Series, finite differences and Taylor expansions · 级数、有限差分与泰勒展开
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| Taylor polynomial/ˈteɪlə ˌpɒlɪˈnəʊmɪəl/ | 泰勒多项式 | tài lēi duō xiàng shì |
Can a polynomial replace an exponential nearby?
- Near a known point, a complicated function can be replaced by a polynomial approximation. The approximation needs a centre and a valid scale.
- This lesson studies Taylor polynomial 泰勒多项式: A polynomial formed from a function's derivatives at a chosen centre.
Choose the mathematical structure
- About x=a, f(x)=f(a)+f prime(a)(x-a)+f double prime(a)(x-a)²/2!+... when the expansion is valid. At a=0 it is a Maclaurin expansion. Finite sums may also simplify by cancellation or standard sum formulae.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines Taylor polynomial? · 哪个描述正确定义了泰勒多项式？
A polynomial formed from a function's derivatives at a chosen centre. · 由函数在某选定中心点的导数构成的多项式。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
The first three terms of e^x are 1+x+x²/2. At x=0.1 this gives 1.105, close to e^0.1≈1.105170. For the sum of k² from k=1 to 5, n(n+1)(2n+1)/6 gives 5×6×11/6=55.
Series, finite differences and Taylor expansions · 级数、有限差分与泰勒展开
About x=a, f(x)=f(a)+f prime(a)(x-a)+f double prime(a)(x-a)²/2!+ · 关于 x=a，f(x)=f(a)+f'(a)(x-a)+f''(a)(x-a)²/2!+
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Use 1+x+x²/2 to estimate e^0.1. · 使用 1+x+x²/2 来估算 e^0.1。
The truncated polynomial is 1+0.1+0.1²/2=1.105. · 截断多项式为 1+0.1+0.1²/2=1.105。
Test a tempting shortcut
- The factorial belongs in the denominator of every Taylor coefficient. The expansion centre is not always zero. An approximation close to its centre can be poor far away.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A Taylor polynomial is always exact for the original function at every input. This claim is false. Explain which definition or assumption it violates.
Find the sum of k² for k=1 to 5. · 求 k² 的和，其中 k 从 1 到 5。
Add the squared terms: 1+4+9+16+25=55. · 将平方项相加：1+4+9+16+25=55。
A Taylor polynomial is always exact for the original function at every input. · 泰勒多项式对原函数在每个输入值处总是精确的。
The factorial belongs in the denominator of every Taylor coefficient. The expansion centre is not always zero. An approximation close to its centre can be poor far away. · 阶乘位于每个泰勒系数的分母中。展开中心并不总是零。靠近中心的近似值在远离中心时可能较差。
Interpret a new situation
- For a telescoping series, write several terms and identify both surviving ends. For a Taylor approximation, state its order and compare with a known value or a remainder estimate where required.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the coefficient of x² in the Maclaurin expansion of e^x. · 求 e^x 的麦克劳林展开式中 x² 的系数。
The second derivative of e^x at zero is 1; divide by 2!=2 to get 1/2. · e^x 在零点的二阶导数为 1；除以 2!=2 得到 1/2。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL further mathematics; official unit FP2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A polynomial formed from a function's derivatives at a chosen centre. Choose the relationship, show the method, check its assumptions and interpret the result.