Series, finite differences and Taylor expansions · Séries, diferenças finitas e expansões de Taylor
| English | Português |
|---|---|
| Taylor polynomial/ˈteɪlə ˌpɒlɪˈnəʊmɪəl/ | polinômio de Taylor |
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? · Qual descrição define corretamente um polinômio de Taylor?
A polynomial formed from a function's derivatives at a chosen centre. · Um polinômio formado pelas derivadas de uma função em um centro escolhido.
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 · Séries, diferenças finitas e expansões de Taylor
About x=a, f(x)=f(a)+f prime(a)(x-a)+f double prime(a)(x-a)²/2!+ · Sobre 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. · Compare o modelo com o caso resolvido e explique uma mudança.
Use 1+x+x²/2 to estimate e^0.1. · Use 1+x+x²/2 para estimar e^0.1.
The truncated polynomial is 1+0.1+0.1²/2=1.105. · O polinômio truncado é 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. · Encontre a soma de k² para k=1 até 5.
Add the squared terms: 1+4+9+16+25=55. · Some os termos quadrados: 1+4+9+16+25=55.
A Taylor polynomial is always exact for the original function at every input. · Um polinômio de Taylor é sempre exato para a função original em cada entrada.
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. · O fatorial pertence ao denominador de cada coeficiente de Taylor. O centro da expansão nem sempre é zero. Uma aproximação próxima ao seu centro pode ser ruim longe dele.
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. · Encontre o coeficiente de x² na expansão de Maclaurin de e^x.
The second derivative of e^x at zero is 1; divide by 2!=2 to get 1/2. · A segunda derivada de e^x em zero é 1; divida por 2!=2 para obter 1/2.
Match each part of a complete solution to its purpose. · Associe cada parte de uma solução completa ao seu propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Uma suposição justifica o modelo; uma verificação testa o resultado; a interpretação conecta-o à pergunta.
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.