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.
If $\text{f(x)}=\begin{cases}\frac{1-\sin\text{x}}{(\pi-2\text{x}^2)}\times\frac{\log\sin\text{x}}{\log(1+\pi^2-4\pi\text{x}+4\text{x}^2)},&\text{x}\neq\frac{\pi}{2}\\\text{k},&\text{x}=\frac{\pi}{2}\end{cases}$ is continuous at $\text{x}=\frac{\pi}{2},$ then k =
- $-\frac{1}{16}$
- $-\frac{1}{32}$
- $-\frac{1}{64}$
- $-\frac{1}{28}$
View full solution →The derivative of the function $\cot^{-1}\Big|(\cos2\text{x})^{\frac{1}{2}}\Big|\text{ at }\text{x}=\frac{\pi}{6}$ is:
- $\Big(\frac{2}{3}\Big)^\frac{1}{2}$
- $\Big(\frac{1}{3}\Big)^\frac{1}{2}$
- $3^\frac{1}{2}$
- $6^\frac{1}{2}$
View full solution →The value of a for which the function $\text{f(x)}=\begin{cases}5\text{x}-4,&\text{if }0<\text{x}\leq1\\4\text{x}^2+3\text{ax},&\text{if }<\text{x}<2\end{cases}$ is continuous at every point of its domain, is:
- $\frac{13}{3}$
- 1
- 0
- -1
View full solution →If $\text{f(x)}=\frac{1-\sin\text{x}}{(\pi-2\text{x})^2},$ when $\text{x}\neq\frac{\pi}{2}=\lambda$ then f(x) will be continuous function at $\text{x}=\frac{\pi}{2},$ where $\lambda=$
- $\frac{1}{8}$
- $\frac{1}{4}$
- $\frac{1}{2}$
- none of these
View full solution →Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion (A) $\frac{\text{d}}{\text{dx}}(\text{x}^2+\text{x}+1)^4=(\text{x}^2+\text{x}+1)^3(2\text{x}+1)$
Reason(R) $(\text{fog}'=\text{f'}[\text{g(x)}].\text{g'(x)}$
- Both A and R are true and R is the correct explanation of A
- Both A and R are true but R is NOT the correct explanation of A.
- A is true but R is false
- A is false but R is true
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason $(s) (R)$ have been put forward.
Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A) \frac{\text{dx}^{\sin\text{x}}}{\text{dx}}=\text{x}^{\sin\text{x}}[(\cos)\log\text{x}+\frac{\sin\text{x}}{\text{x}}]$
Reason $(R)$ if $y = x^{f(x)}$ then $\frac{\text{dy}}{\text{dx}}=\text{x}^\text{f(x)}[\text{f '(x)}\log\text{x}+\frac{\text{f(x)}}{\text{x}}]$
- ✓
Both $A$ and $R$ are true and $R$ is the correct explanation of $A$
- B
Both $A$ and $R$ are true but $R$ is $\text{NOT}$ the correct explanation of $A$.
- C
$A$ is true but $R$ is false
- D
$A$ is false but $R$ is true
Answer: A.
View full solution →Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion (A) f(x) = x - 1 + x - 2 is continuous but not differentiable at x = 1, 2.
Reason(R) Every differentiable function is continuous
- Both A and R are true and R is the correct explanation of A
- Both A and R are true but R is NOT the correct explanation of A.
- A is true but R is false
- A is false but R is true
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion(A): $\text{f(x)}=\sin\text{x}$ is continuous x = 0.
Reason(R): $\sin\text{x}$ is differentiable at x = 0.
- Both A and R are true and R is the correct explanation of A
- Both A and R are true but R is NOT the correct explanation of A
- A is true but R is false.
- A is false but R is true.
View full solution →Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion (A) f(x) = [x] greatest integer function is not differentiable at x = 2
Reason(R) The greatest integer function is not continuous at any integer
- Both A and R are true and R is the correct explanation of A
- Both A and R are true but R is NOT the correct explanation of A.
- A is true but R is false
- A is false but R is true
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward.
Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A): \text{f(x)}=\tan^2\text{x}$ is continuous at $\text{x}=\frac{\pi}{2}$
Reason $(R):\ ?$ is continuous at $\text{x}=\frac{\pi}{2}$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A$
- B
Both $A$ and $R$ are true but $R$ is $\text{NOT}$ the correct explanation of $A$
- C
$A$ is true but $R$ is false.
- ✓
$A$ is false but $R$ is true.
Answer: D.
View full solution →Determine the value of ‘k’ for which the following function is continuous at x = 3:
$\text{f(x)} = \begin{cases} \frac{(\text{x}+3)^2-36}{\text{x}-3}\ \ \ \ ,\ \text{x}\neq3\\ \ \ \ \ \ \ \text{k}\ \ \ \ \ \ \ \ \ \ \ ,\ \text{x}=3 \end{cases}$
View full solution →Determine the value of ‘k’ for which the following function is continuous at x = 3:
$ \text{f(x)} = \begin{cases} \frac{(\text{x + 3)}^{2} \text{ - } 36}{\text{x - 3}} & , & \text{x}\neq 3 \\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{k}& , & \text{x = 3}\\ \end{cases}$
View full solution →For what value of ‘k’ is the function $\text{f(x)} = \begin{cases} \frac{\sin \text{5x}}{\text{3x}} + \cos \text{x}, & \text{if }& \text{x} \neq 0 \\ \text{ }\text{ }\text{ }\text{ } \text{k}, & \text{if } & \text{x = 0}\\ \end{cases}$ continuous at x = 0?
View full solution →Determine the value of the constant ‘k’ so that the function $\text{f}(x) = \begin{cases} \frac{\text{k}x}{| x|}\text{ }\text{ }, & \text{if } x < 0\\ \text{ }3\text{ }\text{ }\text{ }\text{ }, & \text{if } x\geq 0\\ \end{cases}$ is continuous at x = 0.
View full solution →$\text{If} f(x) = x + 7 \text{and g (x)} = x - 7, x \in \text{R, find (fog) (7)} $
View full solution →Find the value of c in Rolle’s theorem for the function $\text{f(x)} = \text{x}^{3} - \text{3x in} [ -\sqrt{3}, 0].$
View full solution →$\text{Find } \frac{\text{dy}}{\text{dx}} \text{ at t} = \frac{2\pi}{3} \text{ when x 10 (t} -\sin \text{t) and y = 12 } (1 - \cos \text{t}).$
View full solution →$\text{If y}=\text{sin}^{-1}(6\text{x}\sqrt{1-9\text{x}^2}), -\frac{1}{3\sqrt{2}}<\text{x}<\frac{1}{3\sqrt{2}},\text{then find}\frac{\text{dy}}{dx}$
View full solution →Differentiate $\tan^{-1}\Big(\frac{1+\cos\text{x}}{\sin\text{x}}\Big)$ with respect to x.
View full solution →If $x = at^2, y = 2at,$ then find $\frac{\text{d}^2\text{y}}{\text{dx}^2}.$
View full solution →Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.$\text{f}(\text{x})=\sin\text{x}-\sin2\text{x}-\text{x}\text{ on }[0,\pi]$
View full solution →Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point '$c$' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = 2x^2 - 3x + 1$ on $[1, 3]$
View full solution →Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$\text{f}(\text{x})=\sqrt{\text{x}^2-4}\text{ on }[2,4]$
View full solution →Find the value of k in this question, so that the function f is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\frac{2^{\text{x}+2}-16}{4^\text{x}-16},&\text{if x}\neq2\\\text{k},&\text{if x}=2\end{cases}$ at x = 2.
View full solution →Differentiate w.r.t. x the function in Exercise:
$\frac{\cos^{-1}\frac{\text{x}}{2}}{\sqrt{2\text{x}+7}},-2<\text{x}<2$
View full solution →$\text{If} x = \text{a } \sin \text{ 2t}(1 + \cos\text{2t}) \text{ and }y = \text{b}\cos\text{2t}(1- \cos \text{2t}), \text{find }\frac{dy}{dx} \text{ at t} = \frac{\pi}{4}$
View full solution →If $e^{y }(x + 1) = 1,$ then show that $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} = \bigg(\frac{\text{dy}}{\text{dx}}\bigg)^{2}.$
View full solution →$\text{ If }\text{ x } =\cos\text{t} (3-2\cos^2\text{t)}\ \text{and}\ \text{y}=\sin\text{t}(3-2\sin\text{ t}),\text{ find the value of}\ \frac{\text{dy}}{\text{dx}} = \text{ at t}=\frac{\pi}{\text{4}}.$
View full solution →$\text{If x = a} (\cos 2\text{t +2t}\sin \text{2t}) \text{and y = a} (\sin \text{2t - 2t}\cos\text{2t}),\text{then find}\frac{\text{d}^{2}{\text{y}}}{\text{dx}^{2}}.$
View full solution →$AB$ is the diameter of a circle and $C$ is any point on the circle. Show that the area of triangle $\text{ABC}$ is maximum, when it is an isosceles triangle.
