Gradients of Curves | 曲线的梯度

📚 Gradients of Curves | 曲线的梯度

In coordinate geometry, the gradient of a straight line is a measure of how steep it is, found by dividing the change in y by the change in x. When we move to a curve, however, the steepness is not constant: it changes from point to point. To describe the gradient of a curve, we need the idea of a tangent line and the limiting process that leads to differentiation.

在坐标几何中，直线的梯度衡量其倾斜程度，用 y 的变化量除以 x 的变化量来求。但当我们研究曲线时，陡峭程度并不是恒定的：它逐点变化。要描述曲线的梯度，我们需要切线以及引出微分的极限过程这两个概念。

1. Straight Line Gradient vs Curve Gradient | 直线梯度与曲线梯度的比较

For a straight line through two points (x₁, y₁) and (x₂, y₂), the gradient is constant and is given by m = (y₂ − y₁) ÷ (x₂ − x₁). No matter which two points you choose on the line, this ratio is always the same.

对于经过两点 (x₁, y₁) 和 (x₂, y₂) 的直线，梯度是恒定的，公式为 m = (y₂ − y₁) ÷ (x₂ − x₁)。无论你选择直线上的哪两个点，这个比值始终相同。

A curve does not have a single gradient. Instead, every point on a smooth curve has its own gradient, defined as the gradient of the tangent to the curve at that point. This requires a limiting argument because we cannot simply take two points on the curve: a chord joining two points is not the tangent.

曲线没有单一的梯度。相反，光滑曲线上的每个点都有自己的梯度，定义为该点处切线的梯度。这需要极限论证，因为我们不能简单地取曲线上的两个点：连接两点的弦并不是切线。

2. Gradient of a Chord | 弦的梯度

Consider a function y = f(x) and two nearby points on the curve: A(x, f(x)) and B(x + h, f(x + h)), where h is a small positive or negative step. The gradient of the chord AB is the difference in y-values divided by the difference in x-values.

考虑函数 y = f(x) 以及曲线上两个邻近的点：A(x, f(x)) 和 B(x + h, f(x + h))，其中 h 是一个小的正步长或负步长。弦 AB 的梯度等于 y 值之差除以 x 值之差。

Gradient of chord = [f(x + h) − f(x)] ÷ h

This expression is often called a difference quotient. It gives the average rate of change of f over the interval from x to x + h. Geometrically, it is the slope of the secant line AB.

这个表达式通常称为差商。它表示 f 在从 x 到 x + h 区间上的平均变化率。从几何上看，它就是割线 AB 的斜率。

For example, if f(x) = x², with x = 1 and h = 0.5, the chord gradient from (1, 1) to (1.5, 2.25) is (2.25 − 1) ÷ 0.5 = 2.5. This is not yet the gradient of the curve at x = 1, but it is an approximation.

例如，如果 f(x) = x²，取 x = 1 和 h = 0.5，则从 (1, 1) 到 (1.5, 2.25) 的弦梯度为 (2.25 − 1) ÷ 0.5 = 2.5。这还不是曲线在 x = 1 处的梯度，但只是一个近似值。

3. From Chords to Tangents | 从弦到切线

As the point B moves closer and closer to A, the value of h becomes smaller and smaller. The gradient of the chord AB approaches the gradient of the tangent at A. This idea of approaching a limit is fundamental to calculus.

当点 B 越来越接近点 A 时，h 的值就变得越来越小。弦 AB 的梯度趋近于 A 点处切线的梯度。这种趋近极限的思想是微积分的基础。

We write this limiting process as h → 0. If the limit exists, it gives the gradient of the curve at the point where x = a. This limiting value is called the derivative of f at x = a.

我们将这个极限过程写为 h → 0。如果极限存在，它就给出曲线在 x = a 处的梯度。这个极限值称为 f 在 x = a 处的导数。

The tangent can therefore be seen as the limiting position of a chord as the two endpoints coincide. The gradient of the tangent is the instantaneous rate of change of y with respect to x.

