Further Calculus | 进阶微积分

📚 Further Calculus | 进阶微积分

Further calculus extends foundational differentiation and integration into a broader toolkit of techniques, essential for solving complex problems in mathematics, physics, and engineering. In the IB Mathematics curriculum, this topic covers implicit differentiation, related rates, high-order derivatives, series expansions, advanced integration methods, improper integrals, and differential equations. Mastering these ideas empowers you to model dynamic systems and tackle challenging calculus questions with confidence.

进阶微积分将基础求导与积分扩展为更丰富的技术工具，是解决数学、物理和工程复杂问题所必需的。在 IB 数学课程中，该主题包括隐函数求导、相关变化率、高阶导数、级数展开、高级积分法、反常积分和微分方程等内容。掌握这些思想将使你能够自信地建立动态系统模型并解决高难度微积分问题。

1. Implicit Differentiation | 隐函数微分法

but defined by an equation relating x and y, we can differentiate implicitly. This means treating y as a differentiable function of x and applying the chain rule to each term involving y.

当函数不是以 y = f(x) 形式给出，而是由一个涉及 x 和 y 的方程定义时，我们可以使用隐函数求导。这意味着将 y 视为 x 的可导函数，并对每一项含 y 的部分使用链式法则。

Procedure: Differentiate both sides of the equation with respect to x, remembering that the derivative of y is dy/dx. Collect all dy/dx terms and solve for dy/dx.

步骤：对方程两边关于 x 求导，记住 y 的导数是 dy/dx。将所有含 dy/dx 的项合并，然后解出 dy/dx。

Example: For x² + y² = 25, differentiating gives 2x + 2y(dy/dx) = 0, hence dy/dx = -x/y.

例如：对于 x² + y² = 25，求导得 2x + 2y(dy/dx) = 0，因此 dy/dx = -x/y.

dy/dx = -x/y

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2. Related Rates | 相关变化率

Related rates problems involve two or more variables that change with respect to time and are linked by an equation. By differentiating the equation with respect to t, we relate the rates of change.

相关变化率问题涉及两个或多个随时间变化的变量，它们通过一个方程联系起来。通过对该方程关于 t 求导，我们就建立了变化率之间的关系。

Identify all given rates and the required rate. Write an equation connecting the variables, differentiate with respect to t, substitute known values, and solve for the unknown rate.

首先确定所有已知变化率和待求变化率。写出变量间的联系方程，对 t 求导，代入已知值，然后解出未知变化率。

hen r = 5 cm. We use V = (4/3)πr³, so dV/dt = 4πr²(dr/dt). Substitute and solve for dr/dt.

例如：一个球形气球的体积以 100 cm³/s 的速率增大。求当半径 r = 5 cm 时半径的增大速率。利用 V = (4/3)πr³，得 dV/dt = 4πr²(dr/dt)。代入并求解 dr/dt。

dr/dt = (1/(4πr²)) dV/dt → dr/dt = 1/(π) cm/s (approx 0.318)

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3. Higher-Order Derivatives | 高阶导数

Higher-order derivatives are obtained by differentiating derivatives successively. The second derivative f''(x) or d²y/dx² measures the rate of change of the first derivative, and physically represents acceleration.

高阶导数通过逐次求导得到。二阶导数 f''(x) 或 d²y/dx² 度量一阶导数的变化率，在物理上表示加速度。

Notation: The n-th derivative is written as f⁽ⁿ⁾(x) or dⁿy/dxⁿ. These derivatives are crucial for understanding curvature, concavity, and series expansions.

记号：n 阶导数写作 f⁽ⁿ⁾(x) 或 dⁿy/dxⁿ。这些导数对于理解曲率、凹凸性和级数展开至关重要。

For example, if y = x⁴ - 3x² + 2x, then y' = 4x³ - 6x + 2, y'' = 12x² - 6, and y''' = 24x.

例如，设 y = x⁴ - 3x² + 2x，那么 y' = 4x³ - 6x + 2，y'' = 12x² - 6，y''' = 24x。

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4. Maclaurin Series | 麦克劳林级数

It expresses a function as an infinite sum of terms calculated from its derivatives at 0.

麦克劳林级数是函数在 a = 0 处的泰勒级数展开。它将函数表示为基于其在零点各阶导数计算出的无穷多项之和。

The general formula is:

其一般公式为：

f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... + f⁽ⁿ⁾(0)xⁿ/n! + ...

Common expansions: eˣ = 1 + x + x²/2! + x³/3! + ... ; sin x = x - x³/3! + x⁵/5! - ... ; cos x = 1 - x²/2! + x⁴/4! - ...

常见展开式：eˣ = 1 + x + x²/2! + x³/3! + ... ; sin x = x - x³/3! + x⁵/5! - ... ; cos x = 1 - x²/2! + x⁴/4! - ...

o approximate functions near x = 0 with polynomials, making numerical calculations easier.

