📚 The slope of a graph | 图像的斜率
Understanding the slope of a graph is fundamental to mastering calculus and analysing relationships in mathematics. Whether you are dealing with a straight line or a complex curve, the slope tells you how one variable changes with respect to another. This revision guide covers everything an IB student needs: from basic gradient calculations to differentiation, tangents, normals, and real-life applications.
理解图像的斜率是掌握微积分和分析数学关系的基础。无论你处理的是直线还是复杂曲线，斜率都告诉你一个变量相对于另一个变量的变化速率。本篇复习指南涵盖了 IB 学生所需的一切内容：从基本的梯度计算到微分、切线、法线以及实际应用。
1. Definition of Slope | 斜率的定义
The slope of a graph at any point measures the steepness and direction of the line or curve. For a straight line, slope is constant; for a curve, slope varies at different points. In mathematical notation, slope is often denoted by m and defined as the ratio of the vertical change (rise) to the horizontal change (run) between two distinct points.
图像上任意一点的斜率衡量的是直线或曲线的陡峭程度和方向。对于直线，斜率是恒定的；对于曲线，不同点的斜率是变化的。在数学符号中，斜率通常用 m 表示，定义为两个不同点之间的垂直变化（纵轴变化）与水平变化（横轴变化）之比。
ans the graph goes upward as x increases, a negative slope means it goes downward, and a zero slope indicates a horizontal line. An undefined slope occurs when the line is vertical.
斜率 = 上升量 / 水平距离 = Δy / Δx，其中 Δy = y₂ − y₁，Δx = x₂ − x₁。正斜率表示随着 x 增大图像向上走，负斜率表示向下走，零斜率表示水平线。当直线垂直时，斜率无定义。
2. Calculating Slope from Two Points | 通过两点计算斜率
nt. For example, for points A(2, 3) and B(5, 9), the slope is (9 − 3) / (5 − 2) = 6/3 = 2.
给定直线上的两点 (x₁, y₁) 和 (x₂, y₂)，斜率 m 的计算公式为 m = (y₂ − y₁) / (x₂ − x₁)。无论你先标哪一个点，只要顺序一致，这个公式都成立。例如，对于点 A(2, 3) 和 B(5, 9)，斜率为 (9 − 3) / (5 − 2) = 6/3 = 2。
often test this with fractions or parameters, so practise simplifying fractions and dealing with algebraic expressions.
使用该公式时，要注意负坐标。如果给定了 C(−1, 4) 和 D(3, −2)，则 m = (−2 − 4) / (3 − (−1)) = −6/4 = −3/2。IB 考试题经常用分数或参数来考查，因此要练习化简分数和处理代数表达式。
3. Slope of a Straight Line Graph | 直线图像的斜率
form Ax + By + C = 0, you can rearrange it to slope-intercept form to identify the slope.
形如 y = mx + c 的线性方程具有斜率 m 和 y 轴截距 c。其图像是一条斜率恒定的直线。例如，y = 3x − 5 的斜率为 3，与 y 轴交于点 (0, −5)。若方程以一般式 Ax + By + C = 0 给出，你可以将其重新整理成斜截式来识别斜率。
er line is vertical. For example, a line perpendicular to y = 2x + 1 must have slope −1/2.
平行线具有相同的斜率：如果直线 L₁ 的斜率为 m₁，L₂ 的斜率为 m₂，那么当 m₁ = m₂ 时它们平行。垂直的直线满足 m₁ × m₂ = −1，前提是两直线都不垂直。例如，与 y = 2x + 1 垂直的直线斜率必须为 −1/2。
4. Slope as a Rate of Change | 作为变化率的斜率
In applied contexts, slope represents a rate of change. In a distance–time graph, the slope gives the speed. In a velocity–time graph, the slope is the acceleration. In economics, the slope of a cost function can represent marginal cost. IB questions frequently ask you to interpret the real-world meaning of the slope.
在应用情境中，斜率代表变化率。在距离–时间图中，斜率表示速度。在速度–时间图中，斜率是加速度。在经济学中，成本函数的斜率可以代表边际成本。IB 试题经常要求你解释斜率在现实世界中的含义。
For a linear model, the slope is constant, so the rate of change is uniform. For a non-linear model, the rate of change is not constant; it depends on the point you choose. This leads directly to the concept of instantaneous rate of change, which is the derivative.
