Let $f: R \to R$ be a differentiable function such that $f(2) = 6$ and $f'(2) = \frac{1}{48}.$ Then $\lim_{x \to 2} \int_{6}^{f(x)} \frac{4t^3}{x - 2} dt$ equals

If $f(x)=\sin \left(\cos ^{-1}\left(\frac{1-2^{2 x}}{1+2^{2 x}}\right)\right)$ and its first derivative with respect to $x$ is $-\frac{ b }{ a } \log _{ e } 2$ when $x =1,$ where $a$ and $b$ are integers,then the minimum value of $\left| a ^{2}- b ^{2}\right|$ is.........

Let $C$ be the curve $y = x^3$ (where $x$ takes all real values). The tangent at $A(t, t^3)$ meets the curve again at $B(T, T^3)$. If the gradient at $B$ is $K$ times the gradient at $A$,then $K$ is equal to

Suppose $a$ is a positive real number such that $a^5-a^3+a=2$. Then,

Let $f$ be a differentiable function and the equation of the normal to the graph of $y = f(x)$ at $x = 3$ is $3y = x + 18$. If $L = \mathop {\lim }\limits_{x \to 1} \frac{{f\left( {3 + {{\left( {4{{\tan }^{ - 1}}x - \pi } \right)}^2}} \right) - f\left( {3 + {{\left( {f\left( 3 \right) - x - 6} \right)}^2}} \right)}}{{{{\sin }^2}\left( {x - 1} \right)}}$,then:

If $f(x) = \begin{cases} 3x^2 + 12x - 1, & -1 \le x \le 2 \\ 37 - x, & 2 < x \le 3 \end{cases}$,then: