List of top Mathematics Questions on Calculus

The set of all points of differentiability of the function $f(x)=e^{-|x|}$ is

The interval in which $f(x) = 2x^{2} - \log x$ increases is

The function $y = xe^{x}$ has}

If there is an error of 3% in the volume of a sphere then the percentage error in its radius is

$Lt_{x \to 0}\frac{e^{x^2}-cos x}{sin^2x}=$

Which of the following functions have finite number of points of discontinuity?

$Lt_{x \to 0}(\frac{|x|}{x}+x+2)=$

For \(a \in \mathbb{R}\), consider the real valued function defined on \((-1,1)\): \[ f(x)= \begin{cases} \dfrac{(1+x)^{1/3}-(1+2x)^{1/4}}{x}, & x\neq0,\\ a,& x=0. \end{cases} \] If \(f\) is differentiable at \(x=0\), then value of \(a+f'(0)\) is:

\[ \lim_{x\to0} \frac{|x|\log_e(1+|\sin2x|)} {x^2(|x|+3)} \]

Find the acute angle at which curves \[ y=(x-2)^2 \] and \[ y=-4+6x-x^2 \] intersect.

Let \(f:\mathbb{R}\to\mathbb{R}\), \(f(x)=|x+1|e^{-x^2}\), then which of the following option is true?

Find the value of \[ \lim_{x\to\infty} \frac{\sqrt{x}} {\sqrt{x+\sqrt{x+\sqrt{x}}}} \]

Let \(f:[0,\infty)\to\mathbb{R}\) be defined by \[ f(x)=\frac{3x^2+4x+1}{x^2+3x+2}. \] Then the value of \((f^{-1})'(2)\) is:

Evaluate \[ \int_{0}^{\frac{\pi}{4}}\frac{dx}{\cos^4x}. \]

If \(f:[0,\infty)\to\mathbb R\), \[ f(x)=\frac{x^2-1}{x^2+1}, \] then find \[ \int_{-1}^{1}f^{-1}(y)\,dy. \]

Find the value of \[ \lim_{x\to 0}\frac{x^2+2\cos x-2}{x\sin 3x} \]

The position vectors of two adjacent sides of a rectangle \(OACB\) are \(\vec{a}\) and \(\vec{b}\) respectively, where \(O\) is the origin. If \(16|\vec{a}\times\vec{b}|=3(|\vec{a}|+|\vec{b}|)^{2}\) and \(\theta\) be the acute angle between the diagonals \(OC\) and \(AB\), then the value of \(\tan\left(\frac{\theta}{2}\right)\) is:

The true set of values of \(K\) for which \(\sin^{-1}\left(\frac{1}{1+\sin^{2}x}\right)=\frac{K\pi}{6}\) may have a solution is:

If \(\int\frac{\csc^{2}x-2010}{\cos^{2010}x}dx=-\frac{f(x)}{(g(x))^{2010}}+c\), where \(f\left(\frac{\pi}{4}\right)=1\), then the number of solutions of the equation \(\frac{f(x)}{g(x)}=\{x\}\) in \([0,2\pi]\) is/are (where \(\{\cdot\}\) represents fractional part function):

Evaluate the integral: \(\int \frac{x}{x + 2} \, dx\).

The equation \(x^2 - Ky^2 - 4x + 6y - 5 = 0\) represents a pair of straight lines. Find the point of intersection.

Find the approximate value of \(\sqrt[3]{63}\).

Find the eigenvalues of the matrix \(A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}\).

Evaluate the integral: \(\int \frac{2x}{x^2 - 5x + 4} \, dx\)