List of top Mathematics Questions on Continuity and differentiability asked in UP Board XII

Is the function \(f(x)=\begin{cases}x+5,&x\le1\\x-5,&x>1\end{cases}\) continuous at \(x=1\)?

Examine the continuity of the function \(f(x)=2x^2-1\) at \(x=3\).

Differentiate \(\log(\log x)\), \(x>1\) with respect to \(x\).

If \(x=a\sin^3t,\ y=b\cos^3t\), then find \(\dfrac{dy}{dx}\) at the point \(t=\dfrac{\pi}{2}\).

Prove that \(f(x)=\tan x\) is a continuous function.

If \(x=at^2,\ y=2at\), find the value of \(\dfrac{dy}{dx}\).

If \(\cos y=x\cos(a+y)\) and \(\cos a\ne\pm1\), then prove that \(\dfrac{dy}{dx}=\dfrac{\cos^2(a+y)}{\sin a}\).

If \(f(x)=\begin{cases}x+2, & x\ne0\\1, & x=0\end{cases}\), then prove that the function is not continuous at \(x=0\).

If \(x-y=\pi\), then the value of \(\dfrac{dy}{dx}\) is:

If \(y=\sin^{-1}x\), then show that \((1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}=0\).

Prove that the function defined by \(f(x)=\begin{cases}x^2\sin\dfrac1x, & x\neq0\\0, & x=0\end{cases}\) is continuous.

Differentiate the function \(\sin^2x\) with respect to the function \(e^{\cos x}\).

Differentiate: \(y=x^{\sin x}+(\cos x)^{\tan x}\).

Find the values of \(a\) and \(b\) for which the function defined by \(f(x)=\begin{cases}ax+1, & x\le3\\bx+3,& x>3\end{cases}\) is continuous at \(x=3\).

For what value of \(\lambda\), is the function \[f(x)=\begin{cases}\lambda(x^2-2x), & x\le0 \\ 4x+1, & x>0\end{cases}\] continuous at \(x=0\)?