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RD Sharma solutions for Class 12 Mathematics chapter 10 - Differentiability

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Chapter 10: Differentiability

Ex. 10.1Ex. 10.2Others

Chapter 10: Differentiability Exercise 10.1 solutions [Pages 10 - 11]

Ex. 10.1 | Q 1 | Page 10

Show that f(x) = |x − 2| is continuous but not differentiable at x = 2. 

Ex. 10.1 | Q 2 | Page 10

Show that f(x) = x1/3 is not differentiable at x = 0.

Ex. 10.1 | Q 3 | Page 10

Show that \[f\left( x \right) =\]`{(12x, -,13, if , x≤3),(2x^2, +,5, if x,>3):}` is differentiable at x = 3. Also, find f'(3).

Ex. 10.1 | Q 4 | Page 10

Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat: 

\[f\left( x \right) = \begin{cases}3x - 2, & 0 < x \leq 1 \\ 2 x^2 - x, & 1 < x \leq 2 \\ 5x - 4, & x > 2\end{cases}\]
Ex. 10.1 | Q 5 | Page 10

Discuss the continuity and differentiability of the 

\[f\left( x \right) = \left| x \right| + \left| x - 1 \right| \text{in the interval} \left( - 1, 2 \right)\]
Ex. 10.1 | Q 6 | Page 10

Find whether the function is differentiable at x = 1 and x = 2 

\[f\left( x \right) = \begin{cases}x & x \leq 1 \\ \begin{array} 22 - x  \\ - 2 + 3x - x^2\end{array} & \begin{array}11 \leq x \leq 2 \\ x > 2\end{array}\end{cases}\]
Ex. 10.1 | Q 7 | Page 10

Show that the function 

\[f\left( x \right) = \begin{cases}x^m \sin\left( \frac{1}{x} \right) & , x \neq 0 \\ 0 & , x = 0\end{cases}\]

(i) differentiable at x = 0, if m > 1
(ii) continuous but not differentiable at x = 0, if 0 < m < 1
(iii) neither continuous nor differentiable, if m ≤ 0

Ex. 10.1 | Q 8 | Page 10

Find the values of a and b so that the function

\[f\left( x \right)\begin{cases}x^2 + 3x + a, & \text { if } x \leq 1 \\ bx + 2 , &\text {  if } x > 1\end{cases}\] is differentiable at each x ∈ R.
Ex. 10.1 | Q 9 | Page 10

Show that the function 

\[f\left( x \right) = \begin{cases}\left| 2x - 3 \right| \left[ x \right], & x \geq 1 \\ \sin \left( \frac{\pi x}{2} \right), & x < 1\end{cases}\] is continuous but not differentiable at x = 1.

Ex. 10.1 | Q 10 | Page 11

If \[f\left( x \right) = \begin{cases}a x^2 - b, & \text { if }\left| x \right| < 1 \\ \frac{1}{\left| x \right|} , & \text { if }\left| x \right| \geq 1\end{cases}\]  is differentiable at x = 1, find a, b.

Ex. 10.1 | Q 11 | Page 11

Find the values of a and b, if the function f defined by 

\[f\left( x \right) = \begin{cases}x^2 + 3x + a & , & x \leqslant 1 \\ bx + 2 & , & x > 1\end{cases}\] is differentiable at = 1.

Chapter 10: Differentiability Exercise 10.2 solutions [Page 16]

Ex. 10.2 | Q 1 | Page 16

If f is defined by f (x) = x2, find f'(2).

Ex. 10.2 | Q 2 | Page 16

If is defined by  \[f\left( x \right) = x^2 - 4x + 7\] , show that \[f'\left( 5 \right) = 2f'\left( \frac{7}{2} \right)\] 

Ex. 10.2 | Q 3 | Page 16

Show that the derivative of the function f given by 

\[f\left( x \right) = 2 x^3 - 9 x^2 + 12x + 9\], at x = 1 and x = 2 are equal.
Ex. 10.2 | Q 4 | Page 16

If for the function 

\[\Phi \left( x \right) = \lambda x^2 + 7x - 4, \Phi'\left( 5 \right) = 97, \text { find } \lambda .\]

Ex. 10.2 | Q 5 | Page 16

If  \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\] 

, find f'(4).

Ex. 10.2 | Q 6 | Page 16

Find the derivative of the function f defined by f (x) = mx + c at x = 0.

Ex. 10.2 | Q 7 | Page 16

Examine the differentialibilty of the function f defined by

\[f\left( x \right) = \begin{cases}2x + 3 & \text { if }- 3 \leq x \leq - 2 \\ \begin{array}xx + 1 \\ x + 2\end{array} & \begin{array} i\text { if } - 2 \leq x < 0 \\\text {  if } 0 \leq x \leq 1\end{array}\end{cases}\] 

Ex. 10.2 | Q 8 | Page 16

Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points.

