極限、導數與函數的遞增遞減：练习题

Find the derivative of the function \( f(x) = \sin(x^3 + 2) \).

Using implicit differentiation, find the value of \( \frac{dy}{dx} \) at the point \( (3, 2) \) for the curve defined by \( xy + y^2 = 10 \).

The region \( R \) is bounded by the curve \( y = \ln x \), the \( y \)-axis, and the horizontal lines \( y = 0 \) and \( y = 1 \). Find the exact volume of the solid generated when the region \( R \) is rotated \( 360^\circ \) about the \( y \)-axis.

Determine the slope of the tangent to the curve \( y = \tan(x) \) at \( x = \frac{\pi}{4} \).

Find the derivative of the function \( f(x) = \ln(\sin^2 x + 1) \) with respect to \( x \).

Evaluate the definite integral \( \int_{1}^{2} (6x^2 - 2) \, dx \).

Determine the derivative of the function \(f(x) = \sin^2(3x)\) with respect to \(x\).

Find the exact coefficient of the \( x^3 \) term in the Maclaurin series expansion of the function \( f(x) = e^{2x} \cos(x) \).

Consider the function \( f(x) = x^3 - 4x^2 + 4x \).

(a) Find the derivative \( f'(x) \).
(b) Find the coordinates of the points on the graph of \( f \) where the gradient is zero.
(c) Determine the \( y \)-intercept of the graph.

The curve \( C \) is defined by the function \( g(x) = e^x (x^2 - 3) \) for \( x \in \mathbb{R} \).

(a) Show that \( g'(x) = e^x (x^2 + 2x - 3) \).
(b) Find the \( x \)-coordinates of the local maximum and local minimum points on \( C \).
(c) Find the equation of the tangent to the curve at the point where \( x = 0 \).