这些级数使我们能够用多项式在 x = 0 附近近似函数，从而简化数值计算。

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5. Taylor Polynomials and Taylor Series | 泰勒多项式与泰勒级数

For expansions around an arbitrary point a, we use the Taylor series. The n-th degree Taylor polynomial gives a local approximation of a function.

对于围绕任意点 a 的展开，我们使用泰勒级数。n 次泰勒多项式给出函数的局部近似。

f(x) ≈ f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... + f⁽ⁿ⁾(a)(x-a)ⁿ/n!

remainder term Rₙ₊₁ indicates the error between the polynomial and the function. Lagrange's form: Rₙ₊₁ = f⁽ⁿ⁺¹⁾(c)(x-a)ⁿ⁺¹/(n+1)! for some c between a and x.

余项 Rₙ₊₁ 表示多项式与函数之间的误差。拉格朗日型余项：Rₙ₊₁ = f⁽ⁿ⁺¹⁾(c)(x-a)ⁿ⁺¹/(n+1)!，其中 c 介于 a 与 x 之间。

roximation improves, and the full series often converges to the function on some interval.

麦克劳林级数是 a = 0 的特殊情形。随着 n 增大，近似程度提高，且完整的级数往往在某个区间内收敛于原函数。

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6. Integration by Substitution | 换元积分法

Substitution, the reverse chain rule, simplifies integrals by replacing a complicated expression with a new variable u. It works when the integrand contains a function and its derivative.

换元法是链式法则的逆运算，通过用新变量 u 替换复杂表达式来简化积分。当被积函数包含某函数及其导数时，这种方法特别有效。

Choose u = g(x), compute du = g'(x) dx, rewrite the integral in terms of u, integrate, and finally substitute back for x.

选取 u = g(x)，计算 du = g'(x) dx，将积分用 u 表示，进行积分，最后再用 x 替换回。

Example: ∫ 2x √x²+1 dx. Let u = x²+1, then du = 2x dx. The integral becomes ∫ √u du = (2/3)u³/² + C = (2/3)(x²+1)³/² + C.

例如：∫ 2x √x²+1 dx。令 u = x²+1，则 du = 2x dx。积分变为 ∫ √u du = (2/3)u³/² + C = (2/3)(x²+1)³/² + C。

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7. Integration by Parts | 分部积分法

Integration by parts is the integral counterpart of the product rule. It transforms the integral of a product into a (hopefully) simpler integral.

分部积分法是乘法求导法则的积分对应。它将一个乘积的积分转化为（希望更简单的）另一个积分。

∫ u dv = uv - ∫ v du

Decide which part of the integrand to set as u (often a polynomial, ln x, or inverse trig) and which as dv (often eˣ, sin x, cos x). LIATE rule helps choose u.

决定被积函数中哪部分设为 u（常为多项式、ln x 或反三角函数），哪部分设为 dv（常为 eˣ、sin x、cos x）。LIATE 规则有助于选择 u。

Example: ∫ x eˣ dx. Let u = x, dv = eˣ dx; then du = dx, v = eˣ. So ∫ x eˣ dx = x eˣ - ∫ eˣ dx = x eˣ - eˣ + C.

例如：∫ x eˣ dx。令 u = x, dv = eˣ dx；则 du = dx, v = eˣ。于是 ∫ x eˣ dx = x eˣ - ∫ eˣ dx = x eˣ - eˣ + C。

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8. Improper Integrals | 反常积分

Improper integrals involve infinite limits of integration or an integrand that becomes unbounded within the interval. They are evaluated using limits.

反常积分涉及无穷积分限或被积函数在区间内无界的情形。它们通过极限来计算。

For an infinite interval: ∫_a^∞ f(x) dx = lim_b→∞ ∫_a^b f(x) dx. If the limit exists finitely, the integral converges; otherwise it diverges.

对于无穷区间：∫_a^∞ f(x) dx = lim_b→∞ ∫_a^b f(x) dx。如果极限存在且有限，积分收敛；否则发散。

a vertical asymptote at x = c, we split the integral and take a one-sided limit, e.g. ∫_a^b f(x) dx = lim_t→c⁻ ∫_a^t f(x) dx + lim_t→c⁺ ∫_t^b f(x) dx.

对于在 x = c 有垂直渐近线的被积函数，我们将积分拆分并取单侧极限，如 ∫_a^b f(x) dx = lim_t→c⁻ ∫_a^t f(x) dx + lim_t→c⁺ ∫_t^b f(x) dx。

Example: ∫_1^∞ 1/x² dx = lim_b→∞ [-1/x]₁^b = 0 - (-1) = 1. Converges.

例子：∫_1^∞ 1/x² dx = lim_b→∞ [-1/x]₁^b = 0 - (-1) = 1。收敛。

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9. Separable Differential Equations | 可分离变量的微分方程

as dy/dx = g(x)h(y). Then we separate variables by dividing by h(y) and multiplying by dx.

一阶微分方程若能写成 dy/dx = g(x)h(y) 的形式，就是可分离变量的。然后通过除以 h(y) 并乘以 dx 来分离变量。

∫ 1/h(y) dy = ∫ g(x) dx

Integrate both sides and include a constant of integration. If possible, solve for y explicitly.

两边积分，并包含积分常数。如有可能，显式解出 y。

Example: dy/dx = xy. Separate: ∫ 1/y dy = ∫ x dx → ln|y| = x²/2 + C → y = A e^x²/2, where A = ±e^C.

例如：dy/dx = xy。分离变量得 ∫ 1/y dy = ∫ x dx → ln|y| = x²/2 + C → y = A e^x²/2，其中 A = ±e^C。

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10. First-Order Linear Differential Equations and Integrating Factors | 一阶线性微分方程与积分因子

dy/dx + P(x)y = Q(x). To solve, we multiply by an integrating factor μ(x) = e^(∫ P(x) dx).

一阶线性常微分方程的形式为 dy/dx + P(x)y = Q(x)。求解时，我们乘以积分因子 μ(x) = e^(∫ P(x) dx)。

The product rule in reverse turns the left side into (μ y)'. Then integrate both sides: μ y = ∫ μ Q(x) dx + C.

逆用乘法法则将左边变为 (μ y)'。然后两边积分：μ y = ∫ μ Q(x) dx + C。

General solution: y = (1/μ(x)) [ ∫ μ(x) Q(x) dx + C ].

通解为：y = (1/μ(x)) [ ∫ μ(x) Q(x) dx + C ].

Example: dy/dx + 2xy = x. Here P(x)=2x, so μ = e^{∫ 2x dx} = e^x². Multiply: e^x² dy/dx + 2x e^x² y = x e^x² → d/dx[y e^x²] = x e^x². Integrate: y e^x² = ½ e^x² + C → y = ½ + C e^-x².

例如：dy/dx + 2xy = x。这里 P(x)=2x，故 μ = e^{∫ 2x dx} = e^x²。两边乘 μ：e^x² dy/dx + 2x e^x² y = x e^x² → d/dx[y e^x²] = x e^x²。积分得 y e^x² = ½ e^x² + C → y = ½ + C e^-x²。

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11. Euler's Method for Numerical Solutions | 欧拉方法求数值解

When an exact solution to a differential equation is difficult to find, Euler's method provides a step-by-step numerical approximation using the derivative.

当微分方程的解析解难以求得时，欧拉方法利用导数提供一种逐步数值近似方法。

Given y' = f(x, y) and a starting point (x₀, y₀), choose a step size h. Then compute successively:

给定 y' = f(x, y) 和起始点 (x₀, y₀)，选择步长 h。然后依次计算：

y_n+1 = y_n + h f(x_n, y_n),   x_n+1 = x_n + h

This method builds a polygonal path approximating the true solution curve. Accuracy improves with smaller h.

此方法构建一条折线来近似真实解曲线。步长 h 越小，精度越高。

Example: y' y(0.1). f(0,1)=1, so y₁ = 1 + 0.1 * 1 = 1.1. Then y(0.2) ≈ 1.1 + 0.1*(0.1+1.1)=1.22, etc.

例如：y' = x + y, y(0) = 1；取 h = 0.1 近似 y(0.1)。f(0,1)=1，得 y₁ = 1 + 0.1×1 = 1.1。接着 y(0.2) ≈ 1.1 + 0.1×(0.1+1.1)=1.22，以此类推。

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12. Applications of Differential Equations | 微分方程的应用

Many real-world processes are modelled by differential equations: population growth, radioactive decay, Newton's law of cooling, and mixing problems.

许多现实过程都可用微分方程建模：人口增长、放射性衰变、牛顿冷却定律以及混合问题等。

Exponential growth/decay: dy/dt = ky has solution y = y₀ e^kt. If k > 0, growth; if k < 0, decay. The doubling time or half-life can be found from the model.

指数增长/衰减：dy/dt = ky 的解为 y = y₀ e^kt。若 k > 0 为增长，k < 0 为衰减。倍增时间或半衰期可由模型求出。

Newton's law of cooling: dT/dt = -k(T - T_env). The temperature difference between the object and its environment decays exponentially.

牛顿冷却定律：dT/dt = -k(T - T_env)。物体与环境之间的温差按指数衰减。

Mixing problems involve a tank with solution flowing in and out; the rate of change of solute is rate in minus rate out, leading to a linear first-order ODE.

混合问题涉及溶液流入和流出的容器；溶质的变化率等于流入速率减流出速率，从而得到一个一阶线性常微分方程。

Solving these models interprets slopes as rates, helping predict future behaviour or determine unknown parameters.

求解这些模型将斜率解释为变化率，有助于预测未来趋势或

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