对于线性模型，斜率是常数，因此变化率是均匀的。对于非线性模型，变化率不是常数，它取决于你所选择的点。这直接引出了瞬时变化率的概念，也就是导数。
5. Introduction to Gradient of Curves | 曲线梯度导论
Unlike straight lines, curves have a slope that changes continuously. The gradient of a curve at a point is defined as the slope of the tangent line to the curve at that point. To estimate this gradient without calculus, you can draw a tangent by hand and calculate its slope, or use a secant through two close points and shrink the interval.
与直线不同，曲线的斜率是不断变化的。曲线在某一点的梯度定义为该点切线的斜率。要在不用微积分的情况下估算该梯度，你可以用手画出切线并计算其斜率，或者通过两个靠近的点作割线并缩小间隔。
f the secant PQ is [f(a+h) − f(a)] / h. As h approaches 0, the secant approaches the tangent, and the difference quotient gives the derivative. This limit idea is the foundation of differential calculus.
若点 P(a, f(a)) 和 Q(a+h, f(a+h)) 在曲线 y = f(x) 上，则割线 PQ 的斜率为 [f(a+h) − f(a)] / h。当 h 趋近于 0 时，割线逼近切线，差商就给出导数。这一极限思想是微分学的基础。
6. Tangent Lines and Instantaneous Slope | 切线与瞬时斜率
ative of the function at that x-value. For y = f(x), the instantaneous slope at x = a is f'(a), read as "f prime of a". This represents the instantaneous rate of change.
曲线在某点的切线在该点处与曲线相切而不穿过（局部地）。其斜率等于函数在该 x 值处的导数。对于 y = f(x)，在 x = a 处的瞬时斜率为 f'(a)，读作“f 撇 a”。这代表瞬时变化率。
To find the tangent line equation, use the point-slope form: y − f(a) = f' en write the linear equation. IB questions often require this in both paper 1 and paper 2.
要找切线方程，可使用点斜式：y − f(a) = f'(a)(x − a)。你必须能够在给定点计算导数值，然后写出线性方程。IB 试题常在第 1 和第 2 试卷中要求这一点。
7. Differentiation: Finding Slope Function | 微分：求斜率函数
Differentiation is the process of finding the derivative function f' nt multiple rule, and the sum/difference rule. For example, if f(x) = 4x³ − 2x + 7, then f'(x) = 12x² − 2.
微分是求导函数 f'(x) 的过程，它给出图像在任意 x 处的斜率。基本法则包括：幂法则 d/dx (xⁿ) = nxⁿ⁻¹、常数倍法则以及和差法则。例如，若 f(x) = 4x³ − 2x + 7，则 f'(x) = 12x² − 2。
IB expects you to know derivatives of sin x, cos x, eˣ /dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴. Always simplify your derivative before evaluating.
IB 要求你掌握 sin x、cos x、eˣ 和 ln x 的导数，以及积、商和链式法则。对于 y = (3x² + 1)⁵，设 u = 3x² + 1，则 dy/dx = 5u⁴ × du/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴。在求值之前，务必先化简导数。
8. Equation of Tangent and Normal | 切线和法线方程
Once you have the slope of the tangent mtan = f' the tangent line at x = a. The normal line is perpendicular to the tangent, so its slope mnorm = −1 / f'(a), provided f'(a) ≠ 0. The normal passes through the same point (a, f(a)).
一旦有了切线斜率 mtan = f'(a)，你就可以写出 x = a 处的切线方程。法线垂直于切线，因此其斜率 mnorm = −1 / f'(a)，前提是 f'(a) ≠ 0。法线经过同一点 (a, f(a))。
For example, find the tangent and normal to y = x² at x = 1. f'(x) = 2x, so f' y − 1 = 2(x − 1) → y = 2x − 1. Normal: slope = −1/2, so y − 1 = −½(x − 1) → y = −½x + 1.5.
例如，求 y = x² 在 x = 1 处的切线和法线。f'(x) = 2x，所以 f'(1) = 2。该点为 (1, 1)。切线：y − 1 = 2(x − 1) → y = 2x − 1。法线：斜率为 −1/2，所以 y − 1 = −½(x − 1) → y = −½x + 1.5。
9. Increasing and Decreasing Functions | 增函数与减函数
he sign of the first derivative tells us where the graph is increasing or decreasing. If f'(x) > 0 on an interval, the function is increasing there; if f'(x) < 0, it is decreasing. f' y points. IB questions may ask you to determine intervals of increase/decrease using a sign diagram.
一阶导数的符号告诉我们图像在哪里增加或减少。如果在某一区间 f'(x) > 0，函数在此区间递增；如果 f'(x) < 0，则递减。f'(x) = 0 表示可能的驻点。IB 试题可能会要求你用符号图来确定递增/递减区间。
For f(x) = x³ − 3x, f' = 3x² − 3 = 3(x² − 1). Critical points at x = −1 and x = 1. Test intervals: for x < −1, f'(x) > 0 (increasing); for −1 < x < 1, f'(x) < 0 (decreasing); for x > 1, f'(x) > 0 (increasing). Always state your reasoning clearly.
对于 f(x) = x³ − 3x，f'(x) = 3x² − 3 = 3(x² − 1)。临界点在 x = −1 和 x = 1。检验区间：当 x < −1 时，f'(x) > 0（递增）；当 −1 < x < 1 时，f'(x) < 0（递减）；当 x > 1 时，f'(x) > 0（递增）。务必清晰地陈述你的推理。
10. Stationary Points and Optimization | 驻点与最优化
Stationary points occur where f' nts of inflection. You can classify them using the first derivative test (sign change of f') or the second derivative test: if f''(a) > 0, it is a minimum; if f''(a) < 0, it is a maximum; if f''(a) = 0, further investigation is needed.
驻点出现在 f'(x) = 0 处。这些可能是局部极大值、局部极小值或拐点。你可以用一阶导数判别法（f' 的符号变化）或二阶导数判别法来分类：若 f''(a) > 0 则为极小值；若 f''(a) < 0 则为极大值；若 f''(a) = 0，则需进一步检验。
Optimization problems involve finding maximum or minimum values of a quantity. Translate the real-life scenario into a function of one variable, differentiate, set derivative to zero, and verify the nature. For example, maximizing the area of a rectangle with a fixed perimeter leads to a square. IB exam questions often present this as a structured problem.
最优化问题涉及求某个量的最大值或最小值。将现实情境转化为一个变量的函数，求导，令导数为零，并验证其性质。例如，在周长固定的情况下最大化矩形面积会得到正方形。IB 考试题通常以结构化问题的形式呈现。
11. Real-life Applications | 实际应用
Slope interpretations appear in many fields. In physics, the slope of a position–time graph is velocity; the slope of a velocity–time graph is acceleration. In chemistry, reaction rates are slopes of concentration–time curves. In biology, population growth rates can be modelled using derivatives.
斜率的解释出现在许多领域。在物理学中，位置–时间图的斜率是速度；速度–时间图的斜率是加速度。在化学中，反应速率是浓度–时间曲线的斜率。在生物学中，种群增长率可以用导数来建模。
. For a quadratic model h(t) = −4.9t² + 20t + 1 (height of a projectile), the derivative h' 20 gives the vertical velocity at any time t. Always include units in your interpretation.
IB 内部评估经常涉及对数据拟合模型并解释梯度。对于二次模型 h(t) = −4.9t² + 20t + 1（抛射体的高度），导数 h'(t) = −9.8t + 20 给出任意时刻 t 的垂直速度。在解释时务必包含单位。
12. Common Mistakes and Tips | 常见错误与技巧
Many students confuse f'(x) notation or forget to simplify before evaluating. When computing slope from two points, always check if the denominator is zero to avoid division-by-zero errors. In tangent/normal problems, remember that the normal slope is negative reciprocal only if the tangent slope is not zero.
许多学生会混淆 f'(x) 符号，或在求值前忘记化简。在通过两点计算斜率时，务必检查分母是否为零，以免出现除零错误。在切线/法线问题中，记住只有在切线斜率不为零时，法线斜率才是负倒数。
When a question asks for "the slope of the graph", determine whether it refers to a straight line or a curve at a specific point. If given a graph, estimate slope by drawing a tangent; if given an equation, use differentiation. Practise IB past paper questions to become familiar with the command terms like "find", "hence", and "determine".
当问题问到“图像的斜率”时，要判断它指的是直线还是曲线某一点的斜率。如果给出的是图像，通过画切线来估算斜率；如果给出的是方程，则使用微分。练习 IB 历年真题，熟悉诸如“find”、“hence”和“determine”等指令词。
Avoid sign errors when differentiating negative powers or using the chain rule. Write out u and du/dx clearly. Finally, always reread the question to ensure you have answered in the required form, such as exact values or to three significant figures.
在对负指数求导或使用链式法则时避免符号错误。清楚写出 u 和 du/dx。最后，务必重读题目，确保你以所要求的形式作答，例如精确值或保留三位有效数字。
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