Ex. 10.2 | Q 9 | Page 16

Discuss the continuity and differentiability of f (x) = |log |x||.

Ex. 10.2 | Q 10 | Page 16

Discuss the continuity and differentiability of f (x) = e|x| .

Ex. 10.2 | Q 11 | Page 16

Discuss the continuity and differentiability of 

\[f\left( x \right) = \begin{cases}\left( x - c \right) \cos \left( \frac{1}{x - c} \right), & x \neq c \\ 0 , & x = c\end{cases}\]
Ex. 10.2 | Q 12 | Page 16

Is |sin x| differentiable? What about cos |x|?

Chapter 10: Differentiability solutions [Page 17]

Q 1 | Page 17

Define differentiability of a function at a point.

 
Q 2 | Page 17

Is every differentiable function continuous?

Q 3 | Page 17

Is every continuous function differentiable?

Q 4 | Page 17

Give an example of a function which is continuos but not differentiable at at a point.

Q 5 | Page 17

If f (x) is differentiable at x = c, then write the value of 

\[\lim_{x \to c} f \left( x \right)\]
Q 6 | Page 17

If f (x) = |x − 2| write whether f' (2) exists or not.

Q 7 | Page 17

Write the points where f (x) = |loge x| is not differentiable.

Q 8 | Page 17

Write the points of non-differentiability of 

\[f \left( x \right) = \left| \log \left| x \right| \right| .\]
Q 9 | Page 17

Write the derivative of f (x) = |x|3 at x = 0.

Q 10 | Page 17

Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.

Q 11 | Page 17

If  \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\]  exists finitely, write the value of  \[\lim_{x \to c} f\left( x \right)\]

Q 12 | Page 17

Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.

Q 13 | Page 17

If  \[f \left( x \right) = \sqrt{x^2 + 9}\] , write the value of

\[\lim_{x \to 4} \frac{f\left( x \right) - f\left( 4 \right)}{x - 4} .\]

Chapter 10: Differentiability solutions [Pages 17 - 20]

Q 1 | Page 17

Let f (x) = |x| and g (x) = |x3|, then

  •  f (x) and g (x) both are continuous at x = 0

  • f (x) and g (x) both are differentiable at x = 0

  • f (x) is differentiable but g (x) is not differentiable at x = 0

  •  f (x) and g (x) both are not differentiable at x = 0

Q 2 | Page 17

The function f (x) = sin−1 (cos x) is

  • discontinuous at x = 0

  • continuous at x = 0

  • differentiable at x = 0

  • none of these

Q 3 | Page 17

The set of points where the function f (x) = x |x| is differentiable is 

 

  • \[\left( - \infty , \infty \right)\]

  • \[\left( - \infty , 0 \right) \cup \left( 0, \infty \right)\]

  • \[\left( 0, \infty \right)\]

  • \[\left[ 0, \infty \right]\]

Q 4 | Page 17

If \[f\left( x \right) = \begin{cases}\frac{\left| x + 2 \right|}{\tan^{- 1} \left( x + 2 \right)} & , x \neq - 2 \\ 2 & , x = - 2\end{cases}\]  then f (x) is

  • continuous at x = − 2

  • not continuous at x = − 2

  • differentiable at x = − 2

  • continuous but not derivable at x = − 2

Q 5 | Page 18

Let \[f\left( x \right) = \left( x + \left| x \right| \right) \left| x \right|\]

  • f is continuous

  •  f is differentiable for some x

  • f' is continuous

  • f'' is continuous

Q 6 | Page 18

The function f (x) = e|x| is

  • continuous everywhere but not differentiable at x = 0

  • continuous and differentiable everywhere

  • not continuous at x = 0

  • none of these

Q 7 | Page 18

The function f (x) = |cos x| is

  • everywhere continuous and differentiable

  • everywhere continuous but not differentiable at (2n + 1) π/2, n ∈ Z

  • neither continuous nor differentiable at (2n + 1) π/2, n ∈ Z

  • none of these

Q 8 | Page 18

If \[f\left( x \right) = \sqrt{1 - \sqrt{1 - x^2}},\text{ then } f \left( x \right)\text {  is }\] 

  • continuous on [−1, 1] and differentiable on (−1, 1)

  • continuous on [−1, 1] and differentiable on
    \[\left( - 1, 0 \right) \cup \left( 0, 1 \right)\]

  • continuous and differentiable on [−1, 1]

  • none of these

Q 9 | Page 18

If \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\] 

  • \[a = b = c = 0\]

  • \[a = 0, b = 0; c \in R\]

     

  • \[b = c = 0, a \in R\]

  • \[c = 0, a = 0, b \in R\]

     

Q 10 | Page 18

If \[f\left( x \right) = x^2 + \frac{x^2}{1 + x^2} + \frac{x^2}{\left( 1 + x^2 \right)} + . . . + \frac{x^2}{\left( 1 + x^2 \right)} + . . . . ,\] 

then at x = 0, f (x)

