Unit 4: Contextual Applications of Differentiation

Local Linearization

Using a tangent line to approximate function values near a point.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Related Rates

Approximating with the Tangent Line

√4 = 2 exactly, but √4.1 has no clean closed form. Before reading on: since 4.1 is so close to 4, could the tangent line to y = √x at x = 4 — a straight line, easy to evaluate anywhere — give a genuinely good estimate of √4.1?

Definition — Local Linearization

The linearization of f at x = a is L(x) = f(a) + f'(a)(x − a) — the equation of the tangent line at a, used to approximate f(x) for x near a. The approximation is exact at x = a itself and grows less accurate the farther x moves away.

L(x) = f(a) + f'(a)(x - a)

Worked Example — Approximating a Square Root

Approximate √4.1 using the linearization of f(x) = √x at a = 4. f(4) = 2, f'(x) = 1/(2√x), f'(4) = 1/4. L(x) = 2 + (1/4)(x − 4). L(4.1) = 2 + (1/4)(0.1) = 2.025. The actual value, √4.1 ≈ 2.024846, is very close — and slightly less than the approximation. That direction isn't an accident: f''(x) = −1/(4x^(3/2)) is negative for x > 0, so f is concave down, meaning its tangent line lies above the curve — a concave-down function's linearization always overestimates nearby.

Worked Example — Approximating a Trig Value (Converting to Radians First)

Approximate cos(61°) using the linearization of f(x) = cos(x) at a = 60° = π/3 radians — calculus derivative formulas for trig functions require radians. f(π/3) = cos(60°) = 0.5, f'(x) = −sin(x), f'(π/3) = −sin(60°) = −√3/2 ≈ −0.8660. The change from 60° to 61° is 1° = π/180 ≈ 0.017453 radians. L = 0.5 + (−0.8660)(0.017453) ≈ 0.48488. The actual value, cos(61°) ≈ 0.48481, is again very close — and again slightly less than the linearization, consistent with cosine also being concave down on this interval (cos''(x) = −cos(x) < 0 for x near 60°).

Derivative Explorer

f(x) =

x^3 - 3*x

Point of tangency

x = a1

Show tangent line

Trace mode

f(x), with tangent line at x = a

Show f''(x)Show f'''(x)

Derivative graph — linked to the graph above; panning or zooming either moves both

What's happening at x = a

At x = 1, f(x) ≈ -2 — the point (1, -2).
The derivative there is f'(1) ≈ 0: the slope of the tangent line, and the instantaneous rate of change of f at this exact point.
That slope is positive, so f is increasing at this point.
The second derivative is positive here, so f is concave up (curving upward) near this point.
Near x = 1, the tangent line y ≈ -2 + 0·(x − 1) is f's best straight-line approximation — the core idea a derivative captures.

Numerical derivative comparison

Forward, backward, and central difference estimates of f'(a) as the step size h shrinks — all three should converge toward the same value.
h	Forward	Backward	Central
1	4	-2	1
0.1	0.31	-0.29	0.01
0.01	0.0301	-0.0299	0.0001
0.001	0.003	-0.003	0

Analysis of f(x)

y-intercept: (0, 0)
x-intercepts: (-1.73, 0), (0, 0), (1.73, 0)
Extrema: local max at (-1, 2); local min at (1, -2)
Inflection points: (0, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-970, 970]

Analysis of f'(x) — its roots are f's critical points

y-intercept: (0, -3)
x-intercepts: (-1, 0), (1, 0)
Extrema: local min at (0, -3)
Inflection points: none found in view
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-3, 297]

Tip

Before evaluating a trig linearization, convert the change in input to radians — every calculus derivative formula for sin, cos, and tan assumes a radian input, so a degree-based Δx has to be converted first.

Common Mistakes

Using Δx in degrees directly inside a trig linearization instead of converting to radians.
f'(a) for a trig function is only correct in radians — plugging in Δx = 1 (as if it meant '1 degree') instead of π/180 radians produces a wildly wrong approximation.
Trusting a linearization far from the point of tangency without checking how far x has moved.
Linearization error grows the farther x moves from a — it's reliable very close to a, but the same tangent line can be a poor approximation even a moderate distance away, especially where the function curves sharply.

Key Takeaways

Linearization L(x) = f(a) + f'(a)(x − a) approximates f near a using its tangent line at a.
A concave-down function's linearization overestimates nearby values; a concave-up function's linearization underestimates them, since the tangent line lies on the opposite side of the curve from its concavity.
Trig linearizations require the input change to be in radians, since calculus derivative formulas for trig functions assume radians throughout.

Summary

Linearization approximates function values near a point. The next lesson uses that same tangent-line thinking to resolve a different kind of problem — limits that resist ordinary algebra.

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