AP® Calculus BC · Unit 4 · Topic 4.6
Approximating Values of a Function Using Local Linearity and Linearization
Local linearity says a smooth curve looks like its tangent line if you zoom in close enough, so near a known point you can use that tangent line, called the linearization, to approximate values you can't compute directly.
Zoom into any differentiable curve near a point and it flattens out until it's indistinguishable from a straight line. That line is the tangent line at that point, and it shares the curve's value and slope right there. So close to the point of tangency, the tangent line's y-values are a good stand-in for the curve's actual y-values. This is local linearity: locally, curve and tangent agree.
The mechanism is the linearization formula: L(x) = f(a) + f'(a)(x - a), built from the point (a, f(a)) and the slope f'(a). Plug in an x near a and L(x) gives you an estimate of f(x) without needing the actual function value there. The further x drifts from a, the more the curve peels away from its tangent, so the error grows — linearization is a local tool, not a global one.
Whether that estimate runs high or low depends entirely on concavity, which connects this topic to the second derivative work right next to it in this unit. Where the curve is concave up, it bends away above the tangent line, so the tangent underestimates. Where concave down, the curve sags below the tangent, so the tangent overestimates. This is also exactly the reasoning behind Newton's method and behind bounding error in later approximation techniques, so getting the concavity argument solid here pays off down the line.
Key terms
- Local linearity
- The property that a differentiable curve looks like a straight line when viewed close enough to a point.
- Linearization L(x)
- The tangent line formula f(a) + f'(a)(x - a), used to approximate f near x = a.
- Point of tangency
- The point (a, f(a)) where the tangent line touches the curve and shares its slope.
- Concavity
- Whether a curve bends upward or downward, determined by the sign of the second derivative.
Watch it explained
- How it worksHow does a tangent line at a known point predict a nearby unknown value?
- Worked exampleFind an approximation for the square root of 4.1 using the tangent line to √x at x = 4.
- Common mistakeWhy isn't a tangent line approximation always an underestimate?
- Going furtherWhat happens to the approximation error as I wander further from the tangent point?
- Ties togetherHow does concavity decide whether the tangent line over- or underestimates?
Check yourself
End of topic.