Differential Calculus Questions (112)

If $a^2 + b^2 = 1$ and $u$ is the minimum value of $\frac{b+1}{a+b-2}$, then find the value of $u^2$.

Let $f$ be a differentiable function on $R$ and satisfying $f(x) = -(x^2 - x + 1)e^x + \int_0^x e^{xy} f'(y)dy$. If $f(1) + f'(1) + f''(1) = ke$, where $k \in N$, then find $k$.

Let $f : R \to R$ be a differentiable function satisfying $f(x) = f(x)f(x-y), \forall x, y \in R$ and $f'(0) = \int_0^{3}\{2x\}dx$, where $\{.\}$ denotes the fractional part function and $f'(-3) = \alpha e^\beta$. Then, $|\alpha + \beta|$ is equal to_____.

In a $\triangle ABC$, angles $A, B, C$ are in A.P. If $f(C) = \lim_{A \to C} \frac{\sqrt{3} - 4\sin A \sin C}{A - C}$, then $f\left(\frac{\pi}{12}\right)$ is equal to______.

Let $a = \min(x^2 + 2x + 3, x \in \mathbb{R})$ and $b = \lim_{x \to 0} \frac{\sin x \cos x}{e^x - e^{-x}}$. Then the value of $\sum_{r=0}^{n} a^r b^{n-r}$ is :

$\lim_{n \to \infty} \left(\frac{\sqrt[n]{p} + \sqrt[n]{q}}{2}\right)^n$, $p, q > 0$ equals :

The values of $\text{Lim}_{x \to 0^+} \frac{f(-x)-x^2}{1-\cos x}$ where $[\cdot]$ denote greatest integer function and $(\cdot)$ denote fraction part function.

If $f\left(\frac{xy}{2}\right) = \frac{f(x).f(y)}{2}, \forall x, y \in R, f(1) = f'(1) = 2$. Then, $\frac{f(3)}{f'(3)}$ is_____.

The third derivative of a function $f(x)$ vanishes for all $x$. If $f(0) = 1, f'(1) = 2$ and $f''(1) = -1$, then find $f''(x)$ at $x = 3$.

Let $f(x)$ be a differentiable function, $f(1) = 0, f'(1) = 2$ then the value of $\lim_{x \to 1}\frac{\int_1^x \sin(t(f(t)))dt}{(x-1)^2}$ is_____.

Let $K > 0$ and $\lambda = \lim_{x \to 0} \frac{K\left(1-4\sqrt{K^2-x^2}\right)}{x^2\sqrt{K^2-x^2}}$ is finite then the value of $\lambda K$ is_____.

If the independent variable $x$ is changed to $y$, then the expression $x\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 - \frac{dy}{dx} = 0$ is transformed to $x\frac{d^2x}{dy^2} + \left(\frac{dx}{dy}\right)^2 = k\frac{dx}{dy}$ then $k$ equals.

$\text{Limit}_{x \to \infty} \frac{\cot^{-1}(\sqrt{x+1}-\sqrt{x})}{\sec^{-1}\left(\frac{2x+1}{x-1}\right)}$ is equal to:

The value of $\lambda$ for which $2\left(\text{Lim}_{x \to 0} f(x^3-x^2)\right) = \lambda \left(\text{Lim}_{x \to 0} f(2x^4-x^3)\right)$ is:

In the interval $(a,b)$ there exists at least one point $c$, for any two differentiable function $f$ and $g$ such that $\begin{vmatrix} f(a) & f(b) \\ \phi(a) & \phi(b) \end{vmatrix} - \lambda^2(b-a)\begin{vmatrix} f(a) & f'(c) \\ \phi(a) & \phi'(c) \end{vmatrix}$, then sum of absolute value of $\lambda$ is_____.

If $f(x) = \begin{cases} \frac{\left(\frac{\pi}{2} - \sin^{-1}\left|1-\{x\}^2\right|\right) \sin^{-1}(1-\{x\})}{\sqrt{2}\left(\{x\} - \{x\}^3\right)} & x > 0 \\ k & x = 0 \\ \frac{A\sin^{-1}(1-\{x\})\cos^{-1}(1-\{x\})}{\sqrt{2}\{x\}(1-\{x\})} & x < 0 \end{cases}$ is continuous at $x = 0$, then the value of $A$ is______. (where $\{.\}$ denotes fractional part of $x$).

If $9 + f''(x) + f'(x) = x^2 + f^2(x)$, where $f(x)$ is twice differentiable function such that $f''(x) \neq 0 \forall x \in R$ and let $P$ be the point of maxima of $f(x)$ then find the number of tangents which can be drawn from $P$ to the circle $x^2 + y^2 = 9$.

Let $f: [-1,1] \to \left[-\frac{\pi}{4}, \tan 1 + \tan^{-1}\right]$ defined by $f(x) = \tan x + \tan^{-1} x$ and the derivative of $f^{-1}(x)$ at $x = 0$ is $'k'$ then the value of $\frac{4}{k}$ is_____.

The number of point where $|f(x)| + |x-2| - 1$ is non-differentiable in $x \in (0, 3\pi)$, where $f(x) = \prod_{k=1}^{n}\frac{\left(1+2\cos\left(\frac{2x}{3^k}\right)\right)}{3}$ is_____.

Let $f : R \to R$ is a function satisfying $f(10-x) = f(x)$ and $f(2-x) = f(2+x), \forall x \in R$. If $f(0) = 101$. Then, the minimum possible number of values of $x$ satisfying $f(x) = 101, x \in [0, 25]$ is_____.

