CUET Mathematics: CalculusContinuity & Differentiability. Free, no login required.

If $e^y(x+1) = 1$, then $\dfrac{d^2y}{dx^2}$ is equal to

The derivative of $x^{5x}$ with respect to $x$ is

If $x = a(\theta+\sin\theta)$, $y = a(1-\cos\theta)$ then $\dfrac{d^2y}{dx^2}$ is equal to:

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

If $x = t^2$ and $y = t\sqrt{t}$, then $\dfrac{d^2y}{dx^2}$ at $t = \dfrac{1}{4}$ is :

If $y^2 = ax + b, a,b \in \mathbb{R}$ where $\mathbb{R}$ is the set of real numbers, then $y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2$ is equal to:

For the function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = \lvert x-3 \rvert + \lvert x+4 \rvert$, Where $\mathbb{R}$ is set of real numbers, which of the following statements are correct?

(A) $f(x)$ is continuous as well as differentiable at $x=3$.

(B) $f(x)$ is continuous but not differentiable at $x=3$ and $x=-4$

(C) $f(x)$ is neither continuous nor differentiable at $x=3$

(D) $f(x)$ is continuous on $\mathbb{R}$

Choose the correct answer from the options given below:

If $y=\tan^{-1}(x+y)$, then $\frac{d^2y}{dx^2}$ is equal to:

If the function $f(x)=\begin{cases}\frac{e^{2\log_e x^2}-1}{x-1}, & x\ne 1\\ a, & x=1\end{cases}$ is continuous at x=1, then the value of a is:

For the function $y(x)=\sqrt{x+1}-\sqrt{x-1}$, which of the following are true?

(A) $\left(\frac{dy}{dx}\right)_{at\,x=2}=\frac{1-\sqrt{3}}{2\sqrt{3}}$

(B) $(x^2-1)\frac{d^2y}{dx^2}-\frac{dy}{dx}+4y=0$

(C) $(x^2-1)\left(\frac{dy}{dx}\right)^2-4y^2=0$

(D) If $P(x)\frac{d^2y}{dx^2}+Q(x)\frac{dy}{dx}+\alpha y=0$, then $P(x)-xQ(x)-4\alpha=0$

Choose the correct answer from the options given below:

If $y=e^{\pi x}$ and $t=x^{\pi}$, then which of following statements is/are TRUE ?

(A) $\frac{d^2y}{dt^2}=\frac{y}{t^2}\left(\frac{\pi(x-1)+1}{\pi}\right)x$

(B) $\frac{d^2y}{dt^2}=y\left(\frac{\pi(x-1)+1}{\pi}\right)x$

(C) $\frac{d^2y}{dt^2}=1$ at $x=1$

(D) $\frac{d^2y}{dt^2}=\frac{e^{\pi}}{\pi}$ at $x=1$

Choose the correct answer from the options given below:

If the function $f(x)=\begin{cases}\frac{1-\cos kx}{\sin^2 x}, & x\ne0\\\frac{1}{2}, & x=0\end{cases}$ is continuous at $x=0$, then the value of $k$ is

Let a function $f(x)=\alpha+(\beta^2+5\beta+6)|x|+\gamma|x|^4$, where $\alpha, \beta$ and $\gamma$ are real number. Then the function $f(x)$ is differentiable at $x=0$ if

If $x=t^3, y=t^4$, then $\frac{d^2y}{dx^2}$ is:

If $y=\log_e\left(\frac{x^2}{e^2}\right)$, then $\frac{d^2y}{dx^2}$ is equal to:

For the function $f(x) = |x + 1| + |x - 1| + |x - 2|$ which of the following are TRUE?

A. $f(x)$ is continous on $\mathbb{R}$

B. $f(x)$ is not differentiable at $x = 1$

C. $f(x)$ is differentiable on $\mathbb{R}$

D. $f(x)$ has no local maxima

Choose the correct answer from the options given below:

If $y = ax^{n+1} + bx^{-n}$, then $x^2 \dfrac{d^2y}{dx^2}$ is equal to

If $y = x\log_e\left(\dfrac{x}{2+x}\right)$, then $x^3\dfrac{d^2y}{dx^2}$ is equal to:

For the function $f(x) = \min\{|x|, |x-2|\}$ which of the following are TRUE?

A. Function $f(x)$ is continous on $\mathbb{R}$

B. Function $f(x)$ is differentible on $\mathbb{R}$

C. Function $f(x)$ is differentible on $\mathbb{R} \sim \{0,1,2\}$

D. Function is continous on $\mathbb{R}$ but not differentiable at $x = \dfrac{1}{2}$

Choose the correct answer from the options given below:

If a function $f(x) = \begin{cases}\lambda(x^2-2x), & x \le 0\\4x+1, & x>0\end{cases}$ then

If $y = \sqrt{12^x(3x^2+2)}$, then $\dfrac{dy}{dx}$ will be

(consider $\log_e x = \log x$)

If $y = \log_x(x^2+1).\log_{(x^2+1)} x$, then $\frac{dy}{dx}$ is

If $x = 3\sin\theta - \sin 3\theta$, $y = 3\cos\theta - \cos 3\theta$, then $\frac{d^2y}{dx^2}$ at $\theta = \frac{\pi}{3}$ is

If $f(x) = \begin{cases} \frac{x^2+3x-10}{x-2}, & x \neq 2 \\ 2k+1, & x = 2 \end{cases}$ is continuous at $x = 2$, then the value of $k$ is

Let $f(x) = |x-1| + |x-2|, x \in R$. Then which of the following statements is/are TRUE?

(A) $f$ is continuous everywhere.

(B) $f$ is continuous everywhere except at $x = 1$ and $x = 2$

(C) $f$ is not differentiable at $x = 1$ and $x = 2$

(D) $f$ is differentiable at $x = 1$ and $x = 2$

Choose the correct answer from the options given below:

If $x = 2t^2$ and $y = 3t^3$, then the value of $\frac{d^2y}{dx^2}$ at $t = 3$ is :

A function $f(x) = \begin{cases} \frac{kx^3}{4} & \text{if } x \leq 1 \\ 8 & \text{if } x > 1 \end{cases}$ for some constant $k$, then $\frac{d^2f(x)}{dx^2}$ at $x = -1$ is

The derivative of $|x|$ with respect to $x$, where $x \in R \sim \{0\}$ is

Observe the graph of a function $y = f(x)$ and identify the correct statements (s):

A. Function is continuous but not differentiable at $x = a$

B. Function is not defined at $x = a$

C. Function is defined at $x = a$ but not differentiable.

D. Given that at $x = a, y = b$.

Choose the correct answer from the options given below:

Match the LIST-I with LIST-II

Choose the correct answer from the options given below: