______.

For continuity, at x = a, each of `lim_(x -> "a"^+) "f"(x)` and `lim_(x -> "a"^-) "f"(x)` is equal to f(a).

y = |x – 1| is a continuous function.

A continuous function can have some points where limit does not exist.

|sinx| is a differentiable function for every value of x.

cos |x| is differentiable everywhere.

Examine the continuity of the function f(x) = x³ + 2x² – 1 at x = 1

f(x) = `{{:(3x + 5",", "if"  x ≥ 2),(x^2",", "if"  x < 2):}` at x = 2

f(x) = `{{:((1 - cos 2x)/x^2",", "if"  x ≠ 0),(5",", "if"  x = 0):}` at x = 0

f(x) = `{{:((2x^2 - 3x - 2)/(x - 2)",", "if"  x ≠ 2),(5",", "if"  x = 2):}` at x = 2

f(x) = `{{:(|x - 4|/(2(x - 4))",", "if"  x ≠ 4),(0",", "if"  x = 4):}` at x = 4

f(x) = `{{:(|x|cos  1/x",", "if"  x ≠ 0),(0",", "if"  x = 0):}` at x = 0

f(x) = `{{:(|x - "a"| sin  1/(x - "a")",",  "if"  x ≠ 0),(0",",  "if"  x = "a"):}` at x = a

f(x) = `{{:(("e"^(1/x))/(1 + "e"^(1/x))",", "if"  x ≠ 0),(0",", "if"  x = 0):}` at x = 0 

f(x) = `{{:(x^2/2",",  "if"  0 ≤ x ≤ 1),(2x^2 - 3x + 3/2",",  "if"  1 < x ≤ 2):}` at x = 1

f(x) = |x| + |x − 1| at x = 1

f(x) = `{{:(3x - 8",",  "if"  x ≤ 5),(2"k"",",  "if"  x > 5):}` at x = 5

f(x) = `{{:((2^(x + 2) - 16)/(4^x - 16)",",  "if"  x ≠ 2),("k"",",  "if"  x = 2):}` at x = 2

f(x) = `{{:((sqrt(1 + "k"x) - sqrt(1 - "k"x))/x",",  "if" -1 ≤ x < 0),((2x + 1)/(x - 1)",",  "if"  0 ≤ x ≤ 1):}` at x = 0

f(x) = `{{:((1 - cos "k"x)/(xsinx)",",   "if"  x ≠ 0),(1/2",",  "if"  x = 0):}` at x = 0

Prove that the function f defined by 
f(x) = `{{:(x/(|x| + 2x^2)",",  x ≠ 0),("k",  x = 0):}`
remains discontinuous at x = 0, regardless the choice of k.

Find the values of a and b such that the function f defined by
f(x) = `{{:((x - 4)/(|x - 4|) + "a"",",  "if"  x < 4),("a" + "b"",",  "if"  x = 4),((x - 4)/(|x - 4|) + "b"",", "if"  x > 4):}`
is a continuous function at x = 4.

Given the function f(x) = `1/(x + 2)`. Find the points of discontinuity of the composite function y = f(f(x))

Find all points of discontinuity of the function f(t) = `1/("t"^2 + "t" - 2)`, where t = `1/(x - 1)`

Show that the function f(x) = |sin x + cos x| is continuous at x = π.

Examine the differentiability of f, where f is defined by
f(x) = `{{:(x[x]",",  "if"  0 ≤ x < 2),((x - 1)x",",  "if"  2 ≤ x < 3):}` at x = 2

Examine the differentiability of f, where f is defined by
f(x) = `{{:(x^2 sin  1/x",",  "if"  x ≠ 0),(0",", "if"  x = 0):}` at x = 0

Examine the differentiability of f, where f is defined by
f(x) = `{{:(1 + x",",  "if"  x ≤ 2),(5 - x",",  "if"  x > 2):}` at x = 2

Show that f(x) = |x – 5| is continuous but not differentiable at x = 5.

A function f: R → R satisfies the equation f( x + y) = f(x) f(y) for all x, y ∈ R, f(x) ≠ 0. Suppose that the function is differentiable at x = 0 and f′(0) = 2. Prove that f′(x) = 2f(x).

`2^(cos^(2_x)`

`8^x/x^8`

`log (x + sqrt(x^2 + "a"))`

`log [log(logx^5)]`

`sin sqrt(x) + cos^2 sqrt(x)`

sinⁿ (ax² + bx + c)

`cos(tan sqrt(x + 1))`

sinx² + sin²x + sin²(x²)

`sin^-1  1/sqrt(x + 1)`

(sin x)^cosx 

sin^mx . cosⁿx

(x + 1)²(x + 2)³(x + 3)⁴

`cos^-1 ((sinx + cosx)/sqrt(2)), (-pi)/4 < x < pi/4`

`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`

`tan^-1 (secx + tanx), - pi/2 < x < pi/2`

`tan^-1 (("a"cosx - "b"sinx)/("b"cosx - "a"sinx)), - pi/2 < x < pi/2` and `"a"/"b" tan x > -1`

