Examine the continuity of the function $f(x)=x^3+2 x^2-1$ at $x=1$.

$f(x)=\left\{\begin{array}{cc}3 x+5, & \text { if } x \geq 2 \\ x^2, & \text { if } x<2\end{array}\right.$ at $x=2$.

$f(x)=\left\{\begin{array}{ll}\frac{1-\cos 2 x}{x^2}, & \text { if } x \neq 0 \\ 5, & \text { if } x=0\end{array}\right.$ at $x=0$.

$f(x)=\left\{\begin{array}{ll}\frac{2 x^2-3 x-2}{x-2}, & \text { if } x \neq 2 \\ 5, & \text { if } x=2\end{array}\right.$ at $x=2$.

$f(x)=\left\{\begin{array}{ll}\frac{|x-4|}{2(x-4)}, & \text { if } x \neq 4 \\ 0, & \text { if } x=4\end{array}\right.$ at $x=4$

$f(x)=\left\{\begin{array}{ll}|x| \cos \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$ at $x=0$.

$f(x)=\left\{\begin{array}{ll}|x-a| \sin \frac{1}{x-a}, & \text { if } x \neq 0 \\ 0, & \text { if } x=a\end{array}\right.$ at $x=a$.

$f(x)=\left\{\begin{array}{ll}\frac{e^{1 / x}}{1+e^{1 / x}}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$ at $x=0$.

$f(x)=\left\{\begin{array}{ll}\frac{x^2}{2}, & \text { if } 0 \leq x \leq 1 \\ 2 x^2-3 x+\frac{3}{2}, & \text { if } 1< x \leq 2\end{array}\right.$ at $x=1$.

$f(x)=|x|+|x-1|$ at $x=1$.

$f(x)=\left\{\begin{aligned} 3 x-8, & \text { if } x \leq 5 \\ 2 k, & \text { if } x>5\end{aligned}\right.$ at $x=5$.

$f(x)=\left\{\begin{array}{cll}\frac{2^{x+2}-16}{4^x-16}, & \text { if } & x \neq 2 \\ k, & \text { if } & x=2\end{array}\right.$ at $x=2$.

$f(x)=\left\{\begin{array}{cl}\frac{\sqrt{1+k x}-\sqrt{1-k x}}{x}, & \text { if }-1 \leq x<0 \\ \frac{2 x+1}{x-1}, & \text { if } 0 \leq x \leq 1\end{array}\right.$ at $x=0$.

$f(x)=\left\{\begin{array}{ll}\frac{1-\cos k x}{x \sin x}, & \text { if } x \neq 0 \\ \frac{1}{2}, & \text { if } x=0\end{array}\right.$ at $x=0$.

Prove that the function $f$ defined by $f(x)= \begin{cases}\frac{x}{|x|+2 x^2}, & \text { if } x \neq 0 \\ k, & \text { if } x=0\end{cases}$ remains discontinuous at $x=0$, regardless the choice of $k$.

Find the values of $a$ and $b$ such that the function $f$ defined by

$$f(x)= \begin{cases}\frac{x-4}{|x-4|}+a, & \text { if } x<4 \\ a+b, & \text { if } x=4 \\ \frac{x-4}{|x-4|}+b, & \text { if } x>4\end{cases}$$

is a continuous function at $x=4$.

If the function $f(x)=\frac{1}{x+2}$, then find the points of discontinuity of the composite function $y=f\{f(x)\}$.

Find all points of discontinuity of the function $f(t)=\frac{1}{t^2+t-2}$, where $$ t=\frac{1}{x-1}.$$

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

Examine the differentiability of $f$, where $f$ is defined by

$$f(x)=\left\{\begin{array}{ll} x[x], & \text { if } 0 \leq x<2 \\ (x-1) x, & \text { if } 2 \leq x<3 \end{array} \text { at } x=2 .\right.$$

$f(x)=\left\{\begin{array}{ll}x^2 \sin \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$ at $x=0$

$f(x)=\left\{\begin{array}{l}1+x, \text { if } x \leq 2 \\ 5-x, \text { if } x>2\end{array}\right.$ at $x=2$.

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

A function $f: R \rightarrow R$ satisfies the equation $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R, f(x) \neq 0$. Suppose that the function is differentiable at $x=0$ and $f^{\prime}(0)=2$, then prove that $f^{\prime}(x)=2 f(x)$.

