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C47 · Unit C10 Sequences and series

Taylor polynomials and Lagrange error

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 10.11–10.12 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

The degree-n Taylor polynomial at a is sum sub k ₌ sub 0 to the power n [f^(k)(a)/k!](x minus a)ᵏ. It matches the function’s derivatives through order n at a. If |f^(n+1)(t)| is less than or equal to M between a and x, Lagrange’s bound is |R_n(x)| is less than or equal to M|x minus a|^(n+1)/(n+1)!. A derivative bound must hold on the whole interval, not just at its center.

Worked example

The quadratic Maclaurin polynomial for eˣ is 1+x+x squared /2. At x equals 0.1 it gives 1.105. Since the third derivative is eᵗ is less than or equal to e to the power 0.1 on [0,0.1], the error is at most e to the power 0.1(0.1) cubed /6, approximately 0.0001842.

Explore the mathematics

Taylor polynomials and Lagrange error

Increase the degree of a polynomial approximation and check its error.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict where a polynomial built at zero approximates eˣ best.xy-2-112eˣ and Taylor degree 3 about 0

Predict before continuing

Predict where a polynomial built at zero approximates eˣ best.

Read the complete explanation
  1. Predict where a polynomial built at zero approximates eˣ best.
  2. Match the value and successive derivatives at the centre.
  3. Pₙ(x)=Σₖ₌₀ⁿxᵏ/k!. Orange is the polynomial; green is eˣ.
  4. At x=1 the error is 0.0516. A Lagrange bound on [0,1] is e/(n+1)!.
See the idea: Taylor polynomials and Lagrange error
See the idea: Taylor polynomials and Lagrange errorThe exponential and quadratic Maclaurin approximation 1+x+x²/2 agree closely near zero. At x=0.1 the polynomial gives 1.105. The Lagrange estimate bounds the error by approximately 0.0001842; it is not the exact error.xy-0.50.5-0.250.82501.150.251.4750.51.8

The exponential and quadratic Maclaurin approximation 1+x+x²/2 agree closely near zero. At x=0.1 the polynomial gives 1.105. The Lagrange estimate bounds the error by approximately 0.0001842; it is not the exact error.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Do not omit the factorial or the shift x minus a. A high polynomial degree does not by itself certify accuracy far from the center.

Investigate and explain

Compare a calculator error with the certified error bound. Explain why a conservative bound can still be useful.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.