Investigation 5 – Gradient Functions | 探究5 – 梯度函数

📚 Investigation 5 – Gradient Functions | 探究5 – 梯度函数

Gradient functions, or derivatives, form the backbone of calculus. In this investigation, we will explore how the gradient of a curve varies, moving from secants to tangents, and derive the gradient function for basic polynomials. Understanding gradient functions helps us analyse rates of change, find optimal points, and interpret the shape of graphs. This investigation is designed to guide you through both graphical and algebraic approaches, reinforcing the key concepts of differentiation through numerical exploration and formal rules.

梯度函数，即导数，是微积分的核心。在本探究中，我们将探索曲线梯度如何变化，从割线到切线，并推导出基本多项式的梯度函数。理解梯度函数有助于我们分析变化率、寻找最优点并解释图像的形状。本探究旨在引导你掌握图形与代数两种方法，并通过数值探索和形式规则来巩固微分的关键概念。

1. What is a Gradient Function? | 什么是梯度函数？

The gradient of a straight line is constant and equals its slope. For a curve y = f(x), the gradient is not constant; it depends on the x‑value. The gradient function f'(x) or dy/dx gives the gradient of the tangent at any point (x, f(x)). In this investigation, we aim to find expressions for f'(x) from a given function f(x).

直线的梯度是常量，等于其斜率。对于曲线 y = f(x)，梯度不是常量，它依赖于 x 值。梯度函数 f'(x) 或 dy/dx 给出任意点 (x, f(x)) 处切线的斜率。在本探究中，我们的目标是从给定的函数 f(x) 求 f'(x) 的表达式。

We can investigate the gradient function graphically by drawing tangents at several points on a curve and estimating their slopes. Alternatively, we can use algebraic secant approximations and a limiting process. Both methods will lead us to the powerful concept of the derivative.

我们可以通过在曲线上多个点处绘制切线并估算其斜率，以图形方式研究梯度函数。也可以使用代数割线逼近和极限过程。这两种方法都将引领我们到达导数这个强大的概念。

For example, consider f(x) = x². If we sketch the tangent at x = 1, its slope is about 2; at x = 2, the slope appears to be 4. This suggests a relationship where the gradient function might be 2x. The investigation will formalise this observation.

例如，考虑 f(x) = x²。若我们画出 x = 1 处的切线，其斜率约为 2；在 x = 2 处，斜率似乎为 4。这提示了一种关系，即梯度函数可能是 2x。本探究将正式建立这种观察。

2. Secant and Tangent Lines | 割线与切线

Consider a curve y = f(x). A secant line passes through two distinct points on the curve: (x, f(x)) and (x+h, f(x+h)). The slope of this secant is given by the difference quotient: [f(x+h) – f(x)] / h. This quotient represents the average rate of change of f over the interval [x, x+h].

考虑曲线 y = f(x)。割线通过曲线上两个不同点：(x, f(x)) 和 (x+h, f(x+h))。这条割线的斜率由差商给出：[f(x+h) – f(x)] / h。该商表示 f 在区间 [x, x+h] 上的平均变化率。

As the step h becomes smaller, the secant line rotates and approaches the tangent line at x. The limiting value of the secant slope as h → 0, if it exists, is precisely the gradient of the tangent. This process is the geometric foundation of differentiation.

随着步长 h 变得越来越小，割线旋转并趋近于点 x 处的切线。当 h → 0 时，割线斜率的极限值（如果存在）正是切线的梯度。这个过程是微分的几何基础。

We can visualise this numerically for f(x) = x² at x = 2. The table below shows secant slopes for decreasing values of h. Observe how the slopes approach the value 4, which will become the exact gradient f'(2).

我们可以用数值方式直观显示 f(x) = x² 在 x = 2 处的这一过程。下表显示了随着 h 值减小，割线斜率的变化。观察斜率如何趋近于 4，这个值将成为精确梯度 f'(2)。

The numerical evidence strongly suggests that the instantaneous rate of change at x = 2 is 4. Repeating this procedure at other x‑values reveals a pattern that leads to the gradient function.

数值证据强有力地表明，在 x = 2 处的瞬时变化率为 4。在其他 x 值处重复此过程会揭示一个规律，从而导向梯度函数。

3. The Limit Definition of Derivative | 导数的极限定义

The derivative of a function f at x is formally defined as:

函数 f 在 x 处的导数正式定义为：

f'(x) = lim (h→0) [f(x+h) – f(x)] / h

provided this limit exists. This is known as differentiation from first principles. The notation f'(x) is attributed to Lagrange, while dy/dx is Leibniz’s notation, both expressing the same idea.

只要该极限存在。这被称为从第一原理求导。记号 f'(x) 源于拉格朗日，而 dy/dx 是莱布尼茨的记号，两者表达同一概念。

Using this definition, we can algebraically derive gradient functions for simple f(x). For f(x) = x², the expansion gives (x+h)² = x² + 2xh + h², so:

利用这一定义，我们可以用代数方法推导简单函数 f(x) 的梯度函数。对于 f(x) = x²，展开 (x+h)² = x² + 2xh + h²，因此：

f'(x) = lim (h→0) [x² + 2xh + h² – x²]/h = lim (h→0) (2x + h) = 2x

For f(x) = x³, the binomial expansion (x+h)³ = x³ + 3x²h + 3xh² + h³ leads to f'(x) = 3x². For f(x) = 1/x (x≠0), careful algebraic manipulation with a common denominator yields f'(x) = –1/x². These first‑principle calculations build confidence in the rules we will soon generalise.

对于 f(x) = x³，二项式展开 (x+h)³ = x³ + 3x²h + 3

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