📚 Applications of Differential Calculus | 微分学的应用
Differential calculus is a cornerstone of IB Mathematics, providing powerful tools to analyse change, motion, and optimisation. In this article we explore key applications: from finding tangents and normals, to locating extreme values, analysing concavity, solving real-world optimisation and related rates problems, describing motion, making linear approximations, and using advanced theorems like the Mean Value Theorem and L'Hopital's Rule. Each topic is paired with clear reasoning and examples, reinforcing both conceptual understanding and exam-ready technique.
微分学是 IB 数学的基石，为分析变化、运动和优化提供了强有力的工具。本文我们将探讨关键应用：从求切线与法线，到定位极值，分析凹凸性，解决现实中的最优化与相关变化率问题，描述运动，做线性近似，以及运用中值定理和洛必达法则等更深入的定理。每个主题都配有清晰的推理与例子，巩固概念理解并提升应试技巧。
1. Tangent and Normal Lines | 切线与法线
The derivative f' f(x) at the point (x₀, f(x₀)). The equation of the tangent can be written as y - f(x₀) = f'(x₀)(x - x₀). A normal line is perpendicular to the tangent, so its slope is -1/f'(x₀) provided f'(x₀) ≠ 0.
导数 f'(x₀) 给出了曲线 y = f(x) 在点 (x₀, f(x₀)) 处切线的斜率。切线方程可写作 y - f(x₀) = f'(x₀)(x - x₀)。法线与切线垂直，因此其斜率为 -1/f'(x₀)（假定 f'(x₀) ≠ 0）。
Example: For f(x) = x² at x₀ = 1, we have f(1)=1 and f' mal has slope -½, giving the equation y - 1 = -½(x - 1), which simplifies to y = -½x + 1½.
示例：对于 f(x) = x² 在 x₀ = 1 处，有 f(1)=1 且 f'(1)=2。切线为 y - 1 = 2(x - 1)，即 y = 2x - 1。法线斜率为 -½，方程为 y - 1 = -½(x - 1)，化简为 y = -½x + 1½。
2. Critical Points and Extrema | 临界点与极值
A critical point (or stationary point) occurs where f' t exist. Local maxima and minima can only occur at critical points if the function is differentiable on an open interval. Not every critical point yields an extremum; some are points of inflection with horizontal tangents.
临界点（驻点）出现在 f'(x) = 0 或导数不存在的位置。若函数在开区间内可导，局部极大值和极小值只可能出现在临界点。并非所有临界点都给出极值，有些是切线水平的拐点。
On a closed interval [a, b], the absolute extrema are found by evaluating f at critical points inside (a, b) and at the endpoints a and b, then comparing values.
在闭区间 [a, b] 上，绝对极值需计算 f 在区间内临界点以及端点 a 和 b 处的值，然后比较得出。
3. First Derivative and Monotonicity | 一阶导数与单调性
e sign of the first derivative tells us where a function is increasing or decreasing. If f'(x) > 0 on an interval, f is increasing there; if f' f is decreasing. This leads to the First Derivative Test: at a critical point x = c, if f' changes from positive to negative, f has a local maximum; if f' ; if f' changes from negative to positive, a local minimum. No sign change means neither a max nor a min.
一阶导数的符号表明函数在何处递增或递减。若在区间上 f'(x) > 0，则 f 递增；若 f'(x) < 0，则 f 递减。由此得到一阶导数检验法：在临界点 x = c 处，若 f' 由正变负，f 取得局部极大值；若 f' 由负变正，则取得局部极小值。符号不变则既非极大也非极小。
4. Second Derivative, Concavity, and Points of Inflection | 二阶导数、凹凸性与拐点
The second derivative f''(x) describes the curvature of the graph. f''(x) > 0 implies the graph is concave up (cup-shaped), while f''(x) < 0 implies concave down (cap-shaped). The Second Derivative Test states that if f'(c) = 0 and f''(c) > 0, then f has a local minimum at c; if f''(c) < 0, a local maximum.
