Taylor Polynomials in IB Mathematics | IB数学：泰勒多项式

📚 Taylor Polynomials in IB Mathematics | IB数学：泰勒多项式

In IB Mathematics Analysis and Approaches Higher Level (AA HL), Taylor polynomials provide a powerful way to approximate complicated functions using polynomials. Since polynomials are easy to evaluate, differentiate and integrate, they turn intractable expressions into manageable ones.

在IB数学分析与方法高阶（AA HL）中，泰勒多项式提供了一种用多项式近似复杂函数的强大方法。由于多项式易于求值、求导和积分，它们能把难以处理的形式转化为可操作的形式。

1. What Is a Taylor Polynomial? | 什么是泰勒多项式？

A Taylor polynomial is a finite-degree polynomial that matches a function at a chosen point, along with as many derivatives as possible. For a function f that is sufficiently differentiable, its Taylor polynomial of degree n centred at x = a is:

泰勒多项式是一个有限次多项式，它在选定点处尽可能多地匹配函数及其各阶导数。对于一个充分可微的函数 f，以 x = a 为中心的 n 次泰勒多项式为：

Pₙ(x) = f(a) + f'(a)(x − a) + f”(a)(x − a)²/2! + f”'(a)(x − a)³/3! + … + f⁽ⁿ⁾(a)(x − a)ⁿ/n!

Here f⁽ⁿ⁾(a) denotes the nth derivative evaluated at a, and n! is n factorial. The polynomial is designed so that Pₙ(a) = f(a), Pₙ'(a) = f'(a), and so on up to the nth derivative.

这里 f⁽ⁿ⁾(a) 表示在 a 处取值的 n 阶导数，n! 是 n 的阶乘。该多项式的设计使得 Pₙ(a) = f(a)，Pₙ'(a) = f'(a)，依此类推直到 n 阶导数。

2. Maclaurin Polynomials | 麦克劳林多项式

A Maclaurin polynomial is simply a Taylor polynomial centred at zero. Setting a = 0 gives the special form:

麦克劳林多项式就是中心在零处的泰勒多项式。令 a = 0，得到特殊形式：

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

This version is especially common in IB exams because many standard functions have simple derivatives at 0, such as e⁰ = 1, sin 0 = 0, cos 0 = 1.

这个版本在IB考试中特别常见，因为许多标准函数在 0 处的导数很简单，例如 e⁰ = 1，sin 0 = 0，cos 0 = 1。

3. Constructing Taylor Polynomials | 构造泰勒多项式

To construct a Taylor polynomial, follow these steps:

构造泰勒多项式，请遵循以下步骤：

Choose the centre a and the degree n.

选择中心 a 和次数 n。
Compute the values f(a), f'(a), f”(a), …, up to f⁽ⁿ⁾(a).

计算 f(a), f'(a), f”(a), …直到 f⁽ⁿ⁾(a) 的值。
Plug these values into the Taylor formula and simplify the coefficients.

将这些值代入泰勒公式并化简系数。

For example, if f(x) = sin x and a = 0, then f'(x) = cos x, f”(x) = −sin x, f”'(x) = −cos x, and so on. Evaluating at 0 gives the pattern 0, 1, 0, −1, 0, 1, …

例如，若 f(x) = sin x，a = 0，则 f'(x) = cos x，f”(x) = −sin x，f”'(x) = −cos x，如此循环。在 0 处求值得到模式 0, 1, 0, −1, 0, 1, …

4. Common Taylor Expansions | 常见泰勒展开

The following Maclaurin series are frequently required in IB AA HL. You should know the first few terms, not necessarily the infinite series.

以下麦克劳林展开在IB AA HL中经常要求掌握。你应该知道前几项，而不一定是无限级数。

Function	Maclaurin polynomial (first terms)	Valid for
eˣ	1 + x + x²/2! + x³/3! + …	all real x
sin x	x − x³/3! + x⁵/5! − …	all real x
cos x	1 − x²/2! + x⁴/4! − …	all real x
ln(1+x)	x − x²/2 + x³/3 − …	−1 < x ≤ 1
(1+x)ᵖ	1 + px + p(p−1)x²/2! + …	\|x\| < 1

Notice that the alternating signs in sin x and cos x come from the periodic derivatives. For ln(1+x), the series converges only for −1 < x ≤ 1, not for all x.

注意 sin x 和 cos x 中的交替符号来自周期性导数。对于 ln(1+x)，级数仅在 −1 < x ≤ 1 时收敛，并非对所有 x 都成立。

5. Applications: Approximation | 应用：近似计算

Taylor polynomials are used to approximate function values near the centre. For example, to estimate e^0.1, use the first four terms of eˣ:

泰勒多项式用于近似中心附近的函数值。例如，要估算 e^0.1，可使用 eˣ 的前四项：

e^0.1 ≈ 1 + 0.1 + (0.1)²/2 + (0.1)³/6 = 1.105167…

Comparing with the true value 1.105170…, the approximation is remarkably accurate. The higher the degree n, the better the approximation near the centre.

与真实值 1.105170… 相比，近似非常精确。次数 n 越高，中心附近近似越好。

Another application is computing limits or integrals that are otherwise difficult. For instance, ∫₀¹ e^(−x²) dx has no elementary antiderivative, but replacing e^(−x²) by its Taylor polynomial gives an easy polynomial integral.