View full solution →Logarithmic differentiation is a powerful technique to differentiate functions of the form $\text{f}(\text{x})=[\text{u}(\text{x})]^{\text{v}(\text{x})},$ where both $u(x)$ and $v(x)$ are differentiable functions and $f$ and $u$ need to be positive functions. Let function $\text{y}=\text{f}(\text{x})=(\text{u}(\text{x}))^{\text{v}(\text{x})},$ then $\text{y}\ '=\text{y}\Big[\frac{\text{v}(\text{x})}{\text{u}(\text{x})}\text{u}\ '(\text{x})+\text{v}\ '(\text{x})\cdot\log[\text{u}(\text{x})]\Big]$ On the basis of above information, answer the following questions.
- Differentiate $x^x \ w.r.t. x.$
- $\text{x}^\text{x}(1+\log\text{x})$
- $\text{x}^\text{x}(1-\log\text{x})$
- $-\text{x}^\text{x}(1+\log\text{x})$
- $\text{x}^\text{x}\log\text{x}$
- Differentiate $x^x + a^{x }+ x^a + a^a \ w.r.t. x.$
- $(1+\log\text{x})+(\text{a}^\text{x}\log\text{a}+\text{ax}^{\text{a}-1})$
- $\text{x}^\text{x}(1+\log\text{x})+\log\text{a}+\text{ax}^{\text{a}-1}$
- $\text{x}^\text{x}(1+\log\text{x})+\text{x}^\text{a}\log\text{x}+\text{ax}^{\text{a}-1}$
- $\text{x}^\text{x}(1+\log\text{x})+\text{a}^\text{x}\log\text{a}+\text{ax}^{\text{a}-1}$
- If $\text{x}=\text{e}^\frac{\text{x}}{\text{y}},$ then find $\frac{\text{dy}}{\text{dx}}.$
- $-\frac{(\text{x}+\text{y})}{\text{x}\log\text{x}}$
- $-\frac{(\text{x}-\text{y})}{\text{x}\log\text{x}}$
- $\frac{(\text{x}+\text{y})}{\text{x}\log\text{x}}$
- $\frac{\text{x}-\text{y}}{\text{x}\log\text{x}}$
- If $y = (2 - x)^3(3 + 2x)^5,$ then find $\frac{\text{dy}}{\text{dx}}.$
- $(2-\text{x})^3(3+2\text{x})^5\Big[\frac{15}{3+2\text{x}}-\frac{8}{2-\text{x}}\Big]$
- $(2-\text{x})^3(3+2\text{x})^5\Big[\frac{15}{3+2\text{x}}+\frac{3}{2-\text{x}}\Big]$
- $(2-\text{x})^3(3+2\text{x})^5\Big[\frac{10}{3+2\text{x}}-\frac{3}{2-\text{x}}\Big]$
- $(2-\text{x})^3(3+2\text{x})^5\cdot\Big[\frac{10}{3+2\text{x}}+\frac{3}{2-\text{x}}\Big]$
- If $\text{y}=\text{x}^\text{x}\cdot\text{e}^{(2\text{x}+5)},$ then find $\frac{\text{dy}}{\text{dx}}.$
- $\text{x}^\text{x}\text{e}^{2\text{x}+5}$
- $\text{x}^\text{x}\text{e}^{2\text{x}+5}(3-\log\text{x})$
- $\text{x}^\text{x}\text{e}^{2\text{x}+5}(1-\log\text{x})$
- $\text{x}^\text{x}\text{e}^{2\text{x}+5}\cdot(3+\log\text{x})$
View full solution →If a relation between $x$ and $y$ is such that $y$ cannot be expressed in terms of $x,$ then $y$ is called an implicit function of $x.$ When a given relation expresses $y$ as an implicit function of $x$ and we want to find $\frac{\text{dy}}{\text{dx}},$ then we differentiate every term of the given relation $w.r.t. x,$ remembering that a tenn in $y$ is first differentiated $w.r.t. y$ and then multiplied by $\frac{\text{dy}}{\text{dx}}.$
Based on the ab:ve information, find the value of $\frac{\text{dy}}{\text{dx}}$ in each of the following questions.
