極限、導數與變化率：練習題

Evaluate the definite integral \( \int_{0}^{2} (3x^2 + 1) dx \).

Find the gradient of the tangent to the curve \(y = x^3 - 2x\) at the point where \(x = 2\).

Find the area enclosed by the curves \(y = x^2\) and \(y = 2x\).

Find the indefinite integral of \( \int (2x + 3) dx \).

An open-topped box is designed with a square base of side length \( x \) cm and a fixed volume of \( 108 \, \text{cm}^3 \). The total surface area of the box, \( A \), is given by the function:
\( A(x) = x^2 + \frac{432}{x} \) for \( x > 0 \).
Find the value of \( x \) that minimizes the total surface area of the box.

Evaluate the definite integral \( \int_{1}^{2} 4x^3 \, dx \).

A curve is defined by the equation \( y = \frac{1}{3}x^3 - 4x \). Find the positive \( x \)-coordinate of the local minimum point.

A particle moves along a horizontal line so that its velocity, \(v\) \(ms^{-1}\), at time \(t\) seconds is given by \(v(t) = 6t^2 - 18t + 12\) for \(0 \le t \le 3\). Determine the total distance traveled by the particle during the first 3 seconds.

Consider the function \( f(x) = 3x^2 + 2x \).
(a) Find the indefinite integral \( \int (3x^2 + 2x) dx \).
(b) Calculate the area of the region enclosed by the curve \( y = f(x) \), the x-axis, and the vertical lines \( x = 1 \) and \( x = 2 \).

The rate at which a population of bacteria grows is modeled by \( \frac{dP}{dt} = 200e^{0.1t} \), where \( P \) is the number of bacteria and \( t \) is time in hours.

a) If the initial population is 1000, find an expression for \( P(t) \).

b) Find the population after 5 hours.

c) Find the time \( t \) when the population reaches 5000.

d) Find the average rate of growth over the first 10 hours.