另一个应用是计算原本困难的极限或积分。例如，∫₀¹ e^(−x²) dx 没有初等原函数，但将 e^(−x²) 替换为其泰勒多项式后，就得到一个容易的积分。

6. Lagrange Error Bound | 拉格朗日误差界

When using a Taylor polynomial to approximate a function, it is essential to know how large the error can be. The Lagrange error bound states that if |f⁽ⁿ⁺¹⁾(t)| ≤ M for all t between a and x, then the remainder Rₙ(x) = f(x) − Pₙ(x) satisfies:

使用泰勒多项式近似函数时，必须知道误差可能有多大。拉格朗日误差界指出：若对所有介于 a 和 x 之间的 t，有 |f⁽ⁿ⁺¹⁾(t)| ≤ M，则余项 Rₙ(x) = f(x) − Pₙ(x) 满足：

|Rₙ(x)| ≤ M|x − a|ⁿ⁺¹ / (n+1)!

Here M is the maximum absolute value of the (n+1)th derivative on the interval. In IB problems, you are often given this bound or asked to find the smallest n such that the error is below a certain value.

这里 M 是区间上 (n+1) 阶导数的最大绝对值。在IB题目中，通常会给出这个界，或者要求找出使误差低于某个值的最小 n。

7. Worked Example: Finding the Degree | 例题：确定次数

Question: Find the smallest degree n for the Maclaurin polynomial of sin x such that the approximation at x = 0.5 has error less than 10⁻³.

问题：求 sin x 的麦克劳林多项式的最小次数 n，使得在 x = 0.5 处的近似误差小于 10⁻³。

Solution: For sin x, all derivatives are bounded by 1, so M = 1. The Lagrange error bound gives:

解答：对于 sin x，所有导数的绝对值都不超过 1，所以 M = 1。拉格朗日误差界给出：

|Rₙ(0.5)| ≤ (0.5)ⁿ⁺¹ / (n+1)!

Test n = 3: (0.5)⁴/24 ≈ 0.00260 > 10⁻³. Test n = 4: (0.5)⁵/120 ≈ 0.000260 < 10⁻³. Since sin x has no x⁴ term, the fourth-degree polynomial is the same as the fifth-degree? Actually, the Lagrange bound for n=4 uses the 5th derivative, so n=4 works. Thus the smallest n is 4.

测试 n = 3：(0.5)⁴/24 ≈ 0.00260 > 10⁻³。测试 n = 4：(0.5)⁵/120 ≈ 0.000260 < 10⁻³。由于 sin x 没有 x⁴ 项，四次多项式与五次多项式相同？实际上，n=4 的拉格朗日界使用5阶导数，所以 n=4 可行。因此最小 n 为4。

8. Common IB Exam Styles | IB常见题型

IB AA HL paper questions on Taylor polynomials typically fall into these categories:

IB AA HL 试卷中泰勒多项式的题目通常分为以下几类：

Finding the Taylor or Maclaurin polynomial of a given degree for a given function.

求给定函数在给定次数下的泰勒或麦克劳林多项式。
Using a Taylor polynomial to estimate a function value and then using the Lagrange error bound to justify the precision.

使用泰勒多项式估算函数值，然后使用拉格朗日误差界证明精度。
Determining the smallest degree n needed to achieve a specified accuracy.

确定达到指定精度所需的最小次数 n。
Recognising Taylor series from a given pattern and using them to sum infinite series or evaluate limits.

从给定模式识别泰勒级数，并用以求和或求极限。

Marks are often awarded for correctly computing derivatives, evaluating them at the centre, and writing the final polynomial in simplified form.

分数通常授予正确计算导数、在中心处求值以及以简化形式写出最终多项式。

9. Common Mistakes and Pitfalls | 常见错误与陷阱

Here are frequent errors students make with Taylor polynomials:

以下是学生在泰勒多项式上常犯的错误：

Forgetting factorials in the denominator. The term with f⁽ⁿ⁾(a) must be divided by n!.

忘记分母中的阶乘。含有 f⁽ⁿ⁾(a) 的项必须除以 n!。
Using the wrong centre. If the problem says “about x = 1”, you must use (x−1) powers, not x.

使用错误的中心。如果题目说“在 x = 1 处展开”，你必须使用 (x−1) 的幂，而不是 x。
Confusing Taylor series with Maclaurin series. Maclaurin is only when a = 0.

混淆泰勒级数与麦克劳林级数。麦克劳林仅在 a = 0 时成立。
Misapplying the Lagrange error bound. M must be the maximum of the (n+1)th derivative, not the nth derivative.

错误应用拉格朗日误差界。M 必须是 (n+1) 阶导数的最大值，而不是 n 阶导数。
Not simplifying coefficients after substitution.

代入后未化简系数。

10. Summary and Revision Tips | 总结与复习建议

Taylor polynomials convert complex functions into polynomial approximations that are easy to work with. The key formula is Pₙ(x) = Σ from k=0 to n of f⁽ᵏ⁾(a)(x−a)ᵏ/k!. For the IB exam, memorise the standard expansions for eˣ, sin x, cos x, ln(1+x) and (1+x)ᵖ, and understand the Lagrange error bound.

泰勒多项式将复杂函数转化为易于处理的多项式近似。关键公式是 Pₙ(x) = 从 k=0 到 n 的求和 Σ f⁽ᵏ⁾(a)(x−a)ᵏ/k!。对于IB考试，记住 eˣ、sin x、cos x、ln(1+x) 和 (1+x)ᵖ 的标准展开，并理解拉格朗日误差界。

When revising, practice writing out the first few derivatives of common functions, then evaluate them at the centre. Work on at least one full exam-style question from each category above. Finally, always check whether the error bound is required, as this is the most frequently tested application.

复习时，练习写出常见函数的前几阶导数，然后在中心处求值。针对上述每一类题型，至少做一道完整的考试风格题目。最后，始终检查是否要求误差界，因为这是最常考的考点。

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Taylor Polynomials in IB Mathematics | IB数学：泰勒多项式