List of top Mathematics Questions on Calculus asked in NIMCET

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