Differentiability Questions (1063)

lim n→∞tan n X r=1 tan−1  1 1 + r + r2 ! is equal to:

Let \(f(x) = x^3 - 9x^2 + 24x - 4\). Let \(g(x)\) be defined as follows: \[g(x) = \begin{cases} f(x+2); & x If \(g(x)\) is continuous for all \(x\), find the value of \(a\).

Let $u(x)$ and $v(x)$ be differentiable functions such that $\dfrac{u(x)}{v(x)} = 7$. If $\dfrac{u'(x)}{v'(x)} = p$ and $\left(\dfrac{u(x)}{v(x)}\right)' = q$, then $\dfrac{p+q}{p-q}$ has the value equal to:

30. Let \(y\) be an implicit function of \(x\) defined by \(x^{2x} - 2x^x \cot y - 1 = 0\). The value of \(y'(1)\), where \(y'\) denotes the first derivative of \(y\), is:

If the function $f(x)=\begin{cases}(1+|\cos x|)^{\frac{\lambda}{|\cos x|}}, & 0<x<\dfrac{\pi}{2}\\\mu, & x=\dfrac{\pi}{2}\\\dfrac{\cot 6x}{e^{\cot 4x}}, & \dfrac{\pi}{2}<x<\pi\end{cases}$ is continuous at $x=\dfrac{\pi}{2}$, then $9\lambda+6\log_e\mu+\mu^6-e^{6\lambda}$ is equal to:

Let \(f(x) = \begin{cases} \left[1 + \ln(c^2 + c + 1)\tan^2(x-1)\right]^{\frac{1}{(\ln x)^2}}, & x \neq 1 \\ 3c, & x = 1 \end{cases}\), where \(c \in R\).If \(\lim_{x \to 1} f(x)\) exists but \(f(x)\) is discontinuous at \(x = 1\), then \(c\) can take the value:

The function \(f(x) = |x-3|, x \geq 1\)\(\frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, x

199. Let \(f(x)\) be a function defined by \(f(x) = (k - x^{10})^{1/10}\) where \(k = 1025\) and \(f'(2) = \dfrac{1}{f'(a)}\) where \(a \in N\), then \(a\) equals:

The value of $\lim_{x \to 0} \left[\frac{3}{2} + \frac{x^2}{\sin x \tan x}\right]$ (where $[\cdot]$ denotes the greatest integer function) is

If $x = 3\tan t$ and $y = 3\sec t$, then the value of $\dfrac{d^2y}{dx^2}$ at $t = \dfrac{\pi}{4}$ is:

For any positive integer n, define fn(x) = n X j=1 tan−1  1 1 + (x + j)(x + j −1)  , x ∈(0, ∞). Then which of the following statement(s) is(are) true?

The value of limx→−1 cos 2−cos 2x x2−|x| is:

252. The number of points where \(f(x) = |x + [x]| - 3[2x] + 4[3x]\) is discontinuous in \([-1, 1]\), is:[Note: \([k]\) denotes greatest integer less than or equal to \(k\).]

If β = limx→0 ex3−(1−x3)1/3+((1−x2)1/2−1) sin x x sin2 x , find 6β.

Let $y(x)=\left(1+x\right)\!\left(1+x^2\right)\!\left(1+x^4\right)\!\left(1+x^8\right)\!\left(1+x^{16}\right)$. Then $y'(1)-y(1)$ is:

If $3f(x) - 2f(1/x) = x$, then $f'(2)$ is equal to:

For n ∈N, let an = Pn k=1 2k and bn = Pn k=1(2k −1). Then limn→∞(√an −√bn) is:

Let f: \mathbb{R}^+ \to \mathbb{R} be a differentiable function satisfying: \frac{f(x)}{y} + \frac{f(y)}{x} + f(xy) = \frac{}{} for all x, y \in \mathbb{R}^+, with f(1) = 0 and f'(1) = 1. Find \lim_{x \to e} \lfloor f(x) \rfloor (where \lfloor \cdot \rfloor denotes greatest integer function).

Let α(a) and β(a) be the roots of the equation ( 3√1 + a−1)x2+(√1 + a−1)x+( 6√1 + a−1) = 0. Find lima→0+ α(a) and lima→0+ β(a).

If x = \cos\theta and y = \sin^3\theta, then find y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 at \theta = \frac{\pi}{2}.

Let \(f\) be a continuous function on \(\mathbb{R}\) such that \(f\left(\frac{1}{n^4}\right) = \frac{(\sin e^{-n^2})e^{-n^2} + \frac{n^2}{n^2+1}}{1}\). Then \(f(0) = \)

If \(f(x) = \cot^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)\) and \(g(x) = \cos^{-1}\left(\frac{1 - x^2}{1 + x^2}\right)\) then \(\lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}\) for \(0

Let $S$ be the set of all points in $(-\pi,\pi)$ at which the function $f(x)=\min(\sin x,\cos x)$ is not differentiable. Then $S$ is a subset of which of the following?

