How Many Digits Should a Linear Estimate Keep?

You have computed a linear approximation and now stare at a calculator result with six decimal places. The question how many digits should a linear estimate keep? is answered by two constraints: the error bound from the second derivative and the precision of your input data. Report only digits that are certain based on the error bound, then truncate at the first uncertain digit. Do not pad with zeros or drop meaningful digits.

For the standard linearization error bound |E(x)| ≤ (M/2)(x, a)², M is an upper bound for |f''| on the interval between a and x. That bound tells you the maximum possible error, not the actual error. Round the estimate so that the last reported digit is in the same decimal place as the error bound. If the bound is 0.0034, the thousandths place is uncertain, so report to the nearest 0.01 (two decimal places). Input precision also limits you: if x is given to 0.1, the estimate cannot be reliably reported beyond 0.1 unless the error bound justifies more.

Take f(x) = √x, a = 4, x = 4.1. f(4) = 2, f′(4) = 1/(2√4) = 0.25. L(4.1) = 2 + 0.25(0.1) = 2.025. Compute the error bound.

f″(x) =, 1/(4x^(3/2)). On [4, 4.1], |f″| is largest at x = 4: |f″(4)| = 1/(4×8) = 1/32 = 0.03125. Then (M/2)(x, a)² = (0.03125/2)×(0.1)² = 0.00015625. The error bound is 0.0002 after rounding up. The estimate 2.025 has its thousandths digit (the 5) in the same place as the bound, so report as 2.025. If you had 2.0254, the extra digit is not justified.

Now consider input precision: x = 4.1 has one decimal. The error bound 0.0002 is smaller than 0.1, so the error bound allows more digits than the input precision demands. Report to three decimal places because the bound supports it. If the bound were 0.5, you would report 2.0, not 2.025.

The calculation follows a two-step rule. First, compute the error bound using the maximum of |f″| on the interval. Second, round the estimate so that its last digit aligns with the bound's first uncertain digit. If the bound is 0.0034, the thousandths are uncertain, so report hundredths. If the bound is 0.00015, report to the nearest 0.0001 (four decimal places).

Input precision overrides only when it is coarser than the error bound. If x is given as 4.1 (one decimal) and the bound is 0.0002, you may keep three decimals because the bound justifies them. If x were 4.1 and the bound were 0.5, you must report 2.0. The tighter constraint wins.

At the point where you would compute a new bound for a different input, you need a rounding calculator to apply this rule efficiently. That tool uses the same formula but automates the comparison between error bound and input precision.

The most common failure is reporting too many digits. If the error bound is 0.5 and you report 2.025, the first decimal place is already uncertain; the thousandths are meaningless. The reader assumes precision that does not exist. Another failure is ignoring input precision: if x is 4.1, reporting 2.0250 implies the input was known to 0.0001, which is false.

A third failure is using the actual error instead of the bound. The actual error might be 0.0001, but if the bound is 0.0034, you still cannot report beyond hundredths. The bound is a guarantee; the actual error can be smaller, but rounding based on the actual error is unreliable because you do not know it without the true function value.

The single thing that most often goes wrong: students compute the linearization to eight decimal places and report them all, ignoring the error bound entirely. Cut at the first uncertain digit and stop.

Linear approximation suits AP Calculus AB/BC and Calculus I students who need to compute linearizations by hand and interpret overestimates or underestimates using concavity. It also suits Calculus III students extending the idea to tangent plane approximations for functions of two variables, and physics students using small angle approximations (sin θ ≈ θ, tan θ ≈ θ) in pendulum and optics derivations. Students learning differentials (dy vs Δy) and how they relate to error propagation will also benefit from mastering the digit count rule.

Anyone needing exact function values, numerical root finding via Newton's method, or higher order approximations such as Taylor polynomials beyond n = 1 should skip linear approximation and use a Taylor series calculator or a root finding resource. The linearization error grows quadratically with distance from the point of tangency, so it is useless for distant inputs regardless of how many digits you report.