Applications of differentiation：練習問題

A particle moves along a straight line. Its displacement, \(s\) meters from a fixed point at time \(t\) seconds, is given by \(s = 2t + \ln(t + 1)\) for \(t \ge 0\). Find the velocity of the particle at \(t = 1\) second.

Find the equation of the normal to the curve \(y = \ln(2x-1)\) at the point where \(x = 1\).

An open-topped rectangular box with a square base of side \(x\) cm is to be made from a thin metal sheet. If the total surface area of the sheet used is \(48 \text{ cm}^2\), find the maximum volume of the box in \(\text{cm}^3\).

The equation of a curve is given by \(y = x^2 - 6x + 8\). Find the \(x\)-coordinate of the stationary point on this curve.

The volume of a cube is increasing at a constant rate of \(12 \text{ cm}^3/\text{s}\). Find the rate of increase of the total surface area of the cube at the instant when the side length is \(2 \text{ cm}\).

The radius of a circular oil spill is increasing at a rate of \(0.1\) m/s. Find the rate of increase of the area when the radius is \(5\) m.

The cost function for a product is \(C(x) = 500 + 10x + 0.05x^2\). Find the level of production \(x\) that minimizes the average cost per unit, \(A(x) = \frac{C(x)}{x}\).

Find the coordinates of the point on the curve \(y = \sqrt{x}\) that is closest to the point \((2, 0)\).

Consider the curve defined by the equation $$y = 2\text{ln}(x) + x^2$$ where $$x > 0$$.

A spherical metal ball is being heated and its volume is expanding at a constant rate of \(8 \text{ mm}^3/\text{s}\). Determine the rate at which the surface area of the ball is increasing when its radius reaches \(4 \text{ mm}\).