Series, finite differences and Taylor expansions
| English | 中文 | Pinyin |
|---|---|---|
| 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?
A polynomial formed from a function's derivatives at a chosen centre.
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
About x=a, f(x)=f(a)+f prime(a)(x-a)+f double prime(a)(x-a)²/2!+
Compare the model with the worked case and explain one change.
Use 1+x+x²/2 to estimate e^0.1.
The truncated polynomial is 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.
Add the squared terms: 1+4+9+16+25=55.
A Taylor polynomial is always exact for the original function at every input.
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.
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.
The second derivative of e^x at zero is 1; divide by 2!=2 to get 1/2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.