Let $y=y(x)$ be a function of $x$ satisfying $y\sqrt{1-x^2}=k-x\sqrt{1-y^2}$ where $k$ is a constant and $y\!\left(\tfrac{1}{2}\right)=-\tfrac{1}{4}$. Then $\dfrac{dy}{dx}$ at $x=\tfrac{1}{2}$:

Let $f$ be defined for all $x \in \mathbb{R}$. If $f$ is differentiable and $f(x^3) = x^5$ for all $x \in \mathbb{R}\,(x\ne0)$, then $f'(27)$ is equal to:

If $y = f(x)$ satisfies $f(x+y) = f(x)+f(y)+2xy-1$ for all $x,y\in\mathbb{R}$ and $f'(0) = \cos\alpha$, then which of the following is/are true?

\(h(x)=\min\{\{x\},1-\{x\}\}\). h is:

Let \(f\), \(g\) and \(h\) be differentiable functions. If \(f(0) = 1\), \(g(0) = 2\), \(h(0) = 3\) and the derivative of their pairwise product at \(x = 0\) are \((fg)'(0) = 6\), \((gh)'(0) = 4\) and \((hf)'(0) = 5\), then compute the value of \((fgh)'(0)\).

771. Let \(f: R \to R\) be a function defined by \(f(x) = \begin{cases} x^3 + 2x^2 + x + c, & \text{if } x \leq b \\ e^x, & \text{if } x > b \end{cases}\), where \(b\) and \(c\) are integers. If \(f(x)\) is differentiable \(\forall\, x \in R\), then find the value of \((b + c)\).

Let \(f(x) = \begin{cases} \dfrac{ax^3 + bx^2 + cx + d}{x}, & x \neq 0 \\ 2, & x = 0 \end{cases}\) where \(a, b, c, d\) are in A.P. If \(f(x)\) is continuous, then the number of local minima of \(y = |f(|x|)|\) is:

Let \(g(x) = \begin{cases} k\sqrt{x+1}; & 0 \leq x \leq 3 \\ mx + 2; & 3

A certain function \(f(x)\) has the property that \(f(3x) = af(x)\) for all positive real values of \(x\) and \(f(x) = 1 - |x - 2|\) for \(1 \le x \le 3\). Find \(\lim_{x \to 2} (f(x))^{\frac{\csc\left(\frac{\pi x}{2}\right)}{1}}\)

If $f(x) = \begin{cases} x\tan^{-1}(1/x), & x \in (-\infty,-1]\cup[1,\infty) \\ 0, & x = 0 \end{cases}$, then $f'(0)$ is:

If \(\alpha,\ \beta,\ (\alpha

Question 23: Given We have f: (-1, 1) → ℝf(0) = -1, f'(0) = 1If g(x) = [f(2f(x) + 2)]^(1/2)Find g'(0)

The value of limx→0(1 + sin x)cos x is:

Evaluate: \(\lim_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac{\sin x}{x - \sin x}}\)

If $f(x) = \sin\!\left[\tan^{-1}\!\left(\dfrac{1-x^2/2}{x}\right)\right]$, $g(x) = \cos\!\left[2\tan^{-1}\!\left(\dfrac{x}{1+\sqrt{1-x^2}}\right)\right]$, and $h(x)=f(x)-g(x)$, then $h'(1/2)$:

\(f(x)=\begin{cases}|x-3| & x\ge 1\\ \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4} & x is:

limx→0 cos(sin x)−cos x x4 is equal to:

Let $f(x)$ be a polynomial function of second degree. If $f(1) = f(-1)$ and $a, b, c$ are in A.P., then $f'(a),\, f'(b),\, f'(c)$ are in:

If $f(\theta) = \tan^{-1}\!\left(\dfrac{\sin\theta-\cos\theta}{\sin\theta+\cos\theta}\right)$, then which of the following are correct?

If limx→0 x2 sin βx αx−sin x = 1, find 6(α + β).

Let $y(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $\dfrac{d}{dx}\!\big[y(x)\big]_{x=1}$ equals: [Integer type]

Let \(|f(x)-f(y)|\le 2|x-y|^{3/2}\), \(f(0)=1\). Find \(\int_0^1 f^2(x)\,dx\).

Given \(f\) piecewise defined. \(f_1(x)=|f(|x|)|\). If \(f\) is continuous in \([-2,10]\), then:

The value of limx→0 | cos(sin 3x)|−1 x2 is:

Evaluate: \(\lim_{x \to 0} \left(\tan \frac{x}{x}\right)^{\frac{1}{x}}\)

Let $f(x)$ be a polynomial function of second degree. If $f(1) = f(-1)$ and $a, b, c$ are in A.P., then $f'(a)$, $f'(b)$, $f'(c)$ are in:

Suppose \(f\) is a derivable function that satisfies the equation \( f(x+y) = f(x) + f(y) + x^2y + xy^2 \) for all real numbers \(x\) and \(y\). Suppose that \( \lim_{x \to 0}\left[\dfrac{f(x)}{x}\right] = 1 \), find \( f(3) = \) __________.