Differential Calculus Questions (112)

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

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

A wire $20cm$ long be divided into two parts, if one part is to be bent into a circle, the other part is to be bent into a square and the two plane figures are to have areas the sum of which is maximum, then side length of square is_____.

If $f(x+h) - f(x) + h f'(x+\theta h), 0 < \theta < 1$, the value of $40$, when $f(x) = Ax^2 + Bx + C$ is_____

The value of $\lim_{x \to \frac{\pi}{2}} \sqrt{\frac{\tan x - \sin\left[\tan^{-1}(\tan x)\right]}{\tan x + \cos^2(\tan x)}}$ is______.

The figure shows two regions in the first quadrant. $A(t)$ is the area under the curve $y = \sin x^2$ from $0$ to $t$ and $B(t)$ is the area of the triangle with vertices $O$, $P$ and $M(t, 0)$. If $\lim_{t \to 0} \frac{A(t)}{B(t)} = \frac{1}{k}$, then $k$ is______.

The figure shows a right triangle with its hypotenuse $OB$ along the $y$-axis and its vertex $A$ on the parabola $y = x^2$. Let $h$ represents the length of the hypotenuse which depends on the $x$-coordinate of the point $A$. The value of $\lim_{x \to 0} (h)$ equals

Let $f$ be a twice differentiable function defined in $[-3,3]$ such that $f(0) = -4, f'(3) = 0, f'(-3) = 12$ and $f''(x) \geq -2\forall x \in [-3,3]$. If $g(x) = \int_0^x f(t)dt$ then find maximum value of $g(x), x \in [-3,3]$

Let $f(x) = x\tan^{-1}(x^2) + x^4$. Let $f^k(x)$ denotes kth derivative of $f(x)$ w.r.t. $x$, $k \in \mathbb{N}$. If $f^{2m}(0) \neq 0$, $m \in \mathbb{N}$, then $m$ equals to______.

If $\int_x^e f(t)dt = \sin x - \cos x - \frac{x^2}{2}$ for all $x \in R$ then value of $-2f\left(\frac{\pi}{6}\right)$ is _____.

y = $\tan^{-1}(\cot x) + \cot^{-1}(\tan x)$, $\frac{\pi}{2} < x < \pi$, then $\left(-\frac{dy}{dx}\right)$ is equal to_____.

Let $f(x) = \tan^{-1} x, |x| \leq 1 = \frac{\pi}{4} \operatorname{sgn} x + \frac{x-1}{2}, |x| > 1$, (where $\operatorname{sgn}$ denotes signum function). Then the value of $4f\left(1^{+}\right)$ equals.