進階微積分：练习题

Find the mean value of the function \( f(x) = \sinh x \) over the interval \( [0, \ln 2] \).

Find the first three non-zero terms in the Maclaurin series expansion for the function \(f(x) = \cosh(2x)\).

Find the coefficient of the \(x^3\) term in the Maclaurin series expansion of the function \(f(x) = \ln(1 + \sin x)\).

Which of the following is the first three non-zero terms of the Maclaurin series for \( f(x) = \sin(2x) \)?

Evaluate the improper integral:
\(\int_{1}^{\infty} \frac{1}{x^3} dx\)

A function is defined by \( f(x) = x^3 \). Calculate the mean value of \( f(x) \) over the interval \( [0, 2] \).

Evaluate the improper integral \(\int_{0}^{8} \frac{1}{\sqrt[3]{x}} dx\), showing the limiting process clearly.

A curve is defined by the parametric equations \( x = 2\cos t \) and \( y = 3\sin t \) for \( 0 \le t \le \frac{\pi}{2} \). Find the exact volume of the solid generated when the region bounded by this curve and the axes is rotated through \( 2\pi \) radians about the \( x \)-axis.

(a) Find the first three non-zero terms in the Maclaurin series expansion of \( \sin(2x) \). [2]

(b) Use your result from part (a) to find an approximation for the value of the integral \( \int_{0}^{0.5} \frac{\sin(2x)}{x} \, dx \). [3]

Consider the function \( f(x) = x^2 \cos(x) \).

(a) Find the first three non-zero terms of the Maclaurin series expansion for \( f(x) \). [3]

(b) Use your answer to part (a) to evaluate the limit \(\lim_{x \to 0} \frac{x^2 - f(x)}{x^4}\). [2]