1.Find the value of $k$ if $f(x) = \frac{\sin 5x}{3x}$ for $x \neq 0$ and $f(x) = k$ for $x = 0$ is continuous at $x = 0$.
2.Discuss the continuity of the function $f(x) = |x-1| + |x-2|$ at $x = 1$ and $x = 2$.
3.Find the relationship between $a$ and $b$ so that the function $f(x) = ax+1$ (if $x \le 3$) and $f(x) = bx+3$ (if $x > 3$) is continuous at $x = 3$.
4.Examine the continuity of the function $f(x) = x \sin\left(\frac{1}{x}\right)$ for $x \neq 0$ and $f(x) = 0$ for $x = 0$ at the point $x = 0$.
5.Find the value of $k$ if $f(x) = \frac{k \cos x}{\pi - 2x}$ for $x \neq \pi/2$ and $f(x) = 3$ for $x = \pi/2$ is continuous at $x = \pi/2$.
6.Discuss the continuity of the greatest integer function $f(x) = [x]$ at $x = 2.5$ and at $x = 3$.
7.Prove that the function $f(x) = \cos(x^2)$ is a continuous function everywhere.
8.Identify the points of discontinuity for the function $f(x) = \frac{x^2 - 4}{x - 2}$.
9.Show that the function $f(x) = |x - 3|$ is continuous but not differentiable at $x = 3$.
10.Find the Left Hand Derivative (LHD) and Right Hand Derivative (RHD) of $f(x) = |x|$ at $x = 0$.
11.Check the differentiability of $f(x) = x^2 \sin(1/x)$ for $x \neq 0$ and $f(0) = 0$ at $x=0$.
12.Let $f(x) = 2x+3$ for $x \le 1$ and $f(x) = x^2+4$ for $x > 1$. Is the function differentiable at $x = 1$?
13.Differentiate $y = \log_7(\log_e x)$ with respect to $x$.
14.Differentiate $y = \cos(\sqrt{x})$ with respect to $x$.
15.Find the derivative of $y = \sin^3(2x+1)$.
16.Find the derivative of $y = \sqrt{\tan \sqrt{x}}$.
17.Find $\frac{dy}{dx}$ if $2x + 3y = \sin y$.
18.Find $\frac{dy}{dx}$ if $x^2 + xy + y^2 = 100$.
19.Find $\frac{dy}{dx}$ if $\sin^2 y + \cos(xy) = k$, where $k$ is a constant.
20.Differentiate $y = \frac{e^x}{\sin x}$ with respect to $x$.
21.Differentiate $y = e^{\sin^{-1}x}$ with respect to $x$.
22.Find $\frac{dy}{dx}$ if $x^3 + x^2y + xy^2 + y^3 = 81$.
23.Find $\frac{dy}{dx}$ for $y = \sin^{-1}\left(\frac{2x}{1+x^2}\right)$.
24.Find $\frac{dy}{dx}$ for $y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)$, where $-\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}$.
25.Find $\frac{dy}{dx}$ for $y = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$, where $0 < x < 1$.
26.Differentiate $y = \sec^{-1}\left(\frac{1}{2x^2-1}\right)$ with respect to $x$.
27.Differentiate $y = \tan^{-1}\left(\frac{\sin x}{1+\cos x}\right)$ with respect to $x$.
28.Differentiate $y = \tan^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)$ with respect to $x$.
29.Differentiate $y = \cos^{-1}(\sin x)$ with respect to $x$.
30.Differentiate $y = \cot^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $x$.
31.Differentiate $y = x^{\sin x}$ with respect to $x$.
32.Differentiate $y = (\sin x)^{\cos x}$ with respect to $x$.
33.Find $\frac{dy}{dx}$ if $x^y = y^x$.
34.Find $\frac{dy}{dx}$ if $y = (\log x)^x + x^{\log x}$.
35.Use logarithmic differentiation to find the derivative of $y = \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)}}$.
36.Find $\frac{dy}{dx}$ if $y^x = e^{y-x}$.
37.Differentiate $y = \log_a x$ with respect to $x$.
38.Find $\frac{dy}{dx}$ if $x^y \cdot y^x = 1$.
39.Find $\frac{dy}{dx}$ if $x = a \cos \theta$ and $y = a \sin \theta$.
40.Find $\frac{dy}{dx}$ if $x = a t^2$ and $y = 2at$.
41.Find $\frac{dy}{dx}$ if $x = a(\theta - \sin \theta)$ and $y = a(1 - \cos \theta)$.
42.Find $\frac{dy}{dx}$ if $x = a(\cos t + t \sin t)$ and $y = a(\sin t - t \cos t)$.
43.If $x = \sqrt{a^{\sin^{-1}t}}$ and $y = \sqrt{a^{\cos^{-1}t}}$, prove that $\frac{dy}{dx} = -\frac{y}{x}$.
44.Find the value of $\frac{dy}{dx}$ at $\theta = \frac{\pi}{4}$ if $x = a \sec^3 \theta$ and $y = a \tan^3 \theta$.
45.Find $\frac{d^2y}{dx^2}$ if $y = x^2 + 3x + 2$.
46.Find $\frac{d^2y}{dx^2}$ if $y = x \cdot \cos x$.
47.If $y = A \sin x + B \cos x$, prove that $\frac{d^2y}{dx^2} + y = 0$.
48.If $y = e^x (\sin x + \cos x)$, prove that $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 2y = 0$.
49.If $y = (\tan^{-1} x)^2$, show that $(x^2+1)^2 y_2 + 2x(x^2+1)y_1 = 2$.
50.Find $\frac{d^2y}{dx^2}$ if $x = a \cos t$ and $y = b \sin t$.
51.Verify Rolle's Theorem for the function $f(x) = x^2 + 2$ in the interval $[-2, 2]$.
52.Verify Rolle's Theorem for the function $f(x) = \sin x$ in the interval $[0, \pi]$.
53.Verify Lagrange's Mean Value Theorem for $f(x) = x^2 - 4x - 3$ in the interval $[1, 4]$.
54.Find a point on the parabola $y = (x-3)^2$ where the tangent is parallel to the chord joining $(3,0)$ and $(4,1)$.
55.Discuss the applicability of Rolle's Theorem for the function $f(x) = |x|$ in $[-1, 1]$.