$2^{\cos ^2 x}$

$\frac{8^x}{x^8}$

$\log \left(x+\sqrt{x^2+a}\right)$

$\log \left[\log \left(\log x^5\right)\right]$

$\sin \sqrt{x}+\cos ^2 \sqrt{x}$

$\sin ^n\left(a x^2+b x+c\right)$

$\cos (\tan \sqrt{x+1})$

$\sin x^2+\sin ^2 x+\sin ^2\left(x^2\right)$

$\sin ^{-1} \frac{1}{\sqrt{x+1}}$

$(\sin x)^{\cos x}$

$\sin ^m x \cdot \cos ^n x$

$(x+1)^2(x+2)^3(x+3)^4 $

$\cos ^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right),-\frac{\pi}{4}< x< \frac{\pi}{4}$

$\tan ^{-1} \sqrt{\frac{1-\cos x}{1+\cos x}},-\frac{\pi}{4}< x<\frac{\pi}{4}$

$\tan ^{-1}(\sec x+\tan x), \frac{-\pi}{2}< x<\frac{\pi}{2}$

$\tan ^{-1}\left(\frac{a \cos x-b \sin x}{b \cos x+a \sin x}\right), \frac{-\pi}{2}< x<\frac{\pi}{2}$ and $\frac{a}{b} \tan x>-1$.

$\sec ^{-1}\left(\frac{1}{4 x^3-3 x}\right), 0< x<\frac{1}{\sqrt{2}}$

$\tan ^{-1}\left(\frac{3 a^2 x-x^3}{a^3-3 a x^2}\right), \frac{-1}{\sqrt{3}}<\frac{x}{a}<\frac{1}{\sqrt{3}}$

$\tan ^{-1}\left[\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right],-1< x<1, x \neq 0$

$x=t+\frac{1}{t}, y=t-\frac{1}{t}$

$x=e^\theta\left(\theta+\frac{1}{\theta}\right), y=e^{-\theta}\left(\theta-\frac{1}{\theta}\right)$

$x=3 \cos \theta-2 \cos ^3 \theta, y=3 \sin \theta-2 \sin ^3 \theta$

$\sin x=\frac{2 t}{1+t^2}, \tan y=\frac{2 t}{1-t^2}$

$x=\frac{1+\log t}{t^2}, y=\frac{3+2 \log t}{t}$

If $x=e^{\cos 2 t}$ and $y=e^{\sin 2 t}$, then prove that $\frac{d y}{d x}=-\frac{y \log x}{x \log y}$.

If $x=a \sin 2 t(1+\cos 2 t)$ and $y=b \cos 2 t(1-\cos 2 t)$, then show that $\left(\frac{d y}{d x}\right)_{t=\pi / 4}=\frac{b}{a}$.

If $x=3 \sin t-\sin 3 t, y=3 \cos t-\cos 3 t$, then find $\frac{d y}{d x}$ at $t=\frac{\pi}{3}$.

Differentiate $\frac{x}{\sin x}$ w.r.t. $\sin x$.

Differentiate $\tan ^{-1} \frac{\sqrt{1+x^2}-1}{x}$ w.r.t. $\tan ^{-1} x$, when $x \neq 0$.

$$\sin (x y)+\frac{x}{y}=x^2-y$$

$\sec (x+y)=x y$

$\tan ^{-1}\left(x^2+y^2\right)=a$

$\left(x^2+y^2\right)^2=x y$

If $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$, then show that $\frac{d y}{d x} \cdot \frac{d x}{d y}=1$.

If $x=e^{x / y}$, then prove that $\frac{d y}{d x}=\frac{x-y}{x \log x}$.

If $y^x=e^{y-x}$, then prove that $\frac{d y}{d x}=\frac{(1+\log y)^2}{\log y}$.

If $y=(\cos x)^{(\cos x)^{\left.(\cos x)^{-\infty}\right)}}$, then show that $\frac{d y}{d x}=\frac{y^2 \tan x}{y \log \cos x-1}$.

If $x \sin (a+y)+\sin a \cdot \cos (a+y)=0$, then prove that $$\frac{d y}{d x}=\frac{\sin ^2(a+y)}{\sin a}$$

If $\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y)$, then prove that $\frac{d y}{d x}=\sqrt{\frac{1-y^2}{1-x^2}}$.

If $y=\tan ^{-1} x$, then find $\frac{d^2 y}{d x^2}$ in terms of $y$ alone.

