Differentiation：練習問題

Find the area of the shaded region bounded by the curve, the \(x\)-axis, and the vertical lines shown in the diagram.

Using the product rule, find the derivative of \(y = x^2 \ln x\) with respect to \(x\).

Find the \(x\)-coordinate of the stationary point of the curve \(y = x^2 - 8x + 12\).

Find the derivative of \(y = \frac{\sin x}{1 + \cos x}\) with respect to \(x\).

Find \(\frac{dy}{dx}\) given that \(y = 3\cos(2x)\).

Find the derivative of the function \(y = x^2 \ln x\) with respect to \(x\).

Find the x-coordinate of the stationary point of the curve \(y = e^{2x} - 4x\).

Calculate the area of the region bounded by the curve \(y = \sin x\), the x-axis, and the vertical lines \(x = 0\) and \(x = \pi\).

A curve has the equation \(y = 3x^2 - 12x + 9\).
(a) Find the coordinates of the stationary point and determine its nature using the second derivative test.
(b) Calculate the area of the region bounded by the curve and the \(x\)-axis.

A solid right circular cylinder of radius \(r\) cm and height \(h\) cm is to be carved such that its total surface area is \(600\pi\) cm\(^2\).
(a) Show that the volume, \(V\) cm\(^3\), of the cylinder is given by \(V = 300\pi r - \pi r^3\).
(b) Given that \(r\) can vary, find the value of \(r\) for which \(V\) has a stationary value.
(c) Determine whether this stationary value is a maximum or a minimum.