Lagrange Error Bound
Bound the next derivative across the whole interval before stating a Lagrange remainder guarantee.
College Board Topic 10.12 · CED p. 192 (PDF p. 197) · CED effective Fall 2020; project course outline 2026–27.
If f has continuous derivatives through order n+1 on the interval between a and x, Taylor’s theorem gives a remainder for some c between them. If throughout that interval, then . Choose a bound over the whole interval, not only the derivative’s value at the center.
The result is an error interval: the true function value lies within the stated bound of your polynomial approximation. M need not be the smallest possible upper bound, but it must be valid. Larger valid M values give weaker guarantees. Pay attention to the derivative order n+1 and factorial (n+1)!.
When the function’s series is alternating with decreasing term magnitudes at the chosen x, the alternating-series bound may be sharper or easier. Use the first omitted nonzero term, not blindly the next power of x. For cos x, a degree-4 polynomial omits the degree-6 term next, so that alternating bound is . A degree-4 Lagrange bound instead uses a fifth derivative.
Do not confuse a bound with the observed difference on a calculator. A computed difference can check a particular case; the theorem guarantees the claimed uncertainty under its hypotheses.
Let's work through it
- For eˣ at x=0.2 with T₂=1+x+x²/2, obtain . Why: The degree-2 Maclaurin coefficients are 1,1,1/2.
- On [0,0.2], the third derivative is eᵗ; a valid given upper bound is M=2. Why: The derivative bound must cover every point between center and evaluation point.
- . Why: Use order n+1=3 and the Lagrange inequality.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
See every recall question with its answer
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Retrieval 1
Worked answer and reason
It is the first derivative order beyond the polynomial degree.
Retrieval 2
Worked answer and reason
The degree-5 coefficient is zero; the next actual series term has degree 6.
Catch the tempting mistake
- Use f⁽ⁿ⁺¹⁾(a) as M automatically.
- A single center value might not bound that derivative throughout the interval. Find or justify a valid interval maximum.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 10 problem 23 · No calculator · analytic
- Unit 10 problem 24 · No calculator · analytic
For the teacher
- Have students name n, a, x, derivative order and M before substituting. Compare Lagrange and alternating bounds for the same polynomial.
Glossary and deeper reading · 6 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Polynomial
A polynomial in one variable is a finite sum of constant multiples of nonnegative integer powers of that variable. The degree is the highest exponent with a nonzero coefficient. For example, 2x³−x+5 is a polynomial of degree 3. Expressions containing 1/x or √x are not polynomials in x.
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Infinite series
An infinite series is the sum of the terms of an infinite sequence, defined through its partial sums. The series converges when those partial sums approach a finite value. For example, 1/2+1/4+1/8+⋯ converges to 1. Terms approaching 0 are necessary for convergence, but that condition alone is insufficient.
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Alternating series error bound
For an alternating series satisfying the decreasing-to-zero magnitude conditions, the error after n terms is at most the next term’s magnitude. The actual sum lies between two consecutive partial sums. For example, stopping 1−1/2+1/3−⋯ after its tenth term gives error at most 1/11. The bound requires the alternating series hypotheses and does not apply automatically to every signed series.