Sample QuestionsContinuity and Differentiability questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The value of $k(k<0)$ for which the function $f$ defined as $f(x)=\left\{\begin{array}{cc}\frac{1-\cos k x}{x \sin x} & , x \neq 0 \\ \frac{1}{2} & , x=0\end{array}\right.$ is continuous at $x=0$ is
View full solution →The derivative of $\sin ^{-1}\left(2 x \sqrt{1-x^2}\right) w.r.t. \sin ^{-1} x \frac{1}{\sqrt{2}} < x <1,$ is
View full solution →If $y=\tan ^{-1}\left(e^{2 x}\right)$, then $\frac{d y}{d x}$ is equal to
View full solution →If $y=5 \cos x-3 \sin x$, then $\frac{d^2 y}{d x^2}$ is equal to
View full solution →If $y=\log \left(\cos e^x\right)$, then $\frac{d y}{d x}$ is
View full solution →Assertion $(A)$ : If $x=a t^2$ and $y=2 a t$, then $\left[\frac{d^2 y}{d x^2}\right]_{t=2}=\frac{-1}{16 a}$
Reason $(R) : \frac{d^2 y}{d x^2}=\left(\frac{d y}{d t}\right)^2 \times\left(\frac{d t}{d x}\right)^2$
View full solution →Assertion (A) : If $y=\log _{10} x+\log _e x$, then $\frac{d y}{d x}=\frac{\log _{10} e}{x}+\frac{1}{x}$.
Reason (R): $\frac{d}{d x}\left(\log _{10} x\right)=\frac{\log x}{\log 10}$ and
$
\frac{d}{d x}\left(\log _e x\right)=\frac{\log x}{\log e}
$
View full solution →Assertion $(A)$: If $e^{x y}+\log (x y)+\cos (x y)+5=0$, then $\frac{d y}{d x}=-\frac{y}{x}$.
Reason $(R) : \frac{d}{d x}(x y)=0 \Rightarrow \frac{d y}{d x}=\frac{-y}{x}$
View full solution →Assertion (A) : If $y=\cot ^{-1}\left(\frac{1+x \sqrt{x}}{\sqrt{x}-x}\right)$, then $\frac{d y}{d x}=\frac{-1}{1+x^2}+\frac{1}{2 \sqrt{x}(1+x)}$
Reason (R) : $\frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^2}$ and $\frac{d}{d x}\left(\tan ^{-1} \frac{1}{x}\right)=1+x^2$.
View full solution →Assertion $(A) :$ If $u=f(\sin x), v=g(\cos x)$ and $f^{\prime}\left(\frac{1}{\sqrt{2}}\right)=2, g^{\prime}\left(\frac{1}{\sqrt{2}}\right)=4$, then $\left(\frac{d u}{d v}\right)_{x=\pi / 4}=\frac{1}{\sqrt{2}}$.
Reason $(R):$ If $u=f(x), v=g(x)$, then the derivative of $f$ with respect to $g$ is $\frac{d u}{d v}=\frac{d u / d x}{d v / d x}$.
View full solution →Find all points of discontinuity of $f,$ where $f$ is defined by $:f(x)=\left\{\begin{array}{ll} {\frac{x}{|x|},} & {\text { if } x<0} \\ {-1,} & {\text { if } x \geq 0} \end{array}\right.$
View full solution →Find all points of discontinuity of $\mathrm{f},$ where $\mathrm{f}$ is defined by $:f(x)=\left\{\begin{array}{l}\frac{|x|}{x}, \text { if } x \neq 0 \\ 0, \text { if } x=0\end{array}\right.$
View full solution →Find all points of discontinuity of $f$, where $f$ is defined by $f(x)=\left\{\begin{array}{ll}|x|+3 & \text { if } x \leq-3 \\ -2 x & \text { if }-3<x<3 \\ 6 x+2 & \text { if } x \geq 3\end{array}\right.$
View full solution →Find all points of discontinuity of $\mathrm{f},$ where $\mathrm{f}$ is defined by $:f(x)=\left\{\begin{array}{l}2 x+3, x \leq 2 \\ 2 x-3, x>2\end{array}\right.$
View full solution →Is the function $f$ defined by $f(x)=\left\{\begin{array}{ll} {x,} & {\text { if } x \leq 1} \\ {5,} & {\text { if } x>1} \end{array}\right.$ continuous at $x = 0$ ? At $x = 1$ ? At $x = 2$ ?
