Applications of differentiation：練習問題

The side of a cube is increasing at a rate of \(0.2 \text{ cm/s}\). Find the rate of change of the volume of the cube when the side length is \(5 \text{ cm}\).

The height of a cylinder is equal to its base diameter. If the volume of the cylinder increases at a constant rate of \( 12\pi \text{ cm}^3/\text{s} \), find the rate of change of the base radius when the radius is \( 2 \text{ cm} \).

Consider the curve \(C\) defined by the equation \( y = \frac{x}{x^2+3} \). Find the sum of the slopes of the tangents to the curve \(C\) at all its points of inflexion.

Determine the interval where the function \( f(x) = x^2 - 6x + 5 \) is strictly increasing.

A sector of a circle has a fixed perimeter of \(20 \text{ cm}\). Find the radius of the sector that maximizes its area.

Find the slope of the normal to the curve \(y = \sqrt{x} + x\) at the point where \(x = 4\).

Find the equation of the normal to the curve \(y = \frac{8}{x}\) at the point where \(x = 2\).

A right circular cylinder is inscribed in a right circular cone with a height of \(12 \text{ cm}\) and a base radius of \(6 \text{ cm}\). If the base of the cylinder lies on the base of the cone, find the maximum possible volume of the cylinder.

Find the coordinates of the local maximum and local minimum points of the curve \(y = \frac{1}{3}x^3 - x^2 - 3x + 5\).

Consider the curve \(C: y = \frac{x^2 + 3}{x - 1}\) for \(x \neq 1\).

(a) Find the equations of all vertical and oblique asymptotes of \(C\).

(b) Find the coordinates of the stationary points of \(C\).