`sec^-1 (1/(4x^3 - 3x)), 0 < x < 1/sqrt(2)`

`tan^-1 ((3"a"^2x - x^3)/("a"^3 - 3"a"x^2)), (-1)/sqrt(3) < x/"a" < 1/sqrt(3)`

`tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2))), -1 < x < 1, x ≠ 0`

x = `"t" + 1/"t"`, y = `"t" - 1/"t"`

x = `"e"^theta (theta + 1/theta)`, y= `"e"^-theta (theta - 1/theta)`

x = 3cosθ – 2cos³θ, y = 3sinθ – 2sin³θ

sin x = `(2"t")/(1 + "t"^2)`, tan y = `(2"t")/(1 - "t"^2)`

x = `(1 + log "t")/"t"^2`, y = `(3 + 2 log "t")/"t"`

If x = e^cos2t and y = e^sin2t, prove that `"dy"/"dx" = (-y log x)/(xlogy)`

If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that `("dy"/"dx")_("at  t" = pi/4) = "b"/"a"`

If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`

Differentiate `x/sinx` w.r.t. sin x

Differentiate `tan^-1 ((sqrt(1 + x^2) - 1)/x)` w.r.t. tan^–1x, when x ≠ 0

`sin xy + x/y` = x² – y

sec(x + y) = xy

tan^–1(x² + y²) = a

(x² + y²)² = xy

If ax² + 2hxy + by² + 2gx + 2fy + c = 0, then show that `"dy"/"dx" * "dx"/"dy"` = 1 

If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.

If y^x = e^{y – x}, prove that `"dy"/"dx" = (1 + log y)^2/logy`

If y = `(cos x)^((cos x)^((cosx)....oo)`, show that `"dy"/"dx" = (y^2 tanx)/(y log cos x - 1)`

If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"a"`

If `sqrt(1 - x^2) + sqrt(1 - y^2) = a(x - y)`, prove that `(dy)/(dx) = sqrt((1 - y^2)/(1 - x^2))`.

If y = tan^–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.

f(x) = x(x – 1)² in [0, 1]

f(x) = `sin^4x + cos^4x` in `[0, pi/2]`

f(x) = log(x² + 2) – log3 in [–1, 1]

f(x) = `x(x + 3)e^((–x)/2)` in [–3, 0]

f(x) = `sqrt(4 - x^2)` in [– 2, 2]

Discuss the applicability of Rolle’s theorem on the function given by f(x) = `{{:(x^2 + 1",",  "if"  0 ≤ x ≤ 1),(3 - x",",  "if"  1 ≤ x ≤ 2):}`

Find the points on the curve y = (cosx – 1) in [0, 2π], where the tangent is parallel to x-axis

Using Rolle’s theorem, find the point on the curve y = x(x – 4), x ∈ [0, 4], where the tangent is parallel to x-axis

f(x) = `1/(4x - 1)` in [1, 4]

f(x) = x³ – 2x² – x + 3 in [0, 1]

f(x) = sinx – sin2x in [0, π]

f(x) = `sqrt(25 - x^2)` in [1, 5]

Find a point on the curve y = (x – 3)², where the tangent is parallel to the chord joining the points (3, 0) and (4, 1)

Using mean value theorem, prove that there is a point on the curve y = 2x² – 5x + 3 between the points A(1, 0) and B(2, 1), where tangent is parallel to the chord AB. Also, find that point

Find the values of p and q so that f(x) = `{{:(x^2 + 3x + "p"",",  "if"  x ≤ 1),("q"x + 2",",  "if"  x > 1):}` is differentiable at x = 1

If x^m . yⁿ = (x + y)^m+n, prove that `"dy"/"dx" = y/x`

If x^m . yⁿ = (x + y)^m+n, prove that `("d"^2"y")/("dx"^2)` = 0

If x = sint and y = sin pt, prove that `(1 - x^2) ("d"^2"y")/("dx"^2) - x "dy"/"dx" + "p"^2y` = 0

Find `"dy"/"dx"`, if y = `x^tanx + sqrt((x^2 + 1)/2)`

If f(x) = 2x and g(x) = `x^2/2 + 1`, then which of the following can be a discontinuous function ______.

The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.

The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.

The function f(x) = cot x is discontinuous on the set ______.

The function f(x) = `"e"^|x|` is ______.

If f(x) = `x^2 sin  1/x` where x ≠ 0, then the value of the function f at x = 0, so that the function is continuous at x = 0, is ______.

If f(x) = `{{:("m"x + 1",",  "if"  x ≤ pi/2),(sin x + "n"",",  "If"  x > pi/2):}`, is continuous at x = `pi/2`, then ______.

Let f(x) = |sin x|. Then ______.

If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.

If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.

The derivative of cos^–1(2x² – 1) w.r.t. cos^–1x is ______.

If x = t², y = t³, then `("d"^2"y")/("dx"^2)` is ______.

The value of c in Rolle’s theorem for the function f(x) = x³ – 3x in the interval `[0, sqrt(3)]` is ______.

For the function f(x) = `x + 1/x`, x ∈ [1, 3], the value of c for mean value theorem is ______.

An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ______.

Derivative of x² w.r.t. x³ is ______.

If f(x) = |cosx|, then `"f'"(pi/4)` = ______.

If f(x) = |cosx – sinx| , then `"f'"(pi/4)` = ______.

For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.

Rolle’s theorem is applicable for the function f(x) = |x – 1| in [0, 2].

If f is continuous on its domain D, then |f| is also continuous on D.

The composition of two continuous function is a continuous function.

Trigonometric and inverse-trigonometric functions are differentiable in their respective domain.

If f.g is continuous at x = a, then f and g are separately continuous at x = a.