Advertisements
Advertisements
प्रश्न
Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.
Advertisements
उत्तर
Any function will not be differentiable if the left-hand limit and the right-hand limit are not equal.
f(x) = [x], 0 < x < 3
(i) At x = 1
Left-side limit:
`lim_(h -> 0) ([1 - h] - [1])/-h`
= `lim_(h -> 0) (0 - 1)/-h`
= `lim_(h -> 0) 1/h`
= Infinite (∞)
Right-hand limit:
`lim_(h -> 0) ([1 + h] - [1])/h`
= `lim_(h -> 0) (1 - 1)/h`
= 0
Left-side limit and right-side limit are not equal.
Hence, f(x) is not differentiable at x = 1.
(ii) At x = 2
Left-side limit:
`lim_(h -> 0) (f(2 + h) - f(2))/h`
= `lim_(h -> 0) ([2 + h]-2)/h`
= `lim_(h -> 0) (2 -2)/h`
= 0
Right-hand limit:
`lim_(h -> 0) (f(2 - h) - f (2))/h`
= `lim_(h -> 0) ([2 - h] - [2])/-h`
= `lim_(h -> 0) (1 - 2)/-h`
= Infinite (∞)
Left-side limit and right-side limit are not equal.
Hence, f(x) is not differentiable at x = 2.
संबंधित प्रश्न
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.
`sec(tan (sqrtx))`
Differentiate the function with respect to x.
`2sqrt(cot(x^2))`
Differentiate the function with respect to x:
(3x2 – 9x + 5)9
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`.
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 f(x) = |x|3, show that f"(x) exists for all real x and find it.
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" = (sec^-1 "x")^2 , "x" > 0 "show that" "x"^2 ("x"^2 - 1) (d^2"y")/(d"x"^2) + (2"x"^3 - "x") (d"y")/(d"x") - 2 = 0`
If f(x) = x + 1, find `d/dx (fof) (x)`
If y = tanx + secx, prove that `("d"^2y)/("d"x^2) = cosx/(1 - sinx)^2`
| COLUMN-I | COLUMN-II |
| (A) If a function f(x) = `{((sin3x)/x, "if" x = 0),("k"/2",", "if" x = 0):}` is continuous at x = 0, then k is equal to |
(a) |x| |
| (B) Every continuous function is differentiable | (b) True |
| (C) An example of a function which is continuous everywhere but not differentiable at exactly one point |
(c) 6 |
| (D) The identity function i.e. f (x) = x ∀ ∈x R is a continuous function |
(d) False |
|sinx| is a differentiable function for every value of x.
cos |x| is differentiable everywhere.
`sin sqrt(x) + cos^2 sqrt(x)`
sinmx . cosnx
`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`
For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.
If sin y = x sin (a + y), then value of dy/dx is
A particle is moving on a line, where its position S in meters is a function of time t in seconds given by S = t3 + at2 + bt + c where a, b, c are constant. It is known that at t = 1 seconds, the position of the particle is given by S = 7 m. Velocity is 7 m/s and acceleration is 12 m/s2. The values of a, b, c are ______.
If f(x) = `{{:((sin(p + 1)x + sinx)/x,",", x < 0),(q,",", x = 0),((sqrt(x + x^2) - sqrt(x))/(x^(3//2)),",", x > 0):}`
is continuous at x = 0, then the ordered pair (p, q) is equal to ______.
The function f(x) = x | x |, x ∈ R is differentiable ______.
If f(x) = | cos x |, then `f((3π)/4)` is ______.
The set of all points where the function f(x) = x + |x| is differentiable, is ______.
Differentiability determines whether a function has what at a particular point?
Which expression defines the derivative of a real function \[f\] at a point \[c\] in its domain?
If \[u\] and \[v\] are differentiable functions, what is \[(uv)'\]?
What is \[\frac{d}{dx}(\tan x)\]?
Which limit is the left-hand derivative at \[x=c\]?
What is the practical condition for differentiability at a point?
For differentiability on a closed interval \[[a,b]\], which derivative is considered at \[b\]?
For \[x\ne c\], which identity is used to prove that differentiability implies continuity?
Which statement correctly describes the converse of “differentiability implies continuity”?
For \[f(x)=|x|\], what is the right-hand derivative at \[x=0\]?
Why is \[|x|\] not differentiable at \[x=0\]?
