10.12 Lagrange Error Bound

What the Lagrange Error Bound measures

When you approximate a function with a Taylor polynomial of degree n centered at a, the Lagrange Error Bound gives a maximum possible value for the remainder (error). It answers: 'How far off could my approximation be?' This lets you bound error without knowing the exact function value.

The formula

The bound is: |R_n(x)| ≤ [M·|x−a|^(n+1)] / (n+1)!, where M is the maximum value of |f^(n+1)(t)| on the interval between a and x. You need the (n+1)th derivative and its max magnitude on that interval — finding M often requires checking endpoints or critical points.

Using the bound in practice

To guarantee an approximation is accurate within a given tolerance, set the error bound expression ≤ the desired tolerance and solve for n or x. Common on FRQs: 'Show the error is less than 0.001' — you plug in the largest possible M and simplify, rather than computing the actual remainder.

Key terms

Lagrange Error Bound formula

|R_n(x)| ≤ M|x−a|^(n+1)/(n+1)!, where M bounds |f^(n+1)(t)| on the interval between a and x.

What is M in the formula?

The maximum value of the absolute value of the (n+1)th derivative on the interval between a and x.

Why use the Lagrange Error Bound?

To find an upper limit on how far a Taylor polynomial approximation could be from the true function value, without knowing the exact remainder.

Which derivative order matters for a degree-n polynomial?

The (n+1)th derivative — one more than the polynomial's degree.

Typical FRQ use of the bound

Set the error bound expression ≤ a given tolerance (like 0.001) and show the inequality is satisfied to prove accuracy.

Try an AP-style question

Which is the Lagrange error bound for a degree-n Taylor polynomial about a?

1. M(n + 1)! / |x − a|ⁿ⁺¹
2. M|x − a|ⁿ⁺¹ / (n + 1)!
3. |f(x) − Pₙ(a)|
4. M|x − a|ⁿ / n!