Linear Approximation Error and Over/Underestimates

Whether a linear approximation overestimates or underestimates the true value is determined by the sign of the second derivative at the point of tangency. If the function is concave up at a (f''(a) > 0), the tangent line lies below the curve, so the linearization underestimates f(x) for x near a. If the function is concave down (f''(a) < 0), the tangent line lies above the curve, and the approximation overestimates the true value. This concavity test is the answer. No guesswork is needed; the direction of the error is baked into the geometry of the curve.

A tangent line matches both the function value and the slope at the point a. But it does not match the curvature. When a function curves away from its tangent, the side it curves to, above or below, fixes the sign of the error.

• Concave up (f'' > 0): the graph sits above its tangent line. The linear approximation L(x) lies below the true curve, so L(x) < f(x). That is an underestimate.
• Concave down (f'' < 0): the graph sits below its tangent line. The linear approximation L(x) lies above the true curve, so L(x) > f(x). That is an overestimate.

The rule holds for any differentiable function as long as x is sufficiently close to a. Move far enough away and the sign can flip if the concavity changes, but for the small intervals where linear approximation is useful, the sign of f'' at a is a reliable guide.

For instance, f(x) = √x has f''(x) = -1/(4x^(3/2)), which is negative on its domain. At a = 4, the function is concave down, so the tangent line overestimates √x. The worked example below confirms this: L(4.1) = 2.025, while √4.1 ≈ 2.02485, an overestimate of about 0.00015.

A diagram of two simple curves makes the rule impossible to forget. For a concave-up function like f(x) = x² at a = 1, the tangent line at (1,1) runs below the parabola. Pick any x near 1 and the line gives a value lower than the curve. That is an underestimate. For a concave-down function like f(x) = -x² at a = 1, the tangent line sits above the parabola, giving a value higher than the curve. That is an overestimate.

Sketch a generic concave-up curve and its tangent: the line is a straight chord below the arc. The arc bulges upward away from the line. Sketch a concave-down curve: the line is above the arc, which dips away from the line. The curvature, not the slope, decides which side the error lands on.

The actual error E(x) = |f(x) - L(x)| grows predictably as you move away from a. But you rarely know the exact error without knowing f(x). What you can do is place a guaranteed upper bound on it using Taylor's inequality (n = 1 case, Stewart §11.11).

The error bound formula is:

|E| ≤ (M/2)|x - a|²

Here M is the maximum absolute value of the second derivative on the closed interval between x and a. You find M by evaluating |f''(t)| for all t between x and a and taking the largest value.

Error Bound Linear Approximation: A Practical Example

Take f(x) = sin x, a = 0, and approximate sin(0.1). The linearization is L(0.1) = 0 + 1*(0.1) = 0.1. The actual sin(0.1) ≈ 0.09983, so the true error is about 0.00017. To bound it, find M. On [0, 0.1], |f''(t)| = |-sin t| ≤ sin(0.1) ≈ 0.09983, but the safest upper bound is sin(0.1) < 0.1. A simpler common choice is M = 1, since |sin t| ≤ 1 for all t. Using M = 1 gives |E| ≤ (1/2)(0.1)² = 0.005. The true error of 0.00017 is well inside that bound. The bound is not tight; it is a guaranteed maximum, not the actual error.

Absolute vs Relative Error

Absolute error is |f(x) - L(x)|, measured in the same units as the function. Relative error is absolute error divided by |f(x)|, often expressed as a percentage. The same absolute error of 0.01 m is tiny for a building height of 100 m (0.01% relative error) but enormous for a screw length of 1 cm (100% relative error). No fixed absolute cut-off is meaningful across different functions and units. Always judge error in context.

Why Error Grows With Distance From a

The error bound depends on |x - a|². Double the distance from a and the worst-case error grows by a factor of four. This quadratic growth is why linear approximation fails quickly outside a narrow neighbourhood of a. A small change in x can produce a large error if the curvature is high. For sin x near 0, the error at 0.1 rad is 0.00017; at 0.5 rad the error jumps to about 0.021, and at 1 rad it reaches roughly 0.16. The approximation is trustworthy only for small |x - a|.

This example walks through a full linear approximation error analysis for f(x) = √x at a = 4, estimating √4.1.

1. Find the linearization. f(4) = 2, f'(x) = 1/(2√x), so f'(4) = 1/4. L(x) = 2 + (1/4)(x - 4). At x = 4.1: L(4.1) = 2 + (1/4)(0.1) = 2.025.
2. Find the actual value. √4.1 ≈ 2.024845... (using a calculator). The actual error is |2.024845-2.025| ≈ 0.000155.
3. Determine over- or underestimate. f''(x) = -1/(4x^(3/2)), which is negative on (4, 4.1). The function is concave down, so the linearization overestimates √4.1. Indeed, 2.025 > 2.024845.
4. Compute the error bound. The maximum of |f''(t)| on [4, 4.1] occurs at t = 4, because f'' is decreasing in magnitude as t increases. |f''(4)| = 1/(4*8) = 1/32 = 0.03125. So M = 0.03125. Then |E| ≤ (0.03125/2)(0.1)² = 0.015625 * 0.01 = 0.00015625. The true error of 0.000155 is just under this bound, showing the bound is close to tight in this case.

The error bound using Taylor's inequality gives a guaranteed maximum of 0.000156, and the actual error is essentially equal to it. This example shows both the direction (overestimate from concave down) and the magnitude (tiny, because |x - a| is small and the second derivative is moderate).

Repeat the process for f(x) = sin x, a = 0, approximating sin(0.1).

1. Linearization. f(0) = 0, f'(x) = cos x, f'(0) = 1. L(0.1) = 0 + 1*(0.1) = 0.1.
2. Actual value. sin(0.1) ≈ 0.09983. Actual error ≈ 0.00017.
3. Over- or underestimate? f''(x) = -sin x. At a = 0, f''(0) = 0, but for any small positive x, f''(x) is negative (since sin x > 0). The function is concave down on (0, 0.1], so the linearization overestimates sin x. 0.1 > 0.09983, confirming an overestimate.
4. Error bound. On [0, 0.1], |f''(t)| = |-sin t| = sin t, max at t = 0.1, so M = sin(0.1) ≈ 0.09983.Then |E| ≤ (0.1/2)(0.1)² = 0.05 * 0.01 = 0.0005. The true error 0.00017 is less than half the bound.

This example shows that even with a conservatively chosen M (0.1), the bound is wider than the actual error. If you use the cruder bound M = 1, the bound becomes 0.005, which is 29 times the actual error. The bound is a guarantee, not a prediction.

Trust a linear approximation when |x - a| is small, typically under 0.1 for most common functions, and when the second derivative is small or bounded. For the small-angle approximation sin θ ≈ θ, the error is less than 0.17% at 0.1 rad (about 5.7°). At 0.5 rad (about 28.6°), the error exceeds 2%. In a physics pendulum derivation, the small-angle approximation is valid for θ ≤ 0.1 rad; the Halliday/Resnick pendulum derivation uses this threshold. For angles above that, the linearisation is too crude.

Walk away from the approximation when |x - a| is large, when the function is not differentiable at a, or when the problem demands exact values. Linearisation is a local tool; do not use it as a global one.