Applications of Differentiation | 导数的应用

📚 Applications of Differentiation | 导数的应用

Differentiation is one of the fundamental pillars of calculus, and its applications stretch far beyond finding slopes of curves. In IB Mathematics, both Analysis & Approaches and Applications & Interpretation, the derivative is used to model real‑world phenomena, optimise quantities, analyse motion, and understand the behaviour of functions. This article explores the key applications of differentiation, from tangents and normals to related rates and optimisation, providing clear explanations and examples to support your revision.

导数是微积分的重要基石，它的应用远远不止求曲线的斜率。在 IB 数学（分析与方法、应用与解释）中，导数被用来建立现实世界现象的模型、优化各种量、分析运动以及理解函数的性态。本文探讨导数的核心应用，涵盖切线与法线、相关变化率和最优化等内容，为你的复习提供清晰的解释和示例。

1. Tangents and Normals | 切线与法线

s the derivative there. For a function y = f(x), the gradient of the tangent at x = a is f'(a). The equation of the tangent can be written as y − f(a) = f'(a)(x − a).

曲线在某一点的切线是一条恰好在该点接触曲线且梯度等于该点导数值的直线。对于函数 y = f(x)，在 x = a 处切线的梯度为 f'(a)。其方程可以表示为 y − f(a) = f'(a)(x − a)。

at the point of contact. Therefore its gradient is the negative reciprocal of the tangent's gradient: mₙₒᵣₘₐₗ = −1 / f'(a), provided f'(a) ≠ 0. The equation of the normal is y − f(a) = −(1 / f'(a))(x − a).

法线在切点处与切线垂直。因此它的梯度是切线梯度的负倒数：mₙₒᵣₘₐₗ = −1 / f'(a)（假设 f'(a) ≠ 0）。法线的方程为 y − f(a) = −(1 / f'(a))(x − a)。

2. Kinematics: Velocity and Acceleration | 运动学：速度与加速度

the derivative of velocity, or the second derivative of displacement: a = dv/dt = d²s/dt².

若物体沿直线运动的位移 s 表示为时间 t 的函数 s = s(t)，则瞬时速度 v 是它的导数：v = ds/dt。加速度 a 是速度的导数，也是位移的二阶导数：a = dv/dt = d²s/dt²。

direction. Negative acceleration can indicate deceleration if the signs of v and a differ.

利用这些关系可以分析运动：速度为正值时物体向前移动，加速度为正值时它在正方向上加速。解方程 v(t) = 0 可以求出物体改变方向的时刻。若速度和加速度符号相反，则负加速度意味着减速。

3. Increasing and Decreasing Functions | 函数的单调性

A function f is increasing on an interval if its derivative f'(x) > 0 for all x in that interval. It is decreasing if f'(x) < 0. Points where f'(x) = 0 or f'(x) is undefined may indicate boundaries for monotonic intervals.

如果对于某区间内的所有 x 都有 f'(x) > 0，则函数 f 在该区间上是递增的。如果 f'(x) < 0，则它是递减的。f'(x) = 0 或 f'(x) 无定义的点可能是单调区间的边界。

To determine where a function rises or falls, we find critical numbers and test the sign of the derivative on each interval. This information is essential for sketching graphs and understanding overall behaviour.

要确定函数的升降，需要找出临界值，并在每个区间上检验导数的符号。这些信息对于绘制草图和理解函数的整体性态至关重要。

4. Stationary Points and Their Classification | 驻点及其分类

Stationary points occur where f' axima, local minima, or points of inflection with a horizontal tangent. The first derivative test uses sign changes of f'(x) around the point: if f' t is a maximum; from negative to positive, it is a minimum; if no sign change, it may be a stationary inflection point.

驻点出现在 f'(x) = 0 处。它们分为局部极大值、局部极小值或具有水平切线的拐点。一阶导数检验法利用该点附近 f'(x) 的符号变化：若 f' 由正变负，则为极大值；由负变正，则为极小值；若符号不变，则可能是驻点型拐点。

The second derivative test provides a quicker alternative when f'' exists: if f''(c) > 0, the point is a local minimum; if f''(c) < 0, it is a local maximum. If f''(c) = 0 the test is inconclusive and the first derivative test should be used.

当二阶导数存在时，二阶导数检验法更快捷：若 f''(c) > 0，则该点是局部极小值；若 f''(c) < 0，则是局部极大值。若 f''(c) = 0，该检验法无法判断，应使用一阶导数检验法。

5. Concavity and Points of Inflection | 凹凸性与拐点

a graph is determined by the second derivative. A curve is concave up on intervals where f''(x) > 0, and concave down where f'' A point of inflection is a point where the concavity changes, and it occurs either where f''(x) = 0 or where f''(x) is undefined, provided there is a genuine change in sign.

图形的凹凸性由二阶导数决定。在 f''(x) > 0 的区间上，曲线是凹向上的；在 f''(x) < 0 的区间上是凹向下的。拐点是凹凸性发生变化的点，它出现在 f''(x) = 0 或 f''(x) 无定义且符号确实改变的地方。

It is important to verify the sign change – not every point where f'' flection points reveal where the rate of change (gradient) transitions from increasing to decreasing, or vice versa.