二阶导数 f''(x) 描述图像的弯曲情况。f''(x) > 0 表示图像凹向上（杯状），f''(x) < 0 表示凹向下（帽状）。二阶导数检验法表明：若 f'(c) = 0 且 f''(c) > 0，则 f 在 c 处取得局部极小值；若 f''(c) < 0，则为局部极大值。
A point of inflection occurs where the concavity changes sign. To locate it, solve f''(x) = 0 or find where the second derivative is undefined, then check that f'' actually changes sign across that point.
拐点出现在凹凸性变号的地方。寻找时可解 f''(x) = 0 或找出二阶导数无定义的点，然后验证 f'' 恰好在该点两侧变号。
5. Optimization Problems | 最优化问题
Optimisation applies derivatives to find maximum or minimum values of a quantity under given constraints. A typical strategy: express the quantity to be optimised as a function of one variable using the constraint, differentiate to find critical points, and test using the first or second derivative. Finally, check for practical domain restrictions.
最优化是利用导数在给定约束下求某个量的最大或最小值。典型策略：利用约束将待优化量表示为一元函数，求导找出临界点，并用一阶或二阶导数检验。最后检查实际定义域的限制。
m that maximises area. Let width be x, length be 10 - x. Area A = x(10 - x) = 10x - x². A' = 10 - 2x, setting to zero gives x = 5. Second derivative A'' = -2 < 0, so a maximum. The optimal shape is a 5 m by 5 m square.
示例：求周长为 20 m 的长方形使其面积最大。设宽为 x，长为 10 - x。面积 A = x(10 - x) = 10x - x²。A' = 10 - 2x，令其为零得 x = 5。二阶导数 A'' = -2 < 0，所以为极大值。最优形状是 5 m × 5 m 的正方形。
6. Related Rates | 相关变化率
Related rates problems involve two or more quantities that change with respect to time, linked by an equation. Differentiating the equation implicitly with respect to time t using the chain rule yields a relationship between their rates of change.
相关变化率问题涉及两个或更多随时间变化的量，它们由一个方程关联。对方程关于时间 t 使用链式法则进行隐函数求导，即得到各量变化率之间的关系。
at a constant rate of 100 cm³/s. How fast is the radius increasing when the radius is 5 cm? /dt). With dV/dt = 100 and r = 5, 100 = 4π(25)(dr/dt), so dr/dt = 1/(π) cm/s ≈ 0.318 cm/s.
示例：一个球形气球的体积以 100 cm³/s 的恒定速率增加。当半径为 5 cm 时，半径的增加速率是多少？体积 V = (4/3)πr³， dV/dt = 4πr² (dr/dt)。代入 dV/dt = 100 及 r = 5，得 100 = 4π(25)(dr/dt)，因此 dr/dt = 1/(π) cm/s ≈ 0.318 cm/s。
7. Kinematics | 运动学
e, let s(t) be the displacement from a fixed point. The instantaneous velocity is v(t) = s'(t) and acceleration is a(t) = v'(t) = s'' a indicates whether the particle is speeding up or slowing down relative to its velocity.
对于沿直线运动的质点，设 s(t) 为相对某固定点的位移。瞬时速度为 v(t) = s'(t)，加速度为 a(t) = v'(t) = s''(t)。v 的符号指示运动方向；a 的符号结合速度可判断质点是在加速还是减速。
n v(t) = 0 and the sign of v changes. The total distance travelled is found by integrating |v(t)| over the time interval.
当 v(t) = 0 且 v 改变符号时，质点改变运动方向。运动总路程等于 |v(t)| 在时间段上的积分。
8. Linear Approximation and Differentials | 线性近似与微分
a, a differentiable function f(x) can be approximated by its tangent line: f(x) ≈ f(a) + f')(x - a). This is the linear approximation, particularly useful for estimating values like √4.1 or sin(0.1). The differential dy = f'(x) dx represents the change in the tangent line and approximates the actual change Δy.
