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

If $x = a\sin 2t(1 + \cos 2t)$ and $y = b\cos 2t(1 - \cos 2t)$, then $(\frac{dy}{dx})_{\text{at } x=\frac{\pi}{4}}$ is equal to

If $f(x) = |x| + |x - 5|$, then which of the following statements are TRUE?

(A) f is a continuous function every where

(B) f is a continuous function except $x = 5$ and $x = 0$

(C) f is a continuous function except $x = 0$ but not differentiable at $x = 5$

(D) f is a continuous function everywhere but not differentiable at $x = 0$ and $x = 5$

Choose the correct answer from the options given below:

Differentiation of $\frac{x^3}{1 - x^3}$ with respect to $x^3$ is equal to:

If $x = \frac{a}{1 + t}$ and $y = \frac{a}{(1 + t)^2}$ where $a > 0$ , then $\frac{d^2y}{dx^2}$ at $t = 1$ is

If $y = (x+1)(x^2+1)(x^4+1)(x^8+1)$ then $\frac{dy}{dx}$ at $x = -1$ is

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{d^2y}{dx^2}$ equals.

If $y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots + \infty}}}$ then $\frac{dy}{dx}$ equals to

If $f(x) = \begin{cases} ax - 1 & {if } x \ > 1\\ \ 2x + 1 & {if } x < 1 \end{cases}$ is continuous at $x = 1$, then $a$ equals

$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals

Derivative of $x^x$ with respect to $x\log x$ is

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{dy}{dx}$ at $\theta = \frac{\pi}{3}$ is

The function $f(x) = \begin{cases} \frac{(\sin 2x)}{x} + \cos x & , if \ x \neq 0 \\ K & , if \ x = 0 \end{cases}$ is continuous at $x = 0$, then the value of K is:

If $e^x + e^y = e^{x+y}$, then $\frac{dy}{dx}$ equals

If $x = e^t$ and $y = e^{2t}$ then $\frac{d^2y}{dx^2} =$

If the function

$f(x) = \begin{cases} ax + 2, & x \leq 1 \\ x^2 + 3x + b, & x > 1 \end{cases}$

is differentiable at $x = 1$, then the value of $(2a + b)$ is

If $y = \log_e\left(\frac{e^2}{x^2}\right)$ for $x \neq 0$, then $\frac{d^2y}{dx^2}$ equals

Match List-I with List-II

Where $\mathbb{R}$ is set of real numbers

Choose the correct answer from the options given below:

Let $x = t^2, y = t^3$. Then $\frac{d^2y}{dx^2}$ is equal to

Which of the following functions $f(x)$ are differentiable at $x = 0$?

(A) $|x|$

(B) $|x - 1|$

(C) $[x]$, where $[t]$ denotes the greatest integer $\leq t$

(D) $|x + 1|$

(E) $x^2$

Choose the correct answer from the options given below:

If $y = \sin^{-1} x + \sin^{-1} \sqrt{1-x^2}, x \in (-1, 0)$, then $\frac{dy}{dx}$ is equal to

Let $[x]$ denote the greatest integer $\leq t$ and $a \mathbb{Z} = [ax: x \in \mathbb{Z}, a \in \mathbb{R}]$ (where $\mathbb{Z}$ is set of integer and $\mathbb{R}$ is set of real number). The set of points of discontinuity of the function $f(x) = [2x]$ is given by

The derivative of $(\log x)^x$ with respect to $\log x$ is

If $x = t^{1/2}$, $y = t^{3/2}$, then $\frac{dy}{dx}$ =

If

$f(x) = \begin{cases} mx + 1,\ x \geq \pi/2 \\sin x + n, x \leq \pi/2, & \end{cases}$

is continuous at $x = \pi/2$, where $m \in \mathbb{Z}$ (set of integers), then $\sin 2n =$

If $y = \sin^{-1} \sqrt\frac{x}{x+1} + \sec^{-1}\sqrt{\frac{x+1}{x}}$, then $\frac{dy}{dx}$ is

Differentiation of $\log[\log(\log x^5)]$ with respect to $x$ is

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

If $y = t - \frac{1}{t}$ and $x = t + \frac{1}{t}$, then $\frac{dy}{dx}$ is equal to

If $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$, where $|x| < 1, |y| < 1$ and $x ≠ y$, then

The greatest integer function $f(x) = [x]$ is differentiable for all values of