验证符号变化非常重要——并非所有 f''(x) = 0 的点都是拐点。拐点揭示了变化率（梯度）从递增转为递减或相反的位置。

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f''(x) sign	Concavity
f''(x) > 0	Concave up (∪)
f''(x) < 0	Concave down (∩)

6. Optimisation Problems | 最优化问题

Optimisation involves finding maximum or minimum values of a quantity within a given context, such as minimising surface area for a given volume or maximising profit. The process includes expressing the quantity to be optimised as a function of one variable, finding its derivative, and solving for stationary points.

最优化是指在给定条件下求某个量的最大值或最小值，比如在体积一定的情况下最小化表面积，或使利润最大化。这一过程包括将要优化的量表示为单变量函数，求出导数，并解出驻点。

Always verify that the stationary point corresponds to the desired extremum by using the first or second derivative test. Also check the endpoints if the domain is restricted, as absolute extrema may occur there.

务必使用一阶或二阶导数检验法验证该驻点对应的是所需极值。如果定义域有限制，还应检查端点，因为绝对极值可能出现在端点。

and express one side in terms of the other using the perimeter constraint. Differentiate, set derivative to zero, and confirm maximum. The result is a square, demonstrating that symmetry often yields optima.

例题：一个矩形的周长 P 固定。要使面积 A 最大，设 A = L × W，并利用周长约束把一个边长用另一个表示。求导，令导数为零并确认最大值。结果是正方形，这体现了对称性常常带来最优解。

7. Related Rates | 相关变化率

In related rates problems, two or more quantities change with time and are linked by an equation. We differentiate implicitly with respect to time to find the rate at which one quantity changes when the rate of change of another is known. For instance, if a ladder slides down a wall, the horizontal speed of the foot is related to the vertical speed of the top.

在相关变化率问题中，两个或多个量随时间变化并彼此关联。我们对方程关于时间进行隐微分，从而在已知一个量的变化率时求出另一个量的变化率。例如，当梯子沿墙壁下滑时，底端的水平速度与顶端的竖直速度相关联。

The key steps are: draw a diagram, assign variables, write an equation relating them, differentiate with respect to t using the chain rule, substitute known values, and solve for the unknown rate. Always consider the signs – a negative rate indicates decreasing quantity.

关键步骤是：画示意图，设定变量，写出它们之间的关系式，使用链式法则对 t 求导，代入已知值，并解出未知变化率。请注意符号——负变化率表示该量在减小。

8. Approximations and Small Changes | 近似计算与微小变化

stimate how a small change in x affects y. The linear approximation formula is Δy ≈ dy = f'(x) dx, or expressed as f(x + Δx) ≈ f(x) + f'(x)Δx. This local linearisation is particularly useful when evaluating functions like √4.1 or sin(0.1) without a calculator.

导数可用于估算 x 的微小变化对 y 的影响。线性近似公式为 Δy ≈ dy = f'(x) dx，也可写作 f(x + Δx) ≈ f(x) + f'(x)Δx。这种局部线性化在不用计算器求像 √4.1 或 sin(0.1) 的值时特别有用。

√4.1 ≈ √4 + (1/(2√4))(0.1) = 2 + 0.025 = 2.025

This method is rooted in the idea that a smooth curve very closely resembles its tangent near the point of tangency. The percentage error can also be approximated using differentials.

该方法基于平滑曲线在切点附近与切线非常相似的思想。利用微分还可以近似计算百分误差。

9. Marginal Analysis in Economics | 经济学中的边际分析

In economics, the derivative provides marginal functions: marginal cost is the derivative of total cost with respect to quantity, marginal revenue is the derivative of total revenue, and marginal profit is the derivative of profit. These gauge the approximate additional cost or revenue from producing one more unit.

在经济学中，导数给出了边际函数：边际成本是总成本关于产量的导数，边际收益是总收益的导数，边际利润是利润的导数。这些指标衡量了多生产一个单位产品所带来的额外成本或收益的近似值。

For example, if the cost function is C(q) = 200 + 15q + 0.02q², then marginal cost MC = C' q = 100, MC = 19, meaning the 101st unit costs approximately 19 currency units to produce.

例如，若成本函数为 C(q) = 200 + 15q + 0.02q²，则边际成本 MC = C'(q) = 15 + 0.04q。在 q = 100 时，MC = 19，意味着生产第 101 件产品的成本约为 19 个货币单位。

Firms use these ideas to determine profit‑maximising output, where marginal revenue equals marginal cost, provided certain conditions hold.

公司利用这些概念来确定利润最大化产量——当边际收益等于边际成本且满足特定条件时即可实现。

10. L'Hôpital's Rule | 洛必达法则

L'Hôpital' )/g(x)) yields an indeterminate form and the derivatives exist, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists or is infinite.

洛必达法则是运用导数求解不定式极限（如 0/0 或 ∞/∞）的有力工具。若 lim (f(x)/g(x)) 是不定式且导数存在，则 lim f(x)/g(x) = lim f'(x)/g'(x)，前提是后面的极限存在或为无穷大。

Example: lim_x→0 (sin x)/x = lim_x→0 (cos x)/1 = 1

The rule can be applied repeatedly if the result remains indeterminate. It is essential to check that the conditions of the rule are met before differentiating the numerator and denominator separately.

如果结果仍是不定式，可重复应用此法则。必须注意在单独对分子和分母求导之前，验证法则所要求的条件是否成立。

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