因此，切线可以看作弦的两个端点重合时的极限位置。切线的梯度就是 y 对 x 的瞬时变化率。

4. Differentiation from First Principles | 从第一性原理求导

To find the gradient function of f(x), we define the derivative by the first-principles formula:

为了求 f(x) 的梯度函数，我们用第一性原理公式定义导数：

f ‘(x) = lim (h → 0) [f(x + h) − f(x)] ÷ h

This formula calculates the gradient at a general point x, so it produces a new function f ‘(x), called the derived function or derivative.

这个公式计算一般点 x 处的梯度，因此它产生一个新函数 f ‘(x)，称为导函数或导数。

Worked example: let f(x) = x². Then f(x + h) = (x + h)² = x² + 2xh + h². Therefore f(x + h) − f(x) = 2xh + h². Dividing by h gives 2x + h. Letting h → 0 gives f ‘(x) = 2x.

例题：设 f(x) = x²。则 f(x + h) = (x + h)² = x² + 2xh + h²。因此 f(x + h) − f(x) = 2xh + h²。除以 h 得 2x + h。令 h → 0，得 f ‘(x) = 2x。

This shows that the gradient of the curve y = x² at any point x is exactly 2x. At x = 1, the gradient is 2, which confirms the earlier chord approximation improves as h decreases.

这表明曲线 y = x² 在任意点 x 处的梯度恰好为 2x。在 x = 1 处，梯度为 2，这验证了当 h 减小时，之前的弦近似会变得越来越准确。

5. Derivative Notation | 导数记号

There are several common notations for the derivative. If y = f(x), the derivative can be written as f ‘(x), dy/dx, or d/dx [f(x)]. Each has the same meaning: the gradient function of the curve.

导数有几种常用记号。如果 y = f(x)，导数可以写成 f ‘(x)、dy/dx 或 d/dx [f(x)]。它们的含义相同：曲线的梯度函数。

The notation dy/dx reminds us that the derivative is the limit of a quotient: a small change in y divided by a small change in x. It is read as ‘dee y by dee x’ and is not a fraction in the ordinary sense, although it can be manipulated in certain contexts such as the chain rule.

记号 dy/dx 提醒我们导数是商的极限：y 的微小变化除以 x 的微小变化。它读作 ‘dee y by dee x’，虽然它通常不是普通意义上的分数，但在链式法则等某些情境中可以灵活处理。

The value of the derivative at a specific point x = a is written f ‘(a) or dy/dx at x = a. It is the numerical gradient of the tangent at that point.

导数在具体点 x = a 处的值写为 f ‘(a) 或在 x = a 时的 dy/dx。它就是该点处切线的数值梯度。

6. Differentiating Powers of x | x 的幂函数的求导

The most important rule for differentiation in A-Level Mathematics is the power rule. If y = xⁿ, where n is any real constant, then dy/dx = n xⁿ⁻¹. This rule applies to positive powers, negative powers, and fractional powers.

A-Level 数学中最重要的求导法则是幂法则。如果 y = xⁿ，其中 n 是任意实常数，那么 dy/dx = n xⁿ⁻¹。该法则适用于正整数幂、负整数幂和分数幂。

Constants differentiate to zero because a constant function has a horizontal graph. If y = k, then dy/dx = 0.

常数求导为零，因为常数函数的图像是水平的。如果 y = k，则 dy/dx = 0。

Function	Derivative
y = k	dy/dx = 0
y = x	dy/dx = 1
y = x²	dy/dx = 2x
y = x³	dy/dx = 3x²
y = x⁻¹ = 1/x	dy/dx = −x⁻² = −1/x²
y = √x = x^½	dy/dx = (1/2) x^−½ = 1/(2√x)

When a function is a sum of terms, differentiate term by term. For example, if y = 4x³ − 2x + 7, then dy/dx = 12x² − 2. The constant 7 disappears.

当函数是若干项的和时，可以逐项求导。例如，如果 y = 4x³ − 2x + 7，那么 dy/dx = 12x² − 2。常数 7 消失。

7. Finding the Gradient at a Point | 求某一点的梯度

To find the gradient of a curve at a specific point, first differentiate the function to obtain dy/dx, then substitute the x-coordinate of the point into the derivative.

