List of top Mathematics Questions on Calculus

If \(\alpha, \beta\) are two distinct eigenvalues of a matrix \(A\), then the corresponding eigenvectors are

\(f:\mathbb{R}\rightarrow\mathbb{R}\) is increasing in \((-\infty,0)\cup(1,\infty)\) and decreasing in \((0,1)\). If \(f(2)=6\) and the continuous curve \(y=f(x)\) passes through \((0,0)\), then

If \( y = \frac{x^2}{1 + x^{b-a} + x^{c-a}} + \frac{x^2}{1 + x^{a-b} + x^{c-b}} + \frac{x^2}{1 + x^{a-c} + x^{b-c}} \), then \( \frac{dy}{dx} \) is:

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

If \( y = \sqrt{x} + \frac{1}{\sqrt{x}} \), then \( 2x \frac{dy}{dx} \) is equal to

फलन f : R $\rightarrow$ R निम्न द्वारा परिभाषित है:
f(x) = |x − 2| + 3|x − 1| + ||x − 2| − 1| .
ऐसे कितने बिन्दु हैं जिन पर f अवकलनीय नहीं है?

वास्तविक संख्याओं a तथा b के लिए निम्न फलन f : R $\rightarrow$ R पर विचार कीजिए:
\[ f(x) = \begin{cases} -ax - b & \text{यदि } x < -1, \\ 5x + 1 & \text{यदि } -1 \le x \le 1, \\ a^2x + 3b & \text{यदि } x > 1. \end{cases} \]
ऐसे कितने युग्म (a, b) संभव हैं जिनके लिए f सभी बिंदुओं पर सतत है?

जहाँ समाकलन सीमा −1 से 2 है, \(\int_{-1}^{2} \min\{1 - x, 1 - x^3\} \, dx\) का मान क्या है?

Let $f : \mathbb{R} \to \mathbb{R}$ be the function given by \[ f(x) = |x - 2| + 3|x - 1| + ||x - 2| - 1| . \] What is the number of points where $f$ is NOT differentiable?

What is the value of $\int_{-1}^{2} \min\{1 - x, 1 - x^3\} \, dx$?

If $z = x^{2}y^{3} + e^{y}\sin x$, then $\frac{\partial^{2}z}{\partial x\partial y} =$

If $y(x) = x^{x}, x > 0$ then $y''(2) - 2y'(2) =$

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

If $2^{x} + 2^{y} = 2^{x+y}$ then $\frac{dy}{dx} =$

If $f(x) = \begin{cases} 4(5^{x}) & x < 0 \\ 8k + x & x \ge 0 \end{cases}$ then $f'(-1) =$

Slope of the tangent to the curve $y = 9x^{2} + 7x^{4} + 5$ at the point $x = 1$ is

If $y = \sqrt{x + \sqrt{x + \sqrt{x + ... \infty}$ then $\frac{dy}{dx} = $

If $y = \frac{(a \cos x + b \sin x + c)}{\sin x}$ then $\frac{dy}{dx} = $

The value of $\lim_{x\rightarrow 0}\frac{\sqrt{1+x} - 1}{x} = $

The value of $\lim_{x\rightarrow\infty}(1 + \frac{1}{x})^{x} = $

The value of $\lim_{x \to 0} \frac{x^2 \sin^2 x}{x^2 - \sin^2 x}$ is:

Let $f$ be a twice differentiable function such that $f(x) = \int_0^x \tan(t-x) dt - \int_0^x f(t) \tan t dt, x \in (-\frac{\pi}{2}, \frac{\pi}{2})$. Then $f''(\frac{\pi}{6}) + 12 f'(-\frac{\pi}{6}) + f(\frac{\pi}{6})$ is equal to ________.

The area of the region \[ R=\{(x,y): xy\le 8,\; 1\le y\le x^2,\; x\ge 0\} \] is:

A rectangle is formed by the lines \( x = 0 \), \( y = 0 \), \( x = 3 \) and \( y = 4 \). Let the line \( L \) be perpendicular to \( 3x + y + 6 = 0 \) and divide the area of the rectangle into two equal parts. Then the distance of the point \( \left(\frac{1}{2}, -5\right) \) from the line \( L \) is equal to :

The value of \[ \lim_{n \to \infty} \left\{ \frac{1}{n+m} + \frac{1}{n+2m} + \frac{1}{n+3m} + \dots + \frac{1}{n+nm} \right\} \] is: