1.Discuss the continuity of $f(x) = [\cos(\pi x)]$ in the closed interval $[0, 2]$, where $[\cdot]$ denotes the greatest integer function. Find the points of discontinuity.
2.Let $f(x) = \min(|x|, |x-2|)$. Find the set of points where $f(x)$ is continuous but not differentiable.
3.If $f(x) = \left\{ \begin{array}{ll} x^2 & \text{for } x \le c \\ ax + b & \text{for } x > c \end{array} \right.$ is differentiable at $x = c$, find the values of $a$ and $b$ in terms of $c$.
4.Find the value of $k$ if the function $f(x) = \frac{1 - \cos(1-\cos x)}{x^4}$ for $x \neq 0$, and $f(x) = k$ for $x = 0$, is continuous at $x = 0$.
5.Find the derivative of $y = \tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $\tan^{-1}x$.
6.If $y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \dots \infty}}}$, prove that $\frac{dy}{dx} = \frac{\cos x}{2y - 1}$.
7.If $x^y = e^{x-y}$, strictly prove that $\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}$.
8.If $x = a(\cos t + \log \tan \frac{t}{2})$ and $y = a\sin t$, evaluate $\frac{d^2y}{dx^2}$ at $t = \frac{\pi}{4}$.
9.Differentiate $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ with respect to $\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ when $x \in (1, \infty)$. Be mindful of the domain intervals.
10.If $y = \left(x + \sqrt{1+x^2}\right)^n$, prove that $(1+x^2)y_2 + x y_1 - n^2y = 0$.
11.If $\cos y = x\cos(a+y)$, with $\cos a \neq \pm 1$, prove that $\frac{dy}{dx} = \frac{\cos^2(a+y)}{\sin a}$.
12.Apply Rolle's Theorem to the function $f(x) = (x-a)^m(x-b)^n$ on the interval $[a, b]$ where $m, n$ are positive integers. Find the value of $c$.
13.Using Lagrange's Mean Value Theorem, prove the inequality: $\frac{b-a}{1+b^2} < \tan^{-1}b - \tan^{-1}a < \frac{b-a}{1+a^2}$, where $0 < a < b$.
14.Find the left hand and right hand derivatives of $f(x) = |x-1| + |x-3|$ at $x = 2$. Is it differentiable at $x=2$?
15.If $y = \frac{\sin^{-1}x}{\sqrt{1-x^2}}$, show that $(1-x^2)\frac{dy}{dx} - xy = 1$.
16.If $\sqrt{1-x^6} + \sqrt{1-y^6} = a^3(x^3 - y^3)$, prove that $\frac{dy}{dx} = \frac{x^2}{y^2}\sqrt{\frac{1-y^6}{1-x^6}}$.
17.If $y = e^{a \sin^{-1}x}$, prove that $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} - a^2y = 0$.
18.Differentiate $y = \tan^{-1}\left(\frac{5x}{1-6x^2}\right)$ with respect to $x$.
19.Find the number of non-differentiable points for the function $f(x) = \max(\sin x, \cos x)$ in the interval $[0, 2\pi]$.
20.Discuss the differentiability of $f(x) = [x^2] \sin(\pi x)$ at $x=1$ and $x=2$, where $[\cdot]$ is the greatest integer function.
21.Using Mean Value Theorem, prove that $e^x > 1 + x$ for all $x > 0$.
22.If $y = x^{x^{x^{\dots \infty}}}$, prove that $\frac{dy}{dx} = \frac{y^2}{x(1 - y\log x)}$.
23.Find the derivative of $y = \log_{x^2}(x^3 + x)$ with respect to $x$.
24.If $y = \frac{ax+b}{cx+d}$, prove that $2y_1 y_3 = 3(y_2)^2$, where $y_n$ denotes the $n^{th}$ derivative of $y$.
25.Find $f''(0)$ for the function $f(x) = |x|^3$.
26.If $y = (x+\sqrt{x^2-1})^m + (x-\sqrt{x^2-1})^m$, prove that $(x^2-1)y_2 + x y_1 - m^2y = 0$.
27.Determine the values of $a$ and $b$ if $f(x) = \frac{x+a}{\sqrt{x}}$ is a solution to the differential equation $2x \frac{d^2y}{dx^2} + 3\frac{dy}{dx} = 0$.
28.Let $f(x) = |x-1|([x] - [-x])$, where $[\cdot]$ denotes the greatest integer function. Investigate the continuity of $f(x)$ at $x=1$.
29.Evaluate $\frac{dy}{dx}$ at $x=e$ for the function $y = (\log_e x)^{\log_e x}$.
30.If $y = \tan^{-1}\left(\frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}}\right)$, find $\frac{dy}{dx}$.