View full solution →Differentiate the function $\cos \left( {a\cos x + b\sin x} \right)$ w.r.t x for some constant a and b.
View full solution →Differentiate the function $(5x)^{3 \cos 2x} \text{w.r.t}$ to $x.$
View full solution →Differentiate the function $\sin^{3 }x + \cos^6 x,$ w.r.t to x.
View full solution →Differentiate the function ${\left( {3{x^2} - 9x + 5} \right)^9}$ w.r.t to x.
View full solution →Find the second-order derivative of the function log x
Differentiate the function $(\sin x – \cos x^{(\sin x – \cos x)}, \frac{\pi}{4}$
View full solution →Differentiate the function $(\log x)^{\log x}, x > 1, \text{w.r.t}$ to $x.$
View full solution →Differentiate the function $\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right], 0<x<\frac{\pi}{2}$ w.r.t. x.
View full solution →Differentiate the function $\frac{\cos ^{-1} \frac{x}{2}}{\sqrt{2 x+7}},-2<x<2$ w.r.t to x.
View full solution →Differentiate the function $sin ^{-1}({x\sqrt x})\ ,{0 \leq x \leq 1}$ w.r.t. to x.
View full solution →If ${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {c^2}$ for some c > 0 prove that $\frac{{{{\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]}^{\frac{3}{2}}}}}{{\frac{{{d^2}y}}{{d{x^2}}}}}$ is a constant independent of a and b.
View full solution →Find $\frac{d y}{d x}$ , if y = 12 (1 - cos t), x = 10 (t - sin t), $-\frac{\pi}{2}<t<\frac{\pi}{2}$
View full solution →Differentiate the function $x^{x^{2}-3}+(x-3)^{x^{2}}, \text { for } x>3 w.r.t to x.$
View full solution →If $y = (\tan^{-1}x)^2$ show that $(x^2 + 1)^2 y_2 + 2x(x^2 + 1)y_1 = 2$
View full solution →If $e^y(x + 1) = 1,$ show that $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \left( \frac { d y } { d x } \right) ^ { 2 }$
View full solution →A function f(x) is said to be continuous in an open interval (a, b), if it is continuous at every point in this interval.
A function f(x) is said to be continuous in the closed interval [a, b), if f(x) is continuous in (a, b) and $\lim\limits_{\text{x}\rightarrow0}\text{f}(\text{a}+\text{h})=\text{f}(\text{a})$ and $\lim\limits_{\text{x}\rightarrow0}\text{f}(\text{b}-\text{h})=\text{f}(\text{b})$
If function $\text{f}(\text{x})=\begin{cases}\frac{\sin(\text{a}+1)\text{x}+\sin\text{x}}{\text{x}}&,\text{x}<0\\\text{c}&,\text{x}=0\\\frac{\sqrt{\text{x}+\text{bx}^2}-\sqrt{\text{x}}}{\text{bx}^{\frac{3}{2}}}&,\text{x}>0\end{cases}$ is continuous at x = 0, then answer the following questions.
- The value of a is:
- $-\frac{3}{2}$
- $0$
- $\frac{1}{2}$
- $-\frac{1}{2}$
- The value of b is:
- 1
- -1
- 0
- Any real number.
- The value of c is:
- $1$
- $\frac{1}{2}$
- $-1$
- $-\frac{1}{2}$
- The value of a + c is:
- 1
- 0
- -1
- -2
- The value of c - a is:
- 1
- 0
- -1
- 2
View full solution →Let f(x) be a real valued function, then its
- Left Hand Derivative (L.H.D.) : $\operatorname{Lf}^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{-h}$
Right Hand Derivative (R.H.D.) : $\operatorname{Rf}^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$
Also, a function f(x) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and are equal.
For the function $\text{f}(\text{x})=\begin{cases}|\text{x}-3|,\text{x}\geq1\\\\\frac{\text{x}^2}{4}-\frac{3\text{x}}{2}+\frac{13}{4},\text{x}<1\end{cases},$ answer the following questions.
- R.H.D. of f(x) at x = 1 is:
- 1
- -1
- 0
- 2
- L.H.D. of f(x) at x = 1 is:
- 1
- -1
- 0
- 2
- f(x) is non-differentiable at:
- x = 1
- x = 2
- x = 3
- x = 4
- Find the value of f'(2).
- 1
- 2
- 3
- -1
- The value of f'(-1) is:
- 2
- 1
- -2
- -1
View full solution →Let x = f(t) and y = g(t) be parametric forms with t as a parameter, then
$\frac{\text{dy}}{\text{dx}}=\frac{\text{dy}}{\text{dt}}\times\frac{\text{dt}}{\text{dx}}=\frac{\text{g}'(\text{t})}{\text{f}'(\text{t})},$ where $\text{f}'(\text{t})\neq0.$
On the basis of above information, answer the following questions.
- The derivative of $\text{f}(\tan\text{x})\text{w.r.t.}\text{ g}(\sec\text{x})\text{ at}\text{ x}=\frac{\pi}{4},$ where f'(1) = 2 and $\text{g}'(\sqrt{2})=4,$ is:
- $\frac{1}{\sqrt{2}}$
- ${\sqrt{2}}$
- 1
- 0
- The derivative of $\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ with respect to $\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big)$ is:
- -1
- 1
- 2
- 4
- The derivative of $\text{e}^{\text{x}^3}$ with respect to log x is:
- $\text{e}^{\text{x}^3}$
- $3\text{x}^22\text{e}^{\text{x}^3}$
- $3\text{x}^3\text{e}^{\text{x}^3}$
- $3\text{x}^2\text{e}^{\text{x}^3}+3\text{x}$
- The derivative of $\cos^{-1}(2\text{x}^2-1)\text{w.r.t.}\cos^{-1}\text{x}$ is:
- $2$
- $\frac{-1}{2\sqrt{1-\text{x}^2}}$
- $\frac{2}{\text{x}}$
- $1-\text{x}^2$
- If $\text{y}=\frac{1}{4}\mu^4$ and $\mu=\frac{2}{3}\text{x}^3+5,$ then $\frac{\text{dy}}{\text{dx}}=$
- $\frac{2}{27}\text{x}^2(2\text{x}^3+15)^3$
- $\frac{2}{7}\text{x}^2(2\text{x}^3+15)^3$
- $\frac{2}{27}\text{x}(2\text{x}^3+5)^3$
- $\frac{2}{7}(2\text{x}^3+15)^3$
View full solution →If a real valued function $f(x)$ is finitely derivable at any point of its domain, it is necessarily continuous at that point. But its converse need not be true.
For example, every polynomial, constant function are both continuous as well as differentiable and inverse trigonometric functions are continuous and differentiable in its domains etc.
Based on the above information, answer the following questions.
- If $\text{f}(\text{x})=\begin{cases}\text{x},\text{for x}\leq0\\0,\text{for x}>0\end{cases},$ then at $x = 0$
- $f(x)$ is differentiable and continuous.
- $f(x)$ is neither continuous nor differentiable.
- $f(x)$ is continuous but not differentiable.
- None of these.
- If $\text{f}(\text{x})=|\text{x}-1|,\text{x }\epsilon\text{ R},$ then at $x = 1$
- $f(x)$ is not continuous.
- $f(x)$ is continuous but not differentiable.
- $f(x)$ is continuous and differentiable.
- None of these.
- $f(x) = x^3$ is:
- Continuous but not differentiable at $x = 3$
- Continuous but not differentiable at $x = 3$
- Neither continuous nor differentiable at $x = 3$
- None of these.
- If $\text{f}(\text{x})=[\sin\text{x}],$ then which of the following is true?
- $f(x)$ is continuous and differentiable at $x = 0$.
- $f(x)$ is discontinuous at $x = 0.$
- $f(x)$ is continuous at $x = 0$ but not differentiable.
