1.
Find the equation of the common tangent to the curves $y = x^2$ and $y = -x^2 + 4x - 4$.
2.
Calculate the shortest distance between the parabola $y = x^2 + 3x + 2$ and the straight line $x - y - 2 = 0$.
3.
Find the value(s) of the parameter $a$ for which the function $f(x) = \sin x - ax + b$ is strictly decreasing on the entire real number line $\mathbb{R}$.
4.
Using the concepts of monotonicity, prove the double inequality: $\frac{x}{1+x} < \ln(1+x) < x$ for all $x > 0$.
5.
A window is designed in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the entire window is fixed at $12\text{ m}$, find the dimensions of the rectangular portion that will maximize the total area of the window.
6.
An inverted conical tank has a base radius of $2\text{ m}$ and a height of $4\text{ m}$. Water is flowing into the tank at the constant rate of $2\text{ m}^3/\text{min}$. At what exact rate is the water level rising at the instant when the depth of the water is $3\text{ m}$?
7.
Determine the absolute maximum and absolute minimum values of the trigonometric function $f(x) = \sin x + \frac{1}{2}\cos 2x$ over the closed interval $[0, \frac{\pi}{2}]$.
8.
Prove analytically that the family of ellipses $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the family of hyperbolas $\frac{x^2}{a^2 + \lambda} + \frac{y^2}{b^2 + \lambda} = 1$ intersect orthogonally at all points of intersection.
9.
For the polynomial function $f(x) = x^4 - 4x^3 + 4x^2$, find all points of local maxima, local minima, and the exact coordinates of any points of inflection.
10.
A manufacturer can sell $x$ items at a price of rupees $(5 - \frac{x}{100})$ each. The total cost price of producing $x$ items is rupees $(\frac{x}{5} + 500)$. Find the number of items he should produce and sell to maximize his overall profit.
11.
Determine the precise intervals of strict monotonicity (intervals of increase and decrease) for the transcendental function $f(x) = x^x$ defined for $x > 0$.
12.
A camera is mounted on the ground $2000\text{ m}$ away from the base of a rocket launch pad. The rocket is rising vertically at a constant velocity of $50\text{ m/s}$. Find the rate of change of the angle of elevation of the camera at the exact moment when the rocket is $1500\text{ m}$ above the ground.
13.
Find the equations of the two distinct tangent lines to the quadratic curve $y = x^2 - 2x + 3$ that can be drawn from the external origin point $(0, 0)$.
14.
Find the points on the curve $y = x^3$ where the slope of the tangent is equal to the $x$-coordinate of the point. Subsequently, find the equation of the normal at the non-zero point.
15.
Show that the maximum volume of a cylinder which can be inscribed in a sphere of radius $R$ is $\frac{1}{\sqrt{3}}$ times the volume of the sphere itself.