$f(x)=x(x-1)^2$ in $[0,1]$

$f(x)=\sin ^4 x+\cos ^4 x$ in $\left[0, \frac{\pi}{2}\right]$

$f(x)=\log \left(x^2+2\right)-\log 3$ in $[-1,1]$

$f(x)=x(x+3) e^{-x / 2}$ in $[-3,0]$

$f(x)=\sqrt{4-x^2}$ in $[-2,2]$

$$\begin{aligned} &\text { Discuss the applicability of Rolle's theorem on the function given by }\\ &f(x)= \begin{cases}x^2+1, & \text { if } 0 \leq x \leq 1 \\ 3-x, & \text { if } 1 \leq x \leq 2\end{cases} \end{aligned}$$

Find the points on the curve $y=(\cos x-1)$ in $[0,2 \pi]$, where the tangent is parallel to $X$-axis.

Using Rolle's theorem, find the point on the curve $y=x(x-4), x \in[0,4]$, where the tangent is parallel to $X$-axis.

$f(x)=\frac{1}{4 x-1}$ in $[1,4]$

$f(x)=x^3-2 x^2-x+3$ in $[0,1]$

$f(x)=\sin x-\sin 2 x$ in $[0, \pi]$

$f(x)=\sqrt{25-x^2}$ 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 $y=2 x^2-5 x+3$ between the points $A(1,0)$ and $B(2,1)$, where tangent is parallel to the chord $A B$. Also, find that point.

Find the values of $p$ and $q$, so that $f(x)=\left\{\begin{array}{ll}x^2+3 x+p, & \text { if } x \leq 1 \\ q x+2, & \text { if } x>1\end{array}\right.$ is differentiable at $x=1$.

If $x^m \cdot y^n=(x+y)^{m+n}$, prove that

(i) $\frac{d y}{d x}=\frac{y}{x}$ and

(ii) $\frac{d^2 y}{d x^2}=0$

$$\begin{aligned} &\text { If } x=\sin t \text { and } y=\sin p t \text {, then prove that }\\ &\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}+p^2 y=0 \end{aligned}$$

Find the value of $\frac{d y}{d x}$, if $y=x^{\tan x}+\sqrt{\frac{x^2+1}{2}}$.

If $f(x)=2 x$ and $g(x)=\frac{x^2}{2}+1$, then which of the following can be a discontinuous function?

The function $f(x)=\frac{4-x^2}{4 x-x^3}$ is

The set of points where the function $f$ given by $f(x)=|2 x-1| \sin x$ is differentiable is

The function $f(x)=\cot x$ is discontinuous on the set

The function $f(x)=e^{|x|}$ is

If $f(x)=x^2 \sin \frac{1}{x}$, where $x \neq 0$, then the value of the function $f$ at $x=0$, so that the function is continuous at $x=0$, is

If $f(x)=\left[\begin{array}{ll}m x+1, & \text { if } x \leq \frac{\pi}{2} \\ \sin x+n, & \text { if } x>\frac{\pi}{2}\end{array}\right.$ is continuous at $x=\frac{\pi}{2}$, then

If $f(x)=|\sin x|$, then

If $y=\log \left(\frac{1-x^2}{1+x^2}\right)$, then $\frac{d y}{d x}$ is equal to

If $y=\sqrt{\sin x+y}$, then $\frac{d y}{d x}$ is equal to

The derivative of $\cos ^{-1}\left(2 x^2-1\right)$ w.r.t. $\cos ^{-1} x$ is

If $x=t^2$ and $y=t^3$, then $\frac{d^2 y}{d x^2}$ is equal to

The value of $c$ in Rolle's theorem for the function $f(x)=x^3-3 x$ in the interval $[0, \sqrt{3}]$ is

For the function $f(x)=x+\frac{1}{x}, x \in[1,3]$, 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 $x^2$ w.r.t. $x^3$ is ............. .

If $f(x)=|\cos x|$, then $f^{\prime}\left(\frac{\pi}{4}\right)$ is equal to ............ .

If $f(x)=|\cos x-\sin x|$, then $f^{\prime}\left(\frac{\pi}{3}\right)$ is equal to ............. .

For the curve $\sqrt{x}+\sqrt{y}=1, \frac{d y}{d x}$ at $\left(\frac{1}{4}, \frac{1}{4}\right)$ 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 $f \cdot g$ is continuous at $x=a$, then $f$ and $g$ are separately continuous at $x=a$.