- $f(x)$ is differentiable but not continuous at $\text{x}=\frac{\pi}{2}.$
- If $\text{f}(\text{x})=\sin^{-1}\text{x},-1\leq\text{x}\leq1,$ then:
- $f(x)$ is both continuous and differentiable.
- $f(x)$ is neither continuous nor differentiable.
- $f(x)$ is continuous but not differentiable.
- None of these.
View full solution →The function f(x) will be discontinuous at x = a if f(x) has
- Discontinuity of first kind : $\lim\limits_{\text{h}\rightarrow0}\text{f}(\text{a}-\text{h})$ and $\lim\limits_{\text{h}\rightarrow0}\text{f}(\text{a}+\text{h})$ both exist but are not equal. If is also known as irremovable discontinuity.
- Discontinuity of second kind : If none of the limits $\lim\limits_{\text{h}\rightarrow0}\text{f}(\text{a}-\text{h})$ and $\lim\limits_{\text{h}\rightarrow0}\text{f}(\text{a}+\text{h})$ exist.
- Removable discontinuity : $\lim\limits_{\text{h}\rightarrow0}\text{f}(\text{a}-\text{h})$ and $\lim\limits_{\text{h}\rightarrow0}\text{f}(\text{a}+\text{h})$ both exist and equal but not equal to f(a).
Based on the above information, answer the following questions.
- If $\text{f}(\text{x})=\begin{cases}\frac{\text{x}^2-9}{\text{x}-3},&\text{for x}\neq3\\4,&\text{for x}=3\end{cases},$ then at x = 3
- f has removable discontinuity.
- f is continuous.
- f has irremovable discontinuity.
- None of these.
- Let $\text{f}(\text{x})=\begin{cases}\text{x}+2,&\text{if x}\leq4\\\text{x}+4,&\text{if x}\geq4\end{cases}$ then at x = 4
- f is continuous.
- f has removable discontinuit.
- f has irremovable discontinuit.
- None of thesee.
- Consider the function f(x) defined as $\text{f}(\text{x})=\begin{cases}\frac{\text{x}^2-4}{\text{x}-2},&\text{for x}\neq2\\5,&\text{for x}=2\end{cases},$ then at x = 2
- f has removable discontinuity.
- f has irremovable discontinuity.
- f is continuous.
- f is continuous if f(2) = 3
- If $\text{f}(\text{x})=\begin{cases}\frac{\text{x}-|\text{x}|}{\text{x}},&\text{x}\neq0\\2,&\text{x}=0\end{cases},$ then at x = 0
- f is continuous.
- f has removable discontinuity.
- f has irremovable discontinuity.
- None of these.
- If $\text{f}(\text{x})=\begin{cases}\frac{\text{e}^\text{x}-1}{\log(1+2\text{x})},&\text{if x}\neq0\\7,&\text{if x}=0\end{cases},$ then at x = 0
- fis continuous if f(0) = 2
- f is continuous
- f has irremovable discontinuity.
- f has removable discontinuity.
View full solution →The greatest integer function defined by f(x) = [x], 0 < x < 2 is not differentiable at x = _________.
Fill in the blanks:
For the curve $\sqrt{\text{x}}+\sqrt{\text{y}}=1,\frac{\text{dy}}{\text{dx}}$ at $\Big(\frac{1}{4},\frac{1}{4}\Big)$ __________.
View full solution →Fill in the blanks :
Derivative of $x^2$ w.r.t. $x^3$ is $...........$
View full solution →Fill in the blanks:
An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is __________.
Fill in the blanks:
If $\text{f(x)}=|\cos\text{x}|,$ then $\text{f}'\Big(\frac{\pi}{4}\Big)=$ _______.
View full solution →State True or False for the statements:
Trigonometric and inverse-trigonometric functions are differentiable in their respective domain.
State True or False for the statements:
If f.g is continuous at x = a, then f and g are separately continuous at x = a.
State True or False for the statements:
Rolle’s theorem is applicable for the function f(x) = |x - 1| in [0, 2].
State True or False for the statements:
The composition of two continuous function is a continuous function.
State True or False for the statements:
If f is continuous on its domain D, then |f| is also continuous on D.