1.
A particle moves along the curve $y = x^2 + 2x$. At what point on the curve are the $x$ and $y$ coordinates of the particle changing at the same rate?
2.
A $5\text{ m}$ long ladder is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2\text{ m/s}$. How fast is the angle $\theta$ between the ladder and the ground changing when the foot of the ladder is $4\text{ m}$ away from the wall?
3.
Water is leaking from a conical funnel at the rate of $5\text{ cm}^3\text{/sec}$. If the radius of the base of the funnel is $5\text{ cm}$ and its height is $10\text{ cm}$, find the rate at which the water level is dropping when the water is $2.5\text{ cm}$ deep.
4.
A man $2\text{ m}$ high walks at a uniform speed of $5\text{ km/h}$ away from a lamp-post $6\text{ m}$ high. Find the rate at which the length of his shadow increases.
5.
Find the intervals in which the function $f(x) = \frac{x^4}{4} - x^3 - 5x^2 + 24x + 12$ is strictly increasing or strictly decreasing.
6.
Find the intervals in which the function $f(x) = (x+1)^3(x-3)^3$ is strictly increasing.
7.
Determine the intervals in which the function $f(x) = \sin^4 x + \cos^4 x$, where $x \in [0, \frac{\pi}{2}]$, is strictly increasing and strictly decreasing.
8.
Prove that the function $f(x) = \frac{4\sin x}{2 + \cos x} - x$ is an increasing function of $x$ in $[0, \frac{\pi}{2}]$.
9.
Find the equation of the tangent to the curve $y = \sqrt{3x-2}$ which is parallel to the line $4x - 2y + 5 = 0$.
10.
Find the points on the curve $y = x^3 - 2x^2 - x$ at which the tangent lines are parallel to the line $y = 3x - 2$.
11.
Find the angle of intersection between the curves $y^2 = x$ and $x^2 = y$ at their non-zero point of intersection.
12.
Find the equations of the normal to the curve $y = x^3 + 2x + 6$ which are parallel to the line $x + 14y + 4 = 0$.
13.
Use differentials to find the approximate value of $\sqrt{0.037}$, given that $\sqrt{0.04} = 0.2$.
14.
Find the approximate value of $f(5.001)$, where $f(x) = x^3 - 7x^2 + 15$.
15.
The radius of a sphere shrinks from $10\text{ cm}$ to $9.8\text{ cm}$. Find the approximate decrease in its volume using differentials.
16.
Find the local maxima and local minima for the function $f(x) = x^3 - 6x^2 + 9x + 15$.
17.
Find the absolute maximum and absolute minimum values of $f(x) = 2x^3 - 15x^2 + 36x + 1$ on the interval $[1, 5]$.
18.
Find the maximum and minimum values of the function $f(x) = x + \sin 2x$ on the interval $[0, 2\pi]$.
19.
Find the local maximum and local minimum values of $f(x) = \frac{x}{2} + \frac{2}{x}$ for $x > 0$.
20.
A wire of length $36\text{ m}$ is cut into two pieces. One piece is bent into a square and the other into an equilateral triangle. Find the lengths of the two pieces so that the combined area of the square and the triangle is minimum.
21.
Find the dimensions of the maximum area rectangle that can be inscribed in a semicircle of radius $R$.
22.
Show that the right circular cylinder of a given volume, which is open at the top, has a minimum total surface area when its height is equal to its base radius.
23.
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10\text{ m}$. Find the dimensions of the window to admit maximum light through the whole opening.
24.
Show that the volume of the greatest cylinder which can be inscribed in a cone of height $h$ and semi-vertical angle $\alpha$ is $\frac{4}{27}\pi h^3 \tan^2\alpha$.