- $x^3 + x^2y + xy^2 + y^3 = 81$
- $\frac{(3\text{x}^2+2\text{xy}+\text{y}^2)}{\text{x}^2+2\text{xy}+3\text{y}^2}$
- $\frac{-(3\text{x}^2+2\text{xy}+\text{y}^2)}{\text{x}^2+2\text{xy}+3\text{y}^2}$
- $\frac{(3\text{x}^2+2\text{xy}-\text{y}^2)}{\text{x}^2-2\text{xy}+3\text{y}^2}$
- $\frac{3\text{x}^2+\text{xy}+\text{y}^2}{\text{x}^2+\text{xy}+3\text{y}^2}$
- $x^y = e^{x-y}$
- $\frac{\text{x}-\text{y}}{(1+\log\text{x})}$
- $\frac{\text{x}+\text{y}}{(1+\log\text{x})}$
- $\frac{\text{x}-\text{y}}{\text{x}(1+\log\text{x})}$
- $\frac{\text{x}+\text{y}}{\text{x}(1+\log\text{x})}$
- $\text{e}^{\sin\text{y}}=\text{xy}$
- $\frac{-\text{y}}{\text{x}(\text{y}\cos\text{y}-1)}$
- $\frac{\text{y}}{\text{y}\cos\text{y}-1}$
- $\frac{\text{y}}{\text{y}\cos\text{y}+1}$
- $\frac{\text{y}}{\text{x}(\text{y}\cos\text{y}-1)}$
- $\sin^2\text{x}+\cos^2\text{y}=1$
- $\frac{\sin2\text{y}}{\sin2\text{x}}$
- $-\frac{\sin2\text{x}}{\sin2\text{y}}$
- $-\frac{\sin2\text{y}}{\sin2\text{x}}$
- $\frac{\sin2\text{x}}{\sin2\text{y}}$
- $\text{y}=(\sqrt{\text{x}})^{\sqrt{\text{x}}^\sqrt{\text{x}}...\infty}$
- $\frac{-\text{y}^2}{\text{x}(2-\text{y}\log\text{x})}$
- $\frac{\text{y}^2}{2+\text{y}\log\text{x}}$
- $\frac{\text{y}^2}{\text{x}(2+\text{y}\log\text{x})}$
- $\frac{\text{y}^2}{\text{x}(2-\text{y}\log\text{x})}$
View full solution →Derivative of $y = f(x) w.r.t. x ($if exists$)$ is denoted by $\frac{\text{dy}}{\text{dx}}$ or $f'(x)$ and is called the first order derivative of $y.$ If we take derivative of $\frac{\text{dy}}{\text{dx}}$ again,
then we get $\frac{\text{d}}{\text{dx}}\Big(\frac{\text{dy}}{\text{dx}}\Big)=\frac{\text{d}^2\text{y}}{\text{dx}^2}$ or $f''(x)$ and is called the second order derivative of $y.$ Similarly, $\frac{\text{d}}{\text{dx}}\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)$ is denoted and defined as $\frac{\text{d}^3\text{y}}{\text{dx}^3}$ or $f'''(x)$ and is known as third order derivative of $y$ and so on.
Based on the above information, answer the following questions.
- If $\text{y}=\tan^{-1}\Big(\frac{\log(\frac{\text{e}}{\text{x}^2})}{\log(\text{ex}^2)}\Big)+\tan^{-1}\Big(\frac{3+2\log\text{x}}{1-6\log\text{x}}\Big),$ then $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ is equal to:
- $2$
- $1$
- $0$
- $-1$
- If $u = x^2 + y^2$ and $x = s + 3t, y = 2s - t,$ then $\frac{\text{d}^2\text{u}}{\text{ds}^2}$ is equal to:
- $12$
- $32$
- $36$
- $10$
- If $\text{f}(\text{x})=2\log\sin\text{x},$ then $f''(x)$ is equal to:
- $2\text{ cosec}^3\text{x}$
- $2\cot^2\text{x}-4\text{x}^2\text{cosec}^2\text{x}^2$
- $2\text{x}\cot\text{x}^2$
- $-2\text{cosec}^2\text{x}$
- If $\text{f}(\text{x})=\text{e}^\text{x}\sin\text{x},$ then $f'''(x) =$
- $2\text{e}^\text{x}(\sin\text{x}+\cos\text{x})$
- $2\text{e}^\text{x}(\cos\text{x}-\sin\text{x})$
- $2\text{e}^\text{x}(\sin\text{x}-\cos\text{x})$
- $2\text{e}^\text{x}\cos\text{x}$
- If $\text{y}^2=\text{ax}^2+\text{bx}+\text{c},$ then $\frac{\text{d}}{\text{dx}}(\text{y}^3\text{y}_2)=$
- $1$
- $-1$
- $\frac{4\text{ac}-\text{b}^2}{\text{a}^2}$
- $0$
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 →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 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:
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.
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:
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.