📚 11A Approximation | 11A 近似
Approximation is a fundamental concept in mathematics that allows us to find near-exact values when exact solutions are difficult or impossible to obtain. In IB Mathematics, approximation techniques include linear approximation using derivatives, the Newton‑Raphson method for solving equations, the trapezoidal rule for definite integrals, and small‑angle approximations in trigonometry. These methods are essential for estimation, numerical analysis, and mathematical modelling.
近似是数学中的一个基本概念，它让我们能够在精确解难以或无法获得时找到接近真实值的数值。在 IB 数学中，近似技术包括利用导数的线性近似、求解方程的牛顿‑拉夫逊法、定积分计算的梯形法则，以及三角学中的小角近似。这些方法对于估计、数值分析和数学建模至关重要。
1. What is Approximation? | 什么是近似？
An approximation replaces an exact value with a nearby value that is easier to use. In many real‑world problems, exact answers involve irrational numbers or complicated expressions, so a carefully chosen approximate value gives sufficient accuracy while saving time. Approximation methods are also at the heart of numerical algorithms used in calculators and computers.
近似就是用更容易使用的邻近值来代替精确值。在许多实际问题中，精确答案涉及无理数或复杂表达式，因此一个仔细选择的近似值既能提供足够的精度，又能节省时间。近似方法也是计算器和计算机中数值算法的核心。
2. Linear Approximation: The Tangent Line | 线性近似：切线
If a function f is differentiable at a, the tangent line at x = a gives a good local approximation to the curve. The idea is to use the tangent line L(x) as a linear model of f near a.
如果函数 f 在 a 处可导，那么 x = a 处的切线给出了曲线的一个良好的局部近似。其思想是把切线 L(x) 作为 f 在 a 附近的线性模型。
f(x) ≈ f(a) + f'(a)(x − a)
f(x) ≈ f(a) + f'(a)(x − a)
3. Linear Approximation Formula and Example | 线性近似公式与示例
To estimate √4.1, let f(x) = √x and choose a = 4. Then f(4) = 2, f'(x) = 1/(2√x) so f'(4) = 1/4. For x (1/4)(0.1) = 2.025. The true value is about 2.024845, so the error is only about 0.000155.
要估算√4.1，设 f(x) = √x 并选择 a = 4。则 f(4) = 2，f'(x) = 1/(2√x)，故 f'(4) = 1/4。对于 x = 4.1，线性近似给出 L(4.1) = 2 + (1/4)(0.1) = 2.025。真实值约为 2.024845，误差仅约 0.000155。
Similarly, to approximate e0.05, use f(x) = ex with a = 0. Then f(0) = 1, f'(0) = 1, so e0.05 ≈ 1 + 0.05 = 1.05, while the exact value is 1.05127 – again a small error.
类似地，要近似 e0.05，用 f(x) = ex 且 a = 0。则 f(0) = 1，f'(0) = 1，所以 e0.05 ≈ 1 + 0.05 = 1.05，而精确值是 1.05127 —— 同样是一个很小的误差。
4. Small‑Angle Approximations | 小角近似
When an angle θ is measured in radians and is close to 0, the sine, cosine and tangent functions can be approximated by simple polynomials. These come from the Maclaurin series and are extremely useful in physics and engineering.
当角度 θ 以弧度为单位且接近 0 时，正弦、余弦和正切函数可以用简单的多项式来近似。这些近似来自麦克劳林级数，在物理和工程中非常有用。
sin θ ≈ θ
sin θ ≈ θ
cos θ ≈ 1 − θ2/2
cos θ ≈ 1 − θ2/2
tan θ ≈ θ
tan θ ≈ θ
5. Applications of Small‑Angle Approximations | 小角近似的应用
For θ = 0.2 rad, sin 0.2 ≈ 0.2 (exact 0.19867, error ≈ 0.0013). Meanwhile, cos 0.2 ≈ 1 − (0.22 ). These approximations are often precise enough for pendulum motion calculations or light‑ray optics.
对于 θ = 0.2 rad，sin 0.2 ≈ 0.2（精确值 0.19867，误差约 0.0013）。同时，cos 0.2 ≈ 1 − (0.22)/2 = 1 − 0.02 = 0.98（精确值 0.98007，误差约 0.00007）。这些近似对于摆的运动或光线光学的计算通常是足够精确的。
Remember that the angle must be in radians; in degrees the same simple forms do not hold because the series expansion changes. For IB problems, always convert to radians before applying small‑angle approximations.
请记住角度必须以弧度为单位；若用角度制，简单的形式不再成立，因为级数展开会改变。对于 IB 题目，在应用小角近似之前一定要先转换成弧度。
6. Newton‑Raphson Method for Finding Roots | 牛顿‑拉夫逊求根法
The Newton‑Raphson method iteratively finds better approximations to the roots of a real‑valued function f. Starting from an initial guess x0, the next approximation is:
牛顿‑拉夫逊法迭代地寻找实值函数 f 的根的更好近似。从初始猜测 x0 开始，下一个近似由下式给出：
xn+1 = xn − f(xn) / f'(xn)
xn+1 = xn − f(xn) / f'(xn)
Geometrically, xn+1 is the x‑intercept of the tangent line to f at (xn, f(xn)). The method converges quickly provided the starting point is reasonably close and f'(xn) ≠ 0.
从几何上看，xn+1 是 f 在 (xn, f(xn)) 处的切线与 x 轴的交点。只要初始点足够接近且 f'(xn) ≠ 0，该方法收敛很快。
7. Newton‑Raphson: Iteration and Convergence | 牛顿法：迭代与收敛
To find √5, set f(x) = x2 − 5. With x0 = 2, the first two iterations are:
要计算 √5，令 f(x) = x2 − 5。取 x0 = 2，前两次迭代如下：
x1 = 2 − (4 − 5)/(2×2) = 2.25
x1 = 2 − (4 − 5)/(2×2) = 2.25
x2 = 2.25 − (5.0625 − 5)/(2×2.25) ≈ 2.236111
x2 = 2.25 − (5.0625 − 5)/(2×2.25) ≈ 2.236111
The true value is 2.236068, so after two steps we already have four decimal places. Possible failures include choosing an x0 where f' is nearly zero, causing the tangent to shoot far away, or landing on a local extremum cycle. In IB problems it is often checked by looking at the graph or calculating successive differences.
真实值是 2.236068，因此两步之后我们已经得到了四位小数精度。可能失败的情况包括选择的 x0 使 f' 接近零，导致切线飞向远处，或陷入局部极值循环。在 IB 题目中，通常通过观察图像或计算连续两次差值的减小来检查收敛性。
8. Numerical Integration: Trapezoidal Rule | 数值积分：梯形法则
When a definite integral ∫ab f(x) dx cannot be evaluated analytically, the trapezoidal rule approximates the area under the curve by dividing [a, b] into n equal strips and summing the areas of trapezoids.
当定积分 ∫ab f(x) dx 无法解析求解时，梯形法则通过将 [a, b] 分成 n 个
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