要求曲线在某一点处的梯度，首先对函数求导得到 dy/dx，然后将该点的 x 坐标代入导数。

Example: find the gradient of y = x³ − 3x + 2 at x = 1. First differentiate: dy/dx = 3x² − 3. Substituting x = 1 gives 3(1)² − 3 = 0, so the gradient is 0. This means the tangent is horizontal at x = 1.

例题：求 y = x³ − 3x + 2 在 x = 1 处的梯度。先求导：dy/dx = 3x² − 3。代入 x = 1 得 3(1)² − 3 = 0，所以梯度为 0。这意味着在 x = 1 处切线是水平的。

If the point is given as a full coordinate pair, check that the point lies on the curve first, then use its x-coordinate. This prevents sign errors and misread coordinates.

如果题目给出的是完整坐标对，先检查该点是否在曲线上，然后使用它的 x 坐标。这样可避免符号错误和坐标误读。

8. Tangents and Normals | 切线与法线

Once the gradient at a point is known, the equation of the tangent can be found using the straight-line formula y − y₁ = m(x − x₁), where m = dy/dx at the point.

一旦知道某点处的梯度，就可以用直线公式 y − y₁ = m(x − x₁) 求切线方程，其中 m 为该点处的 dy/dx。

The normal to a curve at a point is the line perpendicular to the tangent at that point. Its gradient is the negative reciprocal of the tangent gradient: m_normal = −1 ÷ m_tangent, provided m_tangent ≠ 0.

曲线在某点处的法线是在该点处与切线垂直的直线。它的梯度是切线梯度的负倒数：m_normal = −1 ÷ m_tangent，前提是切线梯度不为 0。

Example: find the tangent and normal to y = x² at x = 3. The derivative is dy/dx = 2x, so at x = 3 the gradient is 6. The point is (3, 9). The tangent is y − 9 = 6(x − 3). The normal has gradient −1/6, so its equation is y − 9 = −(1/6)(x − 3).

例题：求 y = x² 在 x = 3 处的切线和法线。导数为 dy/dx = 2x，因此在 x = 3 处梯度为 6。点为 (3, 9)。切线为 y − 9 = 6(x − 3)。法线梯度为 −1/6，因此它的方程是 y − 9 = −(1/6)(x − 3)。

Remember that a horizontal tangent has gradient 0, so its normal is vertical and has equation x = constant. Similarly, a vertical tangent has no defined normal gradient.

请记住，水平切线的梯度为 0，所以它的法线是竖直的，方程为 x = 常数。类似地，竖直切线没有定义的法线梯度。

9. Increasing and Decreasing Functions | 增函数与减函数

The sign of the derivative tells us whether a function is increasing or decreasing. If f ‘(x) > 0 on an interval, the function is increasing there. If f ‘(x) < 0, the function is decreasing.

导数的符号告诉我们函数是递增还是递减。如果在一个区间上 f ‘(x) > 0，则函数在该区间递增。如果 f ‘(x) < 0，则函数递减。

For example, consider f(x) = x³ − 3x. Its derivative is f ‘(x) = 3x² − 3 = 3(x² − 1) = 3(x − 1)(x + 1). The derivative is positive when x < −1 or x > 1, so f is increasing on those intervals. It is negative when −1 < x < 1, so f is decreasing there.

例如，考虑 f(x) = x³ − 3x。它的导数是 f ‘(x) = 3x² − 3 = 3(x² − 1) = 3(x − 1)(x + 1)。当 x < −1 或 x > 1 时导数为正，因此 f 在这些区间上递增。当 −1 < x < 1 时导数为负，因此 f 在该区间上递减。

This information helps us sketch curves and understand the behaviour of a function without plotting every point. The derivative acts as a signpost for the direction of travel along the curve.

这些信息帮助我们在不逐点绘图的情况下描画曲线，并理解函数的行为。导数就像曲线行进方向的指示牌。

10. Stationary Points | 驻点

A stationary point occurs where dy/dx = 0. At such a point, the tangent to the curve is horizontal, so the function is instantaneously neither increasing nor decreasing.

驻点出现在 dy/dx = 0 处。在这样的点上，曲线的切线是水平的，因此函数瞬时既不递增也不递减。

There are three main types of stationary points: local maximum points, local minimum points, and points of inflection with a horizontal tangent. A local maximum is higher than nearby points; a local minimum is lower than nearby points.

驻点主要有三种类型：局部极大点、局部极小点以及具有水平切线的拐点。局部极大点高于附近的点；局部极小点低于附近的点。

To locate stationary points, set dy/dx = 0 and solve for x. Then substitute these x-values into the original equation to find the corresponding y-coordinates.

为了确定驻点，令 dy/dx = 0 并解出 x。然后将这些 x 值代入原方程，求出对应的 y 坐标。

Example: for y = x³ − 3x, setting dy/dx = 3x² − 3 = 0 gives x = ±1. The stationary points are (−1, 2) and (1, −2).

例题：对于 y = x³ − 3x，令 dy/dx = 3x² − 3 = 0，得 x = ±1。驻点为 (−1, 2) 和 (1, −2)。

11. Second Derivative and Nature of Stationary Points | 二阶导数与驻点的性质

The second derivative, written f ”(x) or d²y/dx², is obtained by differentiating the derivative. It measures the rate of change of the gradient, so it tells us whether the gradient is increasing or decreasing.

二阶导数写为 f ”(x) 或 d²y/dx²，是将导数再次求导得到的。它衡量梯度的变化率，因此告诉我们梯度是在增大还是在减小。

To classify a stationary point using the second derivative test: if f ”(a) > 0, the point is a local minimum; if f ”(a) < 0, it is a local maximum. If f ''(a) = 0, the test is inconclusive and the first derivative test should be used.

使用二阶导数判别法给驻点分类：如果 f ”(a) > 0，该点是局部极小点；如果 f ”(a) < 0，该点是局部极大点。如果 f ''(a) = 0，判别法不能给出结论，应使用一阶导数判别法。

For y = x³ − 3x, we have dy/dx = 3x² − 3 and d²y/dx² = 6x. At x = 1, the second derivative is 6 > 0, so (1, −2) is a local minimum. At x = −1, it is −6 < 0, so (−1, 2) is a local maximum.

对于 y = x³ − 3x，有 dy/dx = 3x² − 3 且 d²y/dx² = 6x。在 x = 1 处，二阶导数为 6 > 0，所以 (1, −2) 是局部极小点。在 x = −1 处，它为 −6 < 0，所以 (−1, 2) 是局部极大点。

The second derivative also describes concavity. If d²y/dx² > 0 on an interval, the curve is concave upward. If d²y/dx² < 0, the curve is concave downward.

二阶导数还可以描述凹凸性。如果在某区间上 d²y/dx² > 0，则曲线向上凹。如果 d²y/dx² < 0，则曲线向下凹。

12. Exam Technique and Common Errors | 考试技巧与常见错误

When differentiating in an exam, always simplify the expression first: rewrite roots and fractions as powers of x before applying the power rule. For example, write 1/x³ as x⁻³ and √x as x^½.

考试中求导时，一定要先化简表达式：把根式和分式写成 x 的幂，再应用幂法则。例如，把 1/x³ 写成 x⁻³，把 √x 写成 x^½。

Be careful with signs when differentiating negative powers. The power decreases by 1, so x⁻³ becomes −3x⁻⁴, not −3x⁻². Write each step clearly to avoid dropping a minus sign.

求负指数幂的导数时要小心符号。幂会减 1，所以 x⁻³ 变成 −3x⁻⁴，而不是 −3x⁻²。每一步都写清楚，以免丢掉负号。

For tangent and normal questions, always identify the point and the gradient separately. Many errors come from using the wrong coordinate or forgetting that the normal gradient is the negative reciprocal.

对于切线和法线问题，一定要分别确定点和梯度。许多错误来自使用了错误的坐标，或忘记了法线梯度是负倒数。

Finally, when classifying stationary points, show evidence: write down the second derivative and evaluate it at the stationary point. A conclusion without supporting working may lose marks.

最后，给驻点分类时要展示依据：写出二阶导数并在驻点处求值。没有推导过程支持的结论可能会丢分。

Published by TutorHao | Edexcel A-Level Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Gradients of Curves | 曲线的梯度