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

If $f(x+y) = f(x) + f(y)$ for all $x, y \in R$ (set of real numbers) and $f(x)$ is continuous at $x = 0$, then identify the correct statement ?

For the function $f(x) = -2|\log_e|x||$, which of the following is true?

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

If $y = 500e^{7x} + 600e^{-7x}$, then $\frac{d^2y}{dx^2}$ equals :

At $x = 2$, $f(x) = [x]$ is

[where [.] is greatest integer function]

Match List-I with List-II

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If $y = \cot^{-1}\left(\frac{1-x}{1+x}\right)$ then $\frac{dy}{dx}$ equals

If $x^2y^2 = a^{\pi/2}$, then $\dfrac{d^2y}{dx^2} =$

If $x = 16at^{\frac{1}{4}}$ and $y = 8at^{\frac{3}{4}}$ then $\dfrac{dy}{dx}$ is equal to:

A function defined as $f:R\to R$ satisfies $f(x+y)=f(x)f(y)$ and $f(x)\ne 0$ for all $x,y\in R$, where R is set of real numbers. If $f(x)$ is differentiable at $x=0$ and $f'(0)=2$, then $f'(x)$ is equal to

If the function $f(x)=\begin{cases} x^2, & x\le c \\ ax+b, & x>c \end{cases}$ is differentiable at $x=c$, then the relation between $b$ and $c$ is:

For the greatest integer function $f(x)=[x]$, which of the following statements are correct?

A. It is continuous at all real numbers.

B. It is continuous at $x=2$ and 3.

C. It is continuous and differentiable for $x\in(2,3)$

D. It is continuous and differentiable at $x=4.5$

Choose the correct answer from the options given below:

For the function $(ex)^y=e^x$;

Match the LIST-I with LIST-II

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If $y=\sqrt[5]{\log_e(5x^2)}$, then $\dfrac{dy}{dx}=$

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

If $x=\log_e t$ and $y=\dfrac{1}{t}$, then $\dfrac{d^2y}{dx^2}$ at $t=0.1$ is

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

For the function $f(x) = \sqrt{x}$, which of the following statements are true?

(A) $f(x)$ is continuous for $x \geq 0$.

(B) $f(x)$ is continuous for $x \in R$ (Set of real numbers).

(C) $f(x)$ is differentiable for $x > 0$.

(D) $f(x)$ is differentiable for $x \in R$ (Set of real numbers).

Choose the correct answer from the options given below:

If $x = \sqrt{a^{\tan^{-1}t}}$ and $y = \sqrt{a^{\cot^{-1}t}}$, then $\frac{d^2y}{dx^2}$ is

The value of '$\lambda$' for which the function $f(x) = \begin{cases}\frac{1-\cos 4x}{4x^2}, & \text{if } x \neq 0 \\ \lambda, & \text{if } x = 0\end{cases}$ is continuous at $x=0$, is:

Match List-I with List-II

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The graph of a function $f: R \to R$ is shown below, where $R$ is a real numbers

Then which of following statements is correct?

If $y = \log_e \sin(e^x + 5x + 500)$ then $\dfrac{dy}{dx}$ is equal to

If $f(x) = \begin{cases} \dfrac{1}{|x|}, & x \geq 1 \\ ax^2+b, & x < 1 \end{cases}$ is differentiable at $x=1$ then

The point('s) at which the function $f$ given by $f(x) = \begin{cases} \dfrac{|x|}{2x}, & x<0 \\ -\dfrac{1}{2}, & x\geq0 \end{cases}$ is continuous is (are)

Match the LIST-I with LIST-II

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

If $x\sqrt{1+y}+y\sqrt{1+x}=0, x\neq y$, then $\frac{dy}{dx}$ is equal to

If $f(x)=\begin{cases}\frac{\sin(a-b)x}{x}, & x<0\\ 2a+b, & x=0\\ \frac{-\tan 12x}{bx}, & x>0\end{cases}$, $(b>0)$ is continuous at $x=0$, then

If $y=(\sqrt x)^\pi$ such that $\frac{dy}{dx}=\frac{\pi}{a}y^{\pi+b}$ where $a,b\in\mathbb R$ then:

If $y = \dfrac{\log_e x}{x}$, then $\dfrac{d^2y}{dx^2}$ is :