在点 x = a 附近，可微函数 f(x) 可以用它的切线来近似：f(x) ≈ f(a) + f'(a)(x - a)。这就是线性近似，特别适用于估算诸如 √4.1 或 sin(0.1) 的值。微分 dy = f'(x) dx 表示切线沿高度方向的变化，近似代替实际增量 Δy。
For example, using f(x) = √x at a = 4, f'(x) = 1/(2√x), so f'(4)=1/4. Then √4.1 ≈ 2 + (1/4)(0.1) = 2.025, which is very close to the true value.
例如，取 f(x) = √x 在 a = 4 处，f'(x) = 1/(2√x)，f'(4)=1/4。则 √4.1 ≈ 2 + (1/4)(0.1) = 2.025，这与真实值非常接近。
9. The Mean Value Theorem and Rolle's Theorem | 中值定理与罗尔定理
Rolle' ous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c ∈ (a, b) such that f'(c) = 0. It is a special case of the Mean Value Theorem.
罗尔定理指出：若 f 在 [a, b] 上连续，在 (a, b) 内可导，且 f(a) = f(b)，则至少存在一个 c ∈ (a, b) 使得 f'(c) = 0。它是中值定理的特例。
s the same continuity and differentiability conditions and guarantees the existence of a point c in (a, b) where f' ally, there is a point at which the tangent is parallel to the secant line joining (a, f(a)) and (b, f(b)).
中值定理（MVT）要求相同的连续和可导条件，并保证了在 (a, b) 内存在一点 c 使得 f'(c) = (f(b) - f(a))/(b - a)。几何上，存在某点其切线与连接 (a, f(a)) 和 (b, f(b)) 的割线平行。
10. L'Hopital's Rule | 洛必达法则
L'Hopital' inate forms such as 0/0 or ∞/∞. If limx→a f(x) = 0 and limx→a g(x) = 0, and the limit of f'(x)/g'(x) exists, then limx→a f(x)/g(x) = limx→a f'(x)/g'(x), provided the derivatives exist and g'(x) ≠ 0 near a (a can also be ±∞).
洛必达法则帮助求解产生 0/0 或 ∞/∞ 等不定型的极限。若 limx→a f(x) = 0 且 limx→a g(x) = 0，并且 f'(x)/g'(x) 的极限存在，则 limx→a f(x)/g(x) = limx→a f'(x)/g'(x)，前提是导数在 a 附近存在且 g'(x) ≠ 0（a 也可为 ±∞）。
A classic example is limx→0 sin x / x. Direct substitution gives 0/0. Applying L'Hopital's Rule: differentiate numerator and denominator to get limx→0 cos x / 1 = 1.
经典例子 limx→0 sin x / x。直接代入为 0/0。应用洛必达法则：分子分母分别求导得 limx→0 cos x / 1 = 1。
11. Curve Sketching | 曲线绘制
A complete graph analysis uses f, f', and f'' to determine domain, intercepts, symmetry, asymptotes, intervals of increase/decrease, local extrema, concavity, and points of inflection. Combine these features to produce an accurate sketch.
完整的图像分析利用 f, f' 和 f'' 来确定定义域、截距、对称性、渐近线、增减区间、局部极值、凹凸性以及拐点。综合这些特征即可画出准确的草图。
For rational functions, vertical asymptotes occur where the denominator is zero (and numerator non-zero), while horizontal or oblique asymptotes are found by limits at infinity. With the aid of sign charts for f' and f'', the shape of the graph becomes clear.
对于有理函数，垂直渐近线出现在分母为零且分子非零的地方，而水平或斜渐近线通过无穷远处的极限求得。借助 f' 和 f'' 的符号表，图像的整体形状就变得清晰。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图