  • has no limit

  • is discontinuous

  • is continuous but not differentiable

  • is differentiable

Q 11 | Page 18

If \[f\left( x \right) = \left| \log_e x \right|, \text { then}\]

  • \[f' \left( 1^+ \right) = 1\]

  • \[f' \left( 1 \right) = - 1\]

  • \[f' \left( 1 \right) = 1\]

  • \[f' \left( 1 \right) = - 1\]

Q 12 | Page 18

If \[f\left( x \right) = \left| \log_e |x| \right|\] 

  • f (x) is continuous and differentiable for all x in its domain

  • f (x) is continuous for all for all × in its domain but not differentiable at x = ± 1

  •  (x) is neither continuous nor differentiable at x = ± 1

  • none of these

     

Q 13 | Page 18

Let \[f\left( x \right) = \begin{cases}\frac{1}{\left| x \right|} & for \left| x \right| \geq 1 \\ a x^2 + b & for \left| x \right| < 1\end{cases}\] If f (x) is continuous and differentiable at any point, then

 

 

 

  • \[a = \frac{1}{2}, b = - \frac{3}{2}\]

  • \[a = - \frac{1}{2}, b = \frac{3}{2}\]

  • a = 1, b = − 1

  • none of these

Q 14 | Page 18

The function f (x) = x − [x], where [⋅] denotes the greatest integer function is

  • continuous everywhere

  • continuous at integer points only

  • continuous at non-integer points only

  • differentiable everywhere

Q 15 | Page 18

Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if 

  • a = 2

  • a = 1

  • a = 0

  • a = 1/2

Q 16 | Page 19

Let f (x) = |sin x|. Then,

  • f (x) is everywhere differentiable.

  • f (x) is everywhere continuous but not differentiable at x = n π, n ∈ Z

  • f (x) is everywhere continuous but not differentiable at  \[x = \left( 2n + 1 \right)\frac{\pi}{2}, n \in Z\]

  • none of these

Q 17 | Page 19

Let f (x) = |cos x|. Then,

  • f (x) is everywhere differentable

  •  f (x) is everywhere continuous but not differentiable at x = n π, n ∈ Z

  • f (x) is everywhere continuous but not differentiable at \[x = \left( 2n + 1 \right)\frac{\pi}{2}, n \in Z\].

  • (d) none of these

Q 18 | Page 19

The function f (x) = 1 + |cos x| is

  • continuous no where

  • continuous everywhere

  •  not differentiable at x = 0

  • not differentiable at x = n π, n ∈ Z

Q 19 | Page 19

The function f (x) =  |cos x| is

  • differentiable at x = (2n + 1) π/2, n ∈ Z

  •  continuous but not differentiable at x = (2n + 1) π/2, n ∈ Z

  • neither differentiable nor continuous at x = n ∈ Z

  •  none of these

Q 20 | Page 19

The function \[f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}\] , where [⋅] denotes the greatest integer function, is

  • continuous as well as differentiable for all x ∈ R

  • continuous for all x but not differentiable at some x

  • differentiable for all x but not continuous at some x.

  • none of these

Q 21 | Page 19

Let f (x) = a + b |x| + c |x|4, where a, b, and c are real constants. Then, f (x) is differentiable at x = 0, if

  • a = 0

  • b = 0

  • c = 0

  • none of these

Q 22 | Page 19

If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is

  • continuous and differentiable at x = 3

  • continuous but not differentiable at x = 3

  • differentiable nut not continuous at x = 3

  • neither differentiable nor continuous at x = 3

Q 23 | Page 19

If \[f\left( x \right) = \begin{cases}\frac{1}{1 + e^{1/x}} & , x \neq 0 \\ 0 & , x = 0\end{cases}\]  then f (x) is 

  • continuous as well as differentiable at x = 0

  • continuous but not differentiable at x = 0

  • differentiable but not continuous at x = 0

  • none of these

Q 24 | Page 19

If \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\] 

then at x = 0, f (x) is

  • continuous and differentiable

  • differentiable but not continuous

  • continuous but not differentiable

  • neither continuous nor differentiable

Q 25 | Page 20

The set of points where the function f (x) given by f (x) = |x − 3| cos x is differentiable, is

  • R

  • R − {3}

  • (0, ∞)

  • none of these

Q 26 | Page 20

Let \[f\left( x \right) = \begin{cases}1 , & x \leq - 1 \\ \left| x \right|, & - 1 < x < 1 \\ 0 , & x \geq 1\end{cases}\] Then, f is 

  • continuous at x = − 1

  • differentiable at x = − 1

  • everywhere continuous

  • everywhere differentiable

Chapter 10: Differentiability

Ex. 10.1Ex. 10.2Others

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

RD Sharma solutions for Class 12 Mathematics chapter 10 - Differentiability

RD Sharma solutions for Class 12 Maths chapter 10 (Differentiability) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 10 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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