Let $f_1(x)$ and $f_2(x)$ be twice differentiable function. Where $F(x) = f_1(x) + f_2(x)$ and $G(x) = f_1(x) - f_2(x)$, $\forall x \in \mathbb{R}$. $f_1(0) = 2$ and $f_2(0) = 1$. If $f_1'(x) = f_2(x)$ and $f_2'(x) = f_1(x)$, $\forall x \in \mathbb{R}$, then the number of solutions of the equation $(F(x))^2 = \frac{9x^4}{G(x)}$ is______.

Let $f(n) = \left[\sqrt{n} + \frac{1}{2}\right]$, where $[.]$ denotes greatest integer function, $\forall n \in \mathbb{N}$. Then $\sum_{n=1}^{\infty} \frac{2^{f(n)} + 2^{-f(n)}}{2^n}$ is equal to______.

Let $P(x_0, y_0)$ be a point on the curve $C : (x^2 - 11)(y + 1) + 4 = 0$ where $x_0, y_0 \in N$. If area of the triangle formed by the normal drawn to the curve 'C' at $P$ and the co-ordinate axes is $\left(\frac{a}{b}\right), a,b \in N$ then the least value of $(a - 6b)$.

The set of all real values of $x$ where $f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|$ is not differentiable is

Let $f,g,h:\mathbb{R}\to\mathbb{R}$ be differentiable with $f(x)=x^5+x^3+3x+7$, $g(f(x))=x$ and $h(g(g(x)))=x$. Value of $h'(-1)$ is

Water is filled at rate $\pi$ cm$^3$/s in right circular conical vessel (vertex up) of height 5 cm and diameter 8 cm. When water height is 3 cm, rate of increase of wet conical surface area is (cm$^2$/s)

If $x=\cos\theta$ and $y=\sin^3\theta$, then $\left|y\dfrac{d^2y}{dx^2}+\left(\dfrac{dy}{dx}\right)^2\right|$ at $\theta=\dfrac{\pi}{2}$ is

$\lim_{x \to \infty} \sqrt[3]{(x+a)(x+b)(x+c)} - x =$

If $f(x) = \begin{cases} \frac{[(a-n)nx - \tan x]\sin nx}{x^2} & \text{at } x = 0 \\ 0 & \text{at } x = 0 \end{cases}$, where $n$ is a non-zero real number, and $f$ is continuous at $x = 0$, then $a$ is equal to:

Let $f(x) = x^5 + ax + b, x \in \mathbb{R}, f(0) > 0$ & $f(x)$ has integral roots. Tangent at $\left(\frac{5}{2}, p\right)$ to $y = f(x)$ is parallel to $x$-axis & $g(x) = f(x+1)$. Column 1: (A) $(a + b)$ can be (B) Value of $[p]$ can be (where $[.]$ represents greatest integer function) (C) Number of points where $g(|x|)$ is non differentiable can be (D) Number of points where $|g(|x|)|$ is non differentiable can be Column 2: (p) $-1$ (q) $1$ (r) $3$ (s) $-3$ (t) $5$

Let $f(x)=\displaystyle\int e^x(x-1)(x-2)\,dx$. Then $f(x)$ decreases in the interval

Let $f,g,h:\mathbb{R}\to\mathbb{R}$ be differentiable with $f(x)=x^5+x^3+3x+7$, $g(f(x))=x$ and $h(g(g(x)))=x$. Value of $h'(-1)$ is

Which of the following functions is differentiable at $x=0$?

The set of all real values of $x$ where $f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|$ is not differentiable is

Which of the following functions is differentiable at $x=0$?

Let $f(x)=\displaystyle\int e^x(x-1)(x-2)\,dx$. Then $f(x)$ decreases in the interval

Let $f,g,h:\mathbb{R}\to\mathbb{R}$ be differentiable with $f(x)=x^5+x^3+3x+7$, $g(f(x))=x$ and $h(g(g(x)))=x$. Value of $h'(-1)$ is

The set of all real values of $x$ where $f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|$ is not differentiable is

$\lim_{x \to \infty} \sqrt[3]{(x+a)(x+b)(x+c)} - x =$

Define $f: [0, \pi] \to \mathbb{R}$ by $f(x) = \begin{cases} \tan^2 x + \sqrt{2\sin^2 x + 3\sin x + 4 - \sqrt{\sin^2 x + 6\sin x + 2}} & x \neq \pi/2 \\ k & x = \pi/2 \end{cases}$ is continuous at $x = \pi/2$, then $k$ is equal to:

The function $f(x) = [x] + \sqrt{\{x\}}$, where $[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function respectively, is discontinuous at

Let $f(x) = \begin{cases} \frac{\tan^2 [x]}{x^2 - [x]^2} & \text{for } x > 0 \\ 1 & \text{for } x = 0 \\ \sqrt{[x]} \cot [x] & \text{for } x < 0 \end{cases}$ where $[x]$ is the step up function and $\{x\}$ is the fractional part function of $x$, then :

The function $f(x) = \sqrt{1 - \sqrt{1 - x^2}}$

The function, $f(x) = [x] - [[x]]$, where $[ ]$ denotes greatest integer function:

$f$ is a continuous function in $[a, b]$; $g$ is a continuous function in $[b, c]$. A function $h(x)$ is defined as: $h(x) = f(x)$ for $x \in [a, b]$ $= g(x)$ for $x \in [b, c]$ if $f(b) = g(b)$, then

Which of the following limits vanish?

$\lim_{x \to c} f(x)$ does not exist when:

Let $f(x) = |x - 1|([x] - [-x])$, then which of the following statement(s) is/are correct. (where $[.]$ denotes greatest integer function.)

The function $f(x) = x^2 \left[x^2 - \frac{1}{x^2}\right], x \neq 0$ is ($[x]$ represents the greatest integer $\leq x$)

If $y = f(x)$ defined parametrically by $x = 2t - |t - 1|$ and $y = 2